Partial Di erential Equations of Electrostatic MEMS › wp-content › uploads › 2018 › 10 ›...

Partial Differential Equations ofElectrostatic MEMS

by

Yujin Guo

B.Sc., China Three Gorges University, 2000M.Sc., Huazhong Normal University, 2003

A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OF

Doctor of Philosophy

in

The Faculty of Graduate Studies

(Mathematics)

The University of British Columbia

July 2007

c© Yujin Guo 2007

Abstract

Micro-Electromechanical Systems (MEMS) combine electronics with micro-size mechanical de-vices in the process of designing various types of microscopic machinery, especially those in-volved in conceiving and building modern sensors. Since their initial development in the 1980s,MEMS has revolutionized numerous branches of science and industry. Indeed, MEMS-baseddevices are now essential components of modern designs in a variety of areas, such as in com-mercial systems, the biomedical industry, space exploration, telecommunications, and otherfields of applications.

As it is often the case in science and technology, the quest for optimizing the attributes ofMEMS devices according to their various uses, led to the development of mathematical modelsthat try to capture the importance and the impact of the multitude of parameters involved intheir design and production. This thesis is concerned with one of the simplest mathematicalmodels for an idealized electrostatic MEMS, which was recently developed and popularized ina relatively recent monograph by J. Pelesko and D. Bernstein. These models turned out to bean incredibly rich source of interesting mathematical phenomena.

The subject of this thesis is the mathematical analysis combined with numerical simulationsof a nonlinear parabolic problem ut = ∆u − λf(x)

(1+u)2 on a bounded domain of RN with Dirich-let boundary conditions. This equation models the dynamic deflection of a simple idealizedelectrostatic MEMS device, which consists of a thin dielectric elastic membrane with bound-ary supported at 0 above a rigid ground plate located at −1. When a voltage –representedhere by λ– is applied, the membrane deflects towards the ground plate and a snap-through(touchdown) may occur when it exceeds a certain critical value λ∗ (pull-in voltage). This cre-ates a so-called pull-in instability which greatly affects the design of many devices. In orderto achieve better MEMS design, the elastic membrane is fabricated with a spatially varyingdielectric permittivity profile f(x).

The first part of this thesis is focussed on the pull-in voltage λ∗ and the quantitativeand qualitative description of the steady states of the equation. Applying analytical andnumerical techniques, the existence of λ∗ is established together with rigorous bounds. Weshow the existence of at least one steady state when λ < λ∗ (and when λ = λ∗ in dimensionN < 8), while none is possible for λ > λ∗. More refined properties of steady states –suchas regularity, stability, uniqueness, multiplicity, energy estimates and comparison results– areshown to depend on the dimension of the ambient space and on the permittivity profile.

The second part of this thesis is devoted to the dynamic aspect of the parabolic equation.We prove that the membrane globally converges to its unique maximal negative steady-statewhen λ ≤ λ∗, with a possibility of touchdown at infinite time when λ = λ∗ and N ≥ 8.On the other hand, if λ > λ∗ the membrane must touchdown at finite time T , which cannottake place at the location where the permittivity profile f(x) vanishes. Both larger pull-in distance and larger pull-in voltage can be achieved by properly tailoring the permittivity

ii

Abstract

profile. We analyze and compare finite touchdown times by using both analytical and numericaltechniques. When λ > λ∗, some a priori estimates of touchdown behavior are established, basedon which, we can give a refined description of touchdown profiles by adapting recently developedself-similarity methods as well as center manifold analysis. Applying various analytical andnumerical methods, some properties of the touchdown set –such as compactness, location andshape– are also discussed for different classes of varying permittivity profiles f(x).

iii

Table of Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Statement of Co-Authorship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Electrostatic MEMS devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 PDEs modeling electrostatic MEMS . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Analysis of the elastic problem . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Analysis of electrostatic problem . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Overview and some comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Pull-In Voltage and Steady-States . . . . . . . . . . . . . . . . . . . . . . . . . 142.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 The pull-in voltage λ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.1 Existence of the pull-in voltage . . . . . . . . . . . . . . . . . . . . . . . 172.2.2 Monotonicity results for the pull-in voltage . . . . . . . . . . . . . . . . 19

2.3 Estimates for the pull-in voltage . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.1 Lower bounds for λ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.2 Upper bounds for λ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.3 Numerical estimates for λ∗ . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4 The branch of minimal solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 282.4.1 Spectral properties of minimal solutions . . . . . . . . . . . . . . . . . . 292.4.2 Energy estimates and regularity . . . . . . . . . . . . . . . . . . . . . . 31

2.5 Uniqueness and multiplicity of solutions . . . . . . . . . . . . . . . . . . . . . . 372.5.1 Uniqueness of the solution at λ = λ∗ . . . . . . . . . . . . . . . . . . . 372.5.2 Uniqueness of low energy solutions for small voltage . . . . . . . . . . . 392.5.3 Second solutions around the bifurcation point . . . . . . . . . . . . . . 40

2.6 Radially symmetric case and power-law profiles . . . . . . . . . . . . . . . . . 41

iv

Table of Contents

3 Compactness along Lower Branches . . . . . . . . . . . . . . . . . . . . . . . . 473.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2 Mountain Pass solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.3 Minimal branch for power-law profiles . . . . . . . . . . . . . . . . . . . . . . . 553.4 Compactness along the second branch . . . . . . . . . . . . . . . . . . . . . . . 61

3.4.1 Blow-up analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.4.2 Spectral confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.4.3 Compactness issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.5 The second bifurcation point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.6 The one dimensional problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4 Dynamic Deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.2 Global convergence or touchdown . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.2.1 Global convergence when λ < λ∗ . . . . . . . . . . . . . . . . . . . . . 784.2.2 Touchdown at finite time when λ > λ∗ . . . . . . . . . . . . . . . . . . 794.2.3 Global convergence or touchdown in infinite time for λ = λ∗ . . . . . . 82

4.3 Location of touchdown points . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.4 Estimates for finite touchdown times . . . . . . . . . . . . . . . . . . . . . . . 91

4.4.1 Comparison results for finite touchdown time . . . . . . . . . . . . . . . 914.4.2 Explicit bounds on touchdown times . . . . . . . . . . . . . . . . . . . 93

4.5 Asymptotic analysis of touchdown profiles . . . . . . . . . . . . . . . . . . . . 974.5.1 Touchdown profile: f(x) ≡ 1 . . . . . . . . . . . . . . . . . . . . . . . . 974.5.2 Touchdown profile: variable permittivity . . . . . . . . . . . . . . . . . 99

4.6 Pull-in distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5 Refined Touchdown Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.2 A priori estimates of touchdown behavior . . . . . . . . . . . . . . . . . . . . . 109

5.2.1 Lower bound estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.2.2 Gradient estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.2.3 Upper bound estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.3 Refined touchdown profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.3.1 Refined touchdown profiles for N = 1 . . . . . . . . . . . . . . . . . . . 1245.3.2 Refined touchdown profiles for N ≥ 2 . . . . . . . . . . . . . . . . . . . 126

5.4 Set of touchdown points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.4.1 Radially symmetric case . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.4.2 One dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6 Thesis Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.1 Stationary Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.2 Dynamic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

v

List of Tables

2.1 Numerical values for pull-in voltage λ∗ with the bounds given in Theorem 2.1.1.Here the exponential permittivity profile is chosen as (2.3.21). . . . . . . . . . . 27

2.2 Numerical values for pull-in voltage λ∗ with the bounds given in Theorem 2.1.1.Here the power-law permittivity profile is chosen as (2.3.21). . . . . . . . . . . . 28

4.1 Computations for finite touchdown time T with the bounds T∗, T0,λ, T1,λ andT2,λ given in Proposition 4.4.3. Here the applied voltage λ = 20 and the profileis chosen as (4.4.23). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.2 Numerical values for finite touchdown time T at different applied voltages λ = 5,10, 15 and 20, respectively. Here the constant permittivity profile f(x) ≡ 1 ischosen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.3 Numerical values for pull-in voltage λ∗: Table (a) corresponds to exponentialprofiles, while Table (b) corresponds to power-law profiles. . . . . . . . . . . . . 102

vi

List of Figures

1.1 The simple electrostatic MEMS device. . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Plots of u(0) versus λ for the constant permittivity profile f(x) ≡ 1 defined inthe unit ball B1(0) ⊂ RN with different ranges of N . In the case of N ≥ 8, wehave λ∗ = (6N − 8)/9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Plots of λ∗ versus α for a power-law profile (heavy solid curve) and the exponen-tial profile (solid curve). The left figure corresponds to the slab domain, whilethe right figure corresponds to the unit disk. . . . . . . . . . . . . . . . . . . . . 26

2.3 Plots of u(0) versus λ for profile f(x) = |x|α (α ≥ 0) defined in the slab domain(N = 1). The numerical experiments point to a constant α∗ > 1 (analyticallygiven in (2.6.10)) such that the bifurcation diagrams are greatly different fordifferent ranges of α: 0 ≤ α ≤ 1, 1 < α ≤ α∗ and α > α∗. . . . . . . . . . . . . 43

2.4 Top figure: Plots of u(0) versus λ for 2 ≤ N ≤ 7, where u(0) oscillates aroundthe value λ∗ defined in (2.6.12) and u∗ is regular. Bottom figure: Plots of u(0)versus λ for N ≥ 8: when 0 ≤ α ≤ α∗∗, there exists a unique solution for (S)λwith λ ∈ (0, λ∗) and u∗ is singular; when α > α∗∗, u(0) oscillates around thevalue λ∗ defined in (2.6.12) and u∗ is regular. . . . . . . . . . . . . . . . . . . . 45

3.1 Top figure: plots of u(0) versus λ for the case where f(x) ≡ 1 is defined inthe unit ball B1(0) ⊂ RN with different ranges of dimension N , where we haveλ∗ = (6N−8)/9 for dimension N ≥ 8. Bottom figure: plots of u(0) versus λ forthe case where f(x) ≡ 1 is defined in the unit ball B1(0) ⊂ RN with dimension2 ≤ N ≤ 7, where λ∗ (resp. λ∗2) is the first (resp. second) turning point. . . . . 48

4.1 Left Figure: u versus x for λ = 4.38. Right Figure: u versus x for λ = 4.50.Here we consider (4.3.1) with f(x) = |2x| in the slab domain. . . . . . . . . . . 88

4.2 Left Figure: u versus r for λ = 1.70. Right Figure: u versus r for λ = 1.80.Here we consider (4.3.1) with f(r) = r in the unit disk domain. . . . . . . . . . 88

4.3 Left Figure: plots of u versus x for different f(x) at λ = 8 and t = 0.185736.Right Figure: plots of u versus x for different λ with f(x) = |2x| and t =0.1254864. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.4 Plots of the pull-in distance |u(0)| = |u∗(0)| versus α for the power-law profile(heavy solid curve) and the exponential profile (solid curve). Left figure: the slabdomain. Right figure: the unit disk. . . . . . . . . . . . . . . . . . . . . . . . . . 102

vii

List of Figures

4.5 Left figure: plots of u versus |x| at λ = λ∗ for α = 0, α = 1, α = 3, and α = 10,in the unit disk for the power-law profile. Right figure: plots of u versus |x| atλ = λ∗ for α = 0, α = 2, α = 4, and α = 10, in the unit disk for the exponentialprofile. In both figures the solution develops a boundary-layer structure near|x| = 1 as α is increased. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.6 Bifurcation diagram of w′(0) = −γ versus λ0 from the numerical solution of

(4.6.6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.7 Comparison of numerically computed λ∗ (heavy solid curve) with the asymptotic

result (dotted curve) from (4.6.7) for the unit disk. Left figure: the exponentialprofile. Right figure: the power-law profile. . . . . . . . . . . . . . . . . . . . . . 104

4.8 Left figure: plots of u versus x at λ = λ∗ in the slab domain. Right figure: plotsof u versus |x| at λ = λ∗ in the unit disk domain. . . . . . . . . . . . . . . . . 105

4.9 Left figure: plots of u versus x at λ = λ∗ in the slab domain. Right figure: plotsof u versus |x| at λ = λ∗ in the unit disk domain. . . . . . . . . . . . . . . . . 106

5.1 Left figure: plots of u versus x at different times with f(x) = 1− x2 in the slabdomain, where the unique touchdown point is x = 0. Right figure: plots of uversus r = |x| at different times with f(r) = 1 − r2 in the unit disk domain,where the unique touchdown point is r = 0 too. . . . . . . . . . . . . . . . . . . 129

5.2 Left figure: plots of u versus x at different times with f(x) = e−x2in the slab

domain, where the unique touchdown point is x = 0. Right figure: plots of uversus r = |x| at different times with f(r) = e−r2

in the unit disk domain, wherethe unique touchdown point is r = 0 too. . . . . . . . . . . . . . . . . . . . . . 130

5.3 Left figure: plots of u versus x at different times with f(x) = ex2−1 in the slab

domain, where the unique touchdown point is still at x = 0. Right figure: plotsof u versus r = |x| at different times with f(r) = er

2−1 in the unit disk domain,where the touchdown points satisfy r = 0.51952. . . . . . . . . . . . . . . . . . . 130

5.4 Left figure: plots of u versus x at different times with f(x) = 1/2 − x/2 in theslab domain, where the unique touchdown point is x = −0.10761. Right figure:plots of u versus r = |x| at different times with f(x) = x + 1/2 in the slabdomain, where the unique touchdown point is x = 0.17467. . . . . . . . . . . . 131

5.5 Plots of u versus x at different times in the slab domain, for different permittivityprofiles f [α](x) given by (5.4.5). Top left (a): when α = 0.5, two touchdownpoints are at x = ±0.12631. Top right (b): when α = 1, the unique touchdownpoint is at x = 0. Bottom Left (c): when α = 0.785, touchdown points areobserved to consist of a closed interval [−0.0021255, 0.0021255]. Bottom right(d): local amplified plots of (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

viii

Acknowledgements

My greatest gratitude goes to my supervisor, Nassif Ghoussoub. Without his expert guidance,enormous patience, constant encouragement, and continuing quest for mathematical rigor, thisthesis would not have been possible. I deeply acknowledge his generosity with his time, andhis invaluable suggestions and insightful comments.

I would like to thank my collaborators Michael J. Ward, Zhenguo Pan and PierpaoloEsposito, from whom I have benefited a lot. I am particularly grateful to Michael J. Ward,whose graduate course led me to learn about the PDE models of electrostatic MEMS, andthe pioneering work of J. Pelesko. I would also like to thank Professor Louis Nirenberg forpointing me to the pioneering work of Joseph and Lindgren, and eventually to the extensivemathematical literature on nonlinear eigenvalue problems. I will always be grateful to thePDE team at UBC (whether faculty, postdocs, or graduate students), and in particular IvarEkeland, Tai-Peng Tsai, Stephen Gustafson, Daniele Cassani, Abbas Maomeni, Yu Yan, andMeijiao Guan for their constant support. Many thanks also go to Danny Fan for her greathelp over the past three years.

Financial support from Research Assistantship under Nassif Ghoussoub (2003-2004), Chi-Kit Wat Scholarship (2004-2005), T. K. Lee Scholarship (2005-2006) and U. B. C. GraduateFellowships (2004-2007), is gratefully acknowledged.

Finally, I would like to extend my thanks to my parents and my wife who have given meconstant support and encouragement during my studies at UBC.

ix

Statement of Co-Authorship

The numerical results in sections 2.3.3 were published in:[1] Y. Guo, Z. Pan and M. J. Ward, Touchdown and pull-in voltage behavior of a MEMS

device with varying dielectric properties, SIAM, J. Appl. Math. 66 (2005), 309–338.Most of the results in Chapter 2 have been published in:

[2] N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMSdevices: stationary case, SIAM, J. Math. Anal. 38 (2007), 1423–1449.

The main results of Chapter 3 are due to appear in the following paper in press:[3] P. Esposito, N. Ghoussoub and Y. Guo, Compactness along the branch of semi-stable

and unstable solutions for an elliptic problem with a singular nonlinearity, Comm. Pure Appl.Math., accepted (2006).

Sections 4.5 and 4.6 can be found in the paper [1] mentioned above, and while the main resultsof sections 4.1-4.4 come from the following paper which is also in press:

[4] N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMSdevices II: dynamic case, NoDEA Nonlinear Diff. Eqns. Appl., accepted (2007).

The main results of Chapter 5 can be found in the paper:[5] Y. Guo, On the partial differential equations of electrostatic MEMS devices III: refined

touchdown behavior, submitted (2006).

x

Chapter 1

Introduction

The roots of micro-system technology lie in the technological developments accompanyingWorld War II, and in particular the work around radar stimulated research in the synthesisof pure semiconducting materials. These materials, especially pure silicon, have become themain components of integrated and modern technology of Micro-Electromechanical Systems(MEMS). The advent of MEMS has revolutionized numerous branches of science and industry,and their applications are continuing to flourish, as they are becoming essential components ofmodern sensors in areas as diverse as the biomedical industry, space exploration, and telecom-munications.

A comprehensive overview of the rapidly developing field of MEMS technology can befound in the relatively recent monograph by Pelesko and Bernstein [52]. Not only is this booka rich source of information about the incredibly vast area of applications of MEMS, but alsoit contains justifications and derivations of the fundamental partial differential equations thatmodel such devices. It is the mathematical analysis and the numerical simulations of theseequations that concern us in this thesis.

In this introduction, we shall first briefly recall some of the industrial applications of MEMSwhile referring the reader to the book of Pelesko and Bernstein mentioned above for a morecomprehensive survey. For the convenience of the reader, we shall also include a derivation ofthe simplest PDE modeling electrostatic MEMS devices, which is by now a well known andbroadly accepted mathematical model.

1.1 Electrostatic MEMS devices

The state of the art is best summarized by the following description of Pelesko and Bernstein inthe book [52]: “Spurred by rapid advances in integrated circuit manufacturing, microsystemsprocess technology is already well developed. As a result, researchers are increasingly focusingtheir attention on device engineering questions. Foremost among these is the question ofhow to provide accurate, controlled, stable locomotion for MEMS devices. Just as what hasbeen recognized for some time by several scientists and engineers, it is neither feasible nordesirable to attempt to reproduce modes of locomotion used in the macro world. In fact, theunfavorable scaling of force with device size prohibits this approach in many cases. For example,magnetic forces, which are often used for actuation in the macro world, scale poorly into themicro domain, decreasing in strength by a factor of ten thousand when linear dimensions arereduced by a factor of ten. This unfavorable scaling renders magnetic forces essentially useless.At the micro level, researchers have proposed a variety of new modes of locomotion basedupon thermal, biological, and electrostatic forces”. The use of electrostatic forces to providelocomotion for MEMS devices is behind the mathematical model that we address in this thesis.

Experimental work in this area dates back to 1967 and the work of Nathanson et. al. [49].

1

Chapter 1. Introduction

In their seminal paper, Nathanson and his coworkers describe the modeling and manufactureof, experimentation with, a millimeter-sized resonant gate transistor. These early MEMS de-vices utilized both electrical and mechanical components on the same substrate resulting inimproved efficiency, lowered cost, and reduced system size. Nathanson and his coworkers alsointroduced a simple lumped mass-spring model of electrostatic actuation. In an interestingparallel development, the British scientist, G. I. Taylor [55] investigated electrostatic actuationat about the same time as Nathanson. While Taylor was concerned with electrostatic deflec-tion of soap films rather than the development of MEMS devices, his work spawned a smallbody of the literature with relevance to MEMS. Since Nathanson and Taylor’s seminal work,numerous investigators have been continually exploring new uses of electrostatic actuation,such as Micropumps, Microswitches, Microvalves, Shuffle Motor and etc. See [52] for moredetails on how these devices use electrostatic forces for their operation.

1.2 PDEs modeling electrostatic MEMS

A key component of some MEMS systems is the simple idealized electrostatic device shown inFigure 1.1. The upper part of this device consists of a thin and deformable elastic membranethat is held fixed along its boundary and which lies above a parallel rigid grounded plate. Thiselastic membrane is modeled as a dielectric with a small but finite thickness. The upper surfaceof the membrane is coated with a negligibly thin metallic conducting film. When a voltage Vis applied to the conducting film, the thin dielectric membrane deflects towards the bottomplate. A similar deflection phenomenon, but on a macroscopic length scale, occurs in the fieldof electrohydrodynamics. In this context, Taylor [55] studied the electrostatic deflection of twooppositely charged soap films, and he predicted a critical voltage for which the two soap filmswould touch together.

A similar physical limitation on the applied voltage occurs for the MEMS device of Figure1.1, in that there is a maximum voltage V ∗ –known as pull-in voltage– which can be safelyapplied to the system. More specifically, if the applied voltage V is increased beyond the criticalvalue V ∗, the steady-state of the elastic membrane is lost, and proceeds to snap through ata finite time creating the so-called pull-in instability (cf. [29, 30, 32, 50]). The existence ofsuch a pull-in voltage was first demonstrated for a lumped mass-spring model of electrostaticactuation in the pioneering study of [49], where the restoring force of the deflected membraneis modeled by a Hookean spring. In this lumped model the attractive inverse square lawelectrostatic force between the membrane and the ground plate dominates the restoring forceof the spring for small gap sizes and large applied voltages. This leads to snap-through behaviorwhereby the membrane hits the ground plate when the applied voltage is large enough.

Following closely the analysis in [32, 52], we shall now formulate the partial differentialequations that models the dynamic deflection w = w(x′, y′, t′) of the membrane shown inFigure 1.1.

1.2.1 Analysis of the elastic problem

We shall apply Hamilton’s least action principle and minimize the action S of the system.Here the action consists of the superposition of the kinetic energy, the damping energy andthe potential energy in the system. The pointwise total of these energies is the Lagrangian L

2


D i e l e c t r i c M e m b r a n e w i t h C o n d u c t i n g F i l m a t P o t e n t i a l V S u p p o r t e d B o u n d a r y

F i x e d G r o u n d P l a t e x '

y '

z '

d

L

Figure 1.1: The simple electrostatic MEMS device.

for the system. We then have

S =∫ t2

t1

∫

Ω′LdX ′dt′ = Kinetic Energy + Damping Energy + Potential Energy

= : Ek + Ed + Ep ,

(1.2.1)

where Ω′ is the domain of the membrane with respect to (x′, y′). In this subsection, dX ′

denotes dx′dy′, and the gradient ∇′ (and the Laplace operator ∆′) denotes the differentiationonly with respect to x′ and y′.

For the dynamic deflection w = w(x′, y′, t′) of the membrane, the kinetic energy Ek is

Ek =ρA

2

∫ t2

t1

∫

Ω′w2t′dX

′dt′ , (1.2.2)

where ρ is the mass density per unit volume of the membrane, and A is the thickness of themembrane. The damping energy Ed is assumed to be

Ed =a

2

∫ t2

t1

∫

Ω′w2dX ′dt′ , (1.2.3)

where a is the damping constant.For this model, the potential energy Ep is composed of

Ep = Stretching Energy + Bending Energy . (1.2.4)

It is reasonable to assume that the stretching energy in the elastic membrane is proportional tothe changes in the area of the membrane from its un-stretched configuration. Since we assumethe membrane is held fixed at its boundary, we may write the stretching energy as

Stretching Energy := −µ(∫ t2

t1

∫

Ω′

√1 + |∇′w|2dX ′dt′ − |Ω′|(t2 − t1)

). (1.2.5)

Here the proportionality constant µ, is simply the tension in the membrane. We linearize thisexpression to obtain

Stretching Energy := −µ2

∫ t2

t1

∫

Ω′|∇′w|2dX ′dt′ . (1.2.6)

3


The bending energy is assumed to be proportional to the linearized curvature of the membrane,that is

Bending Energy := −D2

∫ t2

t1

∫

Ω′|∆′w|2dX ′dt′ . (1.2.7)

Here the constant D is the flexural rigidity of the membrane. For the total potential energyEp we now have

Ep = −∫ t2

t1

∫

Ω′

(µ2|∇′w|2 +

D

2|∆′w|2)dX ′dt′ . (1.2.8)

Combining (1.2.1)− (1.2.3) and (1.2.8) now yields that

L =ρA

2w2t′ +

a

2w2 − µ

2|∇′w|2 − D

2|∆′w|2 . (1.2.9)

According to Hamilton’s principle, we should minimize∫ t2

t1

∫

Ω′

(ρA2w2t′ +

a

2w2 − µ

2|∇′w|2 − D

2|∆′w|2)dX ′dt′ , (1.2.10)

which implies –via Euler-Lagrange variational calculus– that the elastic membrane’s deflectionw satisfies

ρA∂2w

∂t′2+ a

∂w

∂t′− µ∆′w +D∆′2w = 0 . (1.2.11)

1.2.2 Analysis of electrostatic problem

We now analyze the electrostatic problem of Figure 1.1 and we allow the dielectric permittivityε2 = ε2(x′, y′) of the elastic membrane to exhibit a spatial variation reflecting the varying di-electric permittivity of the membrane. Therefore, in view of (1.2.11) we assume the membrane’sdeflection w satisfying

ρA∂2w

∂t′2+ a

∂w

∂t′− µ∆′w +D∆′2w = −ε2

2|∇′φ|2 , (1.2.12)

where the term on the right hand side of (1.2.12) denotes the force on the elastic membrane,which is due to the electric field. We suppose that such force is proportional to the normsquared of the gradient of the potential and couples the solution of the elastic problem to thesolution of the electrostatic problem. A derivation of such source term may be found in [42].

We now apply dimensionless analysis to equation (1.2.12). We scale the electrostatic po-tential with the applied voltage V , time with a damping timescale of the system, the x′ andy′ variables with a characteristic length L of the device, and z′ w with the size of the gap dbetween the ground plate and the undeflected elastic membrane. So we define

w =w

d, ψ =

φ

V, x =

x′

L, y =

y′

L, z =

z′

d, t =

µt′

aL2, (1.2.13)

and substitute these into equation (1.2.12) to find

γ∂2w

∂t2+∂w

∂t−∆w + δ∆2w = −λ

(ε2

ε0

)[ε2|∇′ψ|2 +

(∂ψ∂z

)2] in Ω , (1.2.14)

4


where Ω is the dimensionless domain of the elastic membrane. Here the parameter γ satisfies

γ =√ρµA

aL, (1.2.15)

and the parameter δ measures the relative importance of tension and rigidity and it is definedby

δ =D

µL2. (1.2.16)

The parameter ε is the aspect ratio of the system

ε =d

L, (1.2.17)

and the parameter λ is a ratio of the reference electrostatic force to the reference elastic forceand it is defined by

λ =V 2L2ε0

2µd3. (1.2.18)

In view of equation (1.2.12), and in order to further understand membrane’s deflection, weneed to know more about the electrostatic potential φ inside the elastic membrane. In theactual design of a MEMS device there are several issues that must be considered. Typically,one of the primary device design goals is to achieve the maximum possible stable steady-statedeflection, referred to as the pull-in distance, with a relatively small applied voltage V. Anotherconsideration may be to increase the stable operating range of the device by increasing the pull-in voltage V ∗ subject to the constraint that the range of the applied voltage is limited by theavailable power supply. This increase in the stable operating range may be important for thedesign of microresonators. For other devices such as micropumps and microvalves, where snap-through (or called touchdown) behavior is explicitly exploited, it is of interest to decrease thetime for touchdown, thereby increasing the switching speed. One way of achieving larger valuesof V ∗ while simultaneously increasing the pull-in distance, is to use a voltage control schemeimposed by an external circuit in which the device is placed (cf. [53]). This approach leads toa nonlocal problem for the deflection of the membrane. A different approach is to introduce aspatial variation in the dielectric permittivity of the membrane, which was theoretically studiedin [29–32, 50? ? ].

In the following we discuss the electrostatic potential φ by introducing a spatial varyingdielectric permittivity into our simple MEMS model. The idea is to locate the region wherethe membrane deflection would normally be largest under a spatially uniform permittivity, andthen make sure that a new dielectric permittivity ε2 is largest –and consequently the profilef(x, y) smallest– in that region.

We assume that the ground plate, located at z′ = 0, is a perfect conductor. The elasticmembrane is assumed to be a uniform thickness A = 2ι. The deflection of the membrane attime t′ is specified by the deflection of its center plane, located at z′ = w(x′, y′, t′). Hence thetop surface is located at z′ = w(x′, y′, t′) + ι, while the bottom of the membrane is located atz′ = w(x′, y′, t′) − ι. We also assume that the potential between the membrane and groundplate, φ1, satisfies

∆φ1 = 0 , (1.2.19)

5


φ1(x′, y′, 0) = 0 in Ω′ , (1.2.20)

where we assume that the fixed ground plate is held at zero potential. The potential insidethe membrane, φ2, satisfies

∇ · (ε2∇φ2) = 0 , (1.2.21)

φ2(x′, y′, w + ι) = V in Ω′ . (1.2.22)

Defining

ψ1 =φ1

V, ψ2 =

φ2

V(1.2.23)

together with (1.2.13), and applying dimensionless analysis again, the electrostatic problemreduces to

∂2ψ

∂z2+ ε2

(∂2ψ

∂x2+∂2ψ

∂y2

)= 0 , 0 ≤ z ≤ w − ι ; (1.2.24a)

ε2∂2ψ

∂z2+ ε2

( ∂∂x

(ε2∂ψ

∂x

)+

∂

∂y

(ε2∂ψ

∂y

))= 0 , w − ι ≤ z ≤ w + ι ; (1.2.24b)

ψ = 0 , z = 0 (ground plate); ψ = 1 , z = w + ι (upper membrane surface) , (1.2.25)

together with the continuity of the potential and the displacement fields across z = w− ι. Hereψ is the dimensionless potential scaled with respect to the applied voltage V , and, as before,ε ≡ d/L is the device aspect ratio.

In general, we note that one has little hope of finding an exact solution ψ from (1.2.24) and(1.2.25). However, we can simplify the system by examining a restricted parameter regime. Inparticular, we consider the small-aspect ratio limit ε ≡ d/L 1. Physically, this means thatthe lateral dimensions of the device in Figure 1.1 are larger compared to the size of the gapbetween the undeflected membrane and ground plate. In the small-aspect ratio limit ε 1,equation (1.2.24) gives ∂2ψ

∂z2 = 0. Further, the asymptotical solution of ψ which is continuousacross z = w − ι is

ψ =

ψL

zw−ι , 0 ≤ z ≤ w − ι ,

1 + (1−ψL)2ι (z − (w + ι)) , w − ι ≤ z ≤ w + ι .

(1.2.26)

To ensure that the displacement field is continuous across z = w − ι to leading order in ε, weimpose that

ε0∂ψ

∂z|− = ε2

∂ψ

∂z|+ ,

where the plus or minus signs indicate that ∂ψ∂z is to be evaluated on the upper or lower side

of the bottom surface z = w − ι of the membrane, respectively. This condition determines ψLin (1.2.26) as

ψL =[1 +

2ιw − ι

(ε0

ε2

)]−1. (1.2.27)

From (1.2.26) and (1.2.27), we observe that the electric field in the z-direction inside themembrane is independent of z, and is given by

∂ψ

∂z=

ε0

ε2(w − ι)[1 +

2ιw − ι

ε0

ε2

]−1 ∼ ε0

ε2wfor ι 1 . (1.2.28)

6

In engineering parlance, this approximation is equivalent to ignoring fringing fields. Therefore,in the small-aspect ratio limit ε 1, the governing equation (1.2.14) is simplified from (1.2.28)into

γ∂2w

∂t2+∂w

∂t−∆w + δ∆2w = − λε0

ε2w2in Ω . (1.2.29)

We now suppose the membrane is undeflected at the initial time, that is w(x, y, 0) = 1.Since the boundary of the membrane is held fixed, we have w(x, y, t) = 1 on the boundary ofΩ at any time t > 0. We now also assume that the membrane’s thickness A = 2dι satisfiesA = 2dι 1 which gives γ 1 in view of (1.2.15), and assume that the elastic membrane hasno rigidity which gives δ = 0 in view of (1.2.16). Therefore, by using further simplification,the dynamic deflection w = w(x, t) of the membrane on a bounded domain Ω in R2, is foundto satisfy the following parabolic problem

∂w

∂t−∆w = −λf(x)

w2for x ∈ Ω , (1.2.30a)

w(x, t) = 1 for x ∈ ∂Ω , (1.2.30b)w(x, 0) = 1 for x ∈ Ω , (1.2.30c)

where the parameter λ > 0 is called the applied voltage in view of relation (1.2.18), and whilethe nonnegative continuous function f(x) characterizes the varying dielectric permittivity ofthe elastic membrane, in the point of the relation

f(x) =ε0

ε2(Lx,Ly). (1.2.31)

Therefore, understanding dynamic deflection of our MEMS model is equivalent to studyingsolutions of (1.2.30).

1.3 Overview and some comments

The main contents of this thesis are divided into two major parts, each consisting of twochapters.

Part I: Pull-In Voltage and Stationary Deflection

The first part of this thesis is focussed on pull-in voltage and stationary deflection of theelastic membrane satisfying (1.2.30). For convenience, by setting w = 1 − u we study thefollowing semilinear elliptic problem with a singular nonlinearity

−∆u =λf(x)

(1− u)2in Ω,

0 0 denotes the applied voltage and the nonnegative continuous function f(x) char-acterizes the varying dielectric permittivity of the elastic membrane. Mathematically, weconsider the domain Ω ⊂ RN with any dimension N ≥ 1. (S)λ was firstly studied by Pelesko

7

in [50], where the author focussed on lower dimension N = 1 or 2, and he considered eitherf(x) ≥ C > 0 or f(x) = |x|α. In my joint work with Pan and Ward [32], we studied (S)λ for amore general profile f(x) which can vanish somewhere. In the past two years, (S)λ was furtherextended and sharpened in our series work [22, 29].

The main results of Chapter 2 can be found in [29]. In §2.2 we mainly show the existenceof a specific pull-in voltage in the sense

λ∗ := λ∗(Ω, f) = supλ > 0 | (S)λ possesses at least one solution

. (1.3.1)

The definition of λ∗ shows that for λ < λ∗, there exists at least one solution for (S)λ; while forλ > λ∗, there is no solution for (S)λ. In §2.2 we also study pull-in voltage’s dependence on thesize and shape of the domain, as well as on the permittivity profile. These properties will helpus in §2.3 establish some lower and upper bound estimates on the pull-in voltage, see Theorem2.1.1. In particular, we shall prove in Theorem 2.1.1(5) that if f(x) ≡ |x|α with α ≥ 0 and B1

is a unit ball in RN , then we have

λ∗(B1, |x|α) =(2 + α)(3N + α− 4)

9,

provided N ≥ 8 and 0 ≤ α ≤ α∗∗(N) := 4−6N+3√

6(N−2)4 .

In Chapter 2, we also consider issues of uniqueness and multiplicity of solutions for (S)λwith 0 < λ ≤ λ∗. The bifurcation diagrams in Figure 2.1 of §2.1 show the complexity ofthe situation, even in the radially symmetric case. One can observe from Figure 2.1 that thenumber of branches –and of solutions– is closely connected to the space dimension, a fact thatwe analytically discuss in §2.4, by focussing on the very first branch of solutions considered tobe “minimal” in the following way.

Definition 1.3.1. A solution uλ(x) of (S)λ is said to be minimal if for any other solution uof (S)λ we have uλ(x) ≤ u(x) for all x ∈ Ω.

We shall prove that for any 0 ≤ λ < λ∗, there exists a unique minimal solution uλ of (S)λ suchthat µ1,λ(uλ) > 0. Moreover, for each x ∈ Ω, the function λ→ uλ(x) is strictly increasing anddifferentiable on (0, λ∗).

On the other hand, one can introduce for any solution u of (S)λ, the linearized operator atu defined by Lu,λ = −∆− 2λf(x)

(1−u)3 and its eigenvalues µk,λ(u); k = 1, 2, .... The first eigenvalueis then simple and is given by:

µ1,λ(u) = inf⟨Lu,λφ, φ

⟩H1

0 (Ω); φ ∈ C∞0 (Ω),

∫

Ω|φ(x)|2dx = 1

.

Stable solutions (resp., semi-stable solutions) of (S)λ are those solutions u such that µ1,λ(u) > 0(resp., µ1,λ(u) ≥ 0). We note that there already exist in the literature many interesting resultsconcerning the properties of the branch of semi-stable solutions for Dirichlet boundary valueproblems of the form −∆u = λh(u) where h is a regular nonlinearity (for example of the formeu or (1 + u)p for p > 1). See for example the seminal papers [20, 43, 44] and also [15] for asurvey on the subject and an exhaustive list of related references. The singular situation wasconsidered in a very general context in [48], and the analysis of Chapter 2 is completed to allow

8

for a general continuous permittivity profile f(x) ≥ 0. Our main results in this direction arestated in Theorem 2.1.2, where fine properties of steady states –such as regularity, stability,uniqueness, multiplicity, energy estimates and comparison results– are shown to depend on thedimension of the ambient space and on the permittivity profile. More precisely, Theorem 2.1.2gives that if 1 ≤ N ≤ 7 then –by means of energy estimates– one has supλ∈(0,λ∗) ‖ uλ ‖∞< 1and consequently, u∗ = lim

λ↑λ∗uλ exists in C2,α(Ω) with 0 < α < 1 and is a solution for (S)λ∗ such

that µ1,λ∗(u∗) = 0. In particular, u∗ –often referred to as the extremal solution of problem (S)λ–

is unique. On the other hand, if N ≥ 8, f(x) = |x|α with 0 ≤ α ≤ α∗∗(N) := 4−6N+3√

6(N−2)4

and Ω is the unit ball, then the extremal solution is necessarily u∗(x) = 1 − |x| 2+α3 and is

therefore singular.In general, the function u∗ exists in any dimension, does solve (S)λ∗ in a suitable weak

sense, and is the unique solution in an appropriate class. The above result says that it is,however, a classical solution in dimensions 1 ≤ N ≤ 7, and this will allow us to start anotherbranch of non-minimal (unstable) solutions. Indeed, following ideas of Crandall-Rabinowitz[20], we show in §2.5 that, for 1 ≤ N ≤ 7 and for λ close enough to λ∗, there exists a uniquesecond branch Uλ of solutions for (S)λ, bifurcating from u∗, with

µ1,λ(Uλ) < 0 while µ2,λ(Uλ) > 0. (1.3.2)

In §2.6 we present some numerical evidences for various conjectures relating to the casewhere permittivity profile f(x) = |x|α is defined in a unit ball. The bifurcation diagrams showfour possible regimes –at least if the domain is a ball:

A. There is exactly one branch of solution for 0 < λ < λ∗. This regime occurs when N ≥ 8,and if 0 ≤ α ≤ α∗∗(N) := 4−6N+3

√6(N−2)

4 . The results of this section actually show thatin this range, the first branch of solutions “disappears” at λ∗ which happens to be equalto λ∗(α,N) = (2+α)(3N+α−4)

9 .

B. There exists an infinite number of branches of solutions. This regime occurs when

• N = 1 and α ≥ α∗ := −12 + 1

2

√27/2.

• 2 ≤ N ≤ 7 and α ≥ 0;

• N ≥ 8 and α > α∗∗(N) := 4−6N+3√

6(N−2)4 .

In this case, λ∗(α,N) < λ∗ and the multiplicity becomes arbitrarily large as λ approaches–from either side– λ∗(α,N) at which there is a touchdown solution u (i.e., ‖ u ‖∞= 1).

C. There exists a finite number of branches of solutions. In this case, we have again thatλ∗(α,N) < λ∗, but now the branch approaches the value 1 monotonically, and thenumber of solutions increase but remains finite as λ approaches λ∗(α,N). This regimeoccurs when N = 1 and 1 < α ≤ α∗ := −1

2 + 12

√27/2.

D. There exist exactly two branches of solutions for 0 < λ < λ∗ and one solution for λ = λ∗.The bifurcation diagram vanishes when it returns to λ = 0. This regime occurs whenN = 1 and 0 ≤ α ≤ 1.

9

The main results of Chapter 3 are available in [22]. Note from Chapter 2 that the com-pactness of minimal branch solutions of (S)λ holds in 1 ≤ N ≤ 7 for any profile f(x). In §3.3we extend such compactness to higher dimension N ≥ 8, provided that Ω is a unit ball andf(x) = |x|α satisfies α > α∗∗(N) := 4−6N+3

√6(N−2)

4 . Analytically, this provides a clear dis-tinction between the case where the permittivity profile f(x) is bounded away from zero, andwhere it is allowed to vanish somewhere. Once the compactness of minimal branch solutionsfor (S)λ holds, then the standard Crandall-Rabinowitz theory [20] implies the existence of asecond solution Uλ of (S)λ on the deleted left neighborhood of λ∗, where Uλ satisfies

µ1,λ(Uλ) < 0 while µ2,λ(Uλ) > 0. (1.3.3)

In §3.2 using truncation we shall provide the mountain pass variational characterization ofsuch branch Uλ.

In Chapter 3, we are also interested in continuing the second branch till the second bifur-cation point, by means of the implicit function theorem. Suppose 2 ≤ N ≤ 7 and f ∈ C(Ω)satisfies

f(x) =( k∏

i=1

|x− pi|αi)g(x) , g(x) ≥ C > 0 in Ω (1.3.4)

for some points pi ∈ Ω and exponents αi ≥ 0. Let (λn)n be a sequence such that λn → λ ∈[0, λ∗] and let un be an associated solution of (S)λn such that

µ2,n := µ2,λn(un) ≥ 0. (1.3.5)

Then in §3.4 we shall use blow-up analysis of elliptic PDE to prove the compactness of un. Weexpect that such a result should be true for radial solutions on the unit ball for N ≥ 8, α > αN ,and f ∈ C(Ω) as in (3.3.1). As far as we know, there are no compactness results of this type inthe case of regular nonlinearities, marking a substantial difference with the singular situation.

We define the second bifurcation point in the following way for (S)λ:

λ∗2 = infβ > 0 : ∃ a curveVλ ∈ C([β, λ∗];C2(Ω)) of solutions for (S)λs.t. µ2,λ(Vλ) ≥ 0, Vλ ≡ Uλ ∀λ ∈ (λ∗ − δ, λ∗).

Assume f ∈ C(Ω) to be of the form (1.3.4). Then for 2 ≤ N ≤ 7 we shall prove in §3.5 thatλ∗2 ∈ (0, λ∗) and for any λ ∈ (λ∗2, λ

∗) there exist at least two solutions uλ and Vλ for (S)λ, sothat

µ1,λ(Vλ) < 0 while µ2,λ(Vλ) ≥ 0.

In particular, for λ = λ∗2, there exists a second solution, namely V ∗ := limλ↓λ∗2

Vλ so that

µ1,λ∗2(V ∗) < 0 and µ2,λ∗2(V ∗) = 0.

We note that the second branch cannot approach the value λ = 0 as illustrated by the bifur-cation diagram in Figure 3.1.

Now let Vλ, λ ∈ (β, λ∗) be one of the curves appearing in the definition of λ∗2. By (1.3.3),we have that LVλ,λ is invertible for λ ∈ (λ∗ − δ, λ∗) and, as long as it remains invertible, we

10

can use the Implicit Function Theorem to find Vλ as the unique smooth extension of the curveUλ (in principle Uλ exists only for λ close to λ∗). We now define λ∗∗ in the following way

λ∗∗ = infβ > 0 : ∀λ ∈ (β, λ∗) ∃ Vλ solution of (S)λ so thatµ2,λ(Vλ) > 0, Vλ ≡ Uλ forλ ∈ (λ∗ − δ, λ∗).

Then, λ∗2 ≤ λ∗∗ and there exists a smooth curve Vλ for λ ∈ (λ∗∗, λ∗) so that Vλ is the uniquemaximal extension of the curve Uλ. This is what the second branch is supposed to be. If nowλ∗2 < λ∗∗, then for λ ∈ (λ∗2, λ

∗∗) there is no longer uniqueness for the extension and the “secondbranch” is defined only as one of potentially many continuous extensions of Uλ.

In dimension 1, we have a stronger but somewhat different compactness result. Recall thatµk,λn(un) is the k−th eigenvalue of Lun,λn counted with their multiplicity. Let I be a boundedinterval in R and f ∈ C1(I) be such that f ≥ C > 0 in I. Suppose (un)n is a solution sequencefor (S)λn on I, where λn → λ ∈ (0, λ∗]. Assume for any n ∈ N and k large enough, we haveµk,n := µk,λn(un) ≥ 0. Then in §3.6 we shall prove the compactness of un. Note that themultiplicity result in dimension 2 ≤ N ≤ 7 holds also in dimension 1 for any λ ∈ (λ∗2, λ

∗).

Part II: Dynamic Deflection and Touchdown Behavior

The second part of this thesis is devoted to the dynamic deflection and touchdown behaviorof (1.2.30). When f(x) ≡ 1, there already exist some results for touchdown (quenching)behavior of (1.2.30) since 1980s, see [38, 39, 46] and references therein. However, since theprofile f(x) is assumed to be varying and vanish somewhere for MEMS models, the dynamicbehavior of (1.2.30) turns out to be a more rich source of interesting mathematical phenomena.So far the dynamic behavior of (1.2.30) with varying profile has been investigated in [26, 30–32].

Based on [30, 32], in Chapter 4 we focus on the dynamic problem of (1.2.30) in the form

∂u

∂t−∆u =

λf(x)(1− u)2

for x ∈ Ω , (1.3.6a)

u(x, t) = 0 for x ∈ ∂Ω, u(x, 0) = 0 for x ∈ Ω . (1.3.6b)

Recall that a point x0 ∈ Ω is said to be a touchdown point for a solution u(x, t) of (1.3.6), iffor some T ∈ (0,+∞], we have lim

tn→Tu(x0, tn) = 1. T is then said to be a –finite or infinite–

touchdown time. For each such solution, we define its corresponding –possibly infinite– “firsttouchdown time”:

Tλ(Ω, f, u) = inft ∈ (0,+∞]; sup

x∈Ωu(x, t) = 1

.

In §4.2, we analyze the relationship between the applied voltage λ, the permittivity profile f ,and the solution u of (1.3.6). More precisely, for λ∗ defined as in (1.3.1), we show in §§4.2.1& 4.2.3 that if λ ≤ λ∗, then the unique dynamic solution of (1.3.6) must globally converge toits unique minimal steady-state; while we shall prove in §4.2.2 that if λ > λ∗, then the uniquedynamic solution of (1.3.6) must touchdown at finite time. Note that in the case where theunique minimal steady-state of (1.3.6) at λ = λ∗ is singular, which can happen if N ≥ 8, aboveanalysis shows the possibility of touchdown at infinite time.

In §4.3 we first compute global convergence or touchdown behavior of (1.3.6) for differentapplied voltage λ, and we then prove rigorously the following surprising fact exhibited by the

11

numerical simulations: the permittivity profile f cannot vanish in any isolated set of finite-timetouchdown points.§4.4 is focussed on the analysis and estimate of finite touchdown time, which often translates

into useful information concerning the speed of the operation for many MEMS devices such asRF switches or micro-valves.

In §4.5 we discuss touchdown profiles by the method of asymptotic analysis, and our pur-pose is to look insights into the refined touchdown rate. §4.6 is devoted to the pull-in distanceof MEMS devices, referred to as the maximum stable deflection of the elastic membrane beforetouchdown occurs. We provide numerical results for pull-in distance with some explicit exam-ples, from which one can observe that both larger pull-in distance and pull-in voltage can beachieved by properly tailoring the permittivity profile. Some interesting phenomena are alsoobserved there.

The purpose of Chapter 5 is to discuss the refined touchdown behavior of (5.1.1) (i.e.,(1.2.30)) at finite touchdown time. In §5.2 we shall derive some a priori estimates of touchdownprofiles under the assumption that touchdown set of u is a compact subset of Ω. Note thatwhether the compactness of touchdown set holds for any f(x) satisfying (5.1.2) is a quitechallenging problem. In §5.2 we first prove that the compactness of touchdown set holds forthe case where the domain Ω is convex and f(x) satisfies the additional condition

∂f∂ν ≤ 0 on Ωc

δ := x ∈ Ω : dist(x, ∂Ω) ≤ δ for some δ > 0. (1.3.7)

where ν is the outward unit norm vector to ∂Ω. Whether the assumption (1.3.7) can be removedis still open. Under the compactness assumption of touchdown set, in §5.2.1 we establish thelower bound estimate of touchdown profiles and we also prove an interesting phenomenon:finite-time touchdown point of u is not the zero point of f(x), see Theorem 5.1.1. In §5.2.2we estimate the derivatives of touchdown solution u, see Lemma 5.2.4; and as a byproduct, anintegral estimate is also given in Theorem 5.2.5 of §5.2.2.

Motivated by Theorem 5.1.1, the key point of studying touchdown profiles is a similarityvariable transformation of (5.1.1). For the touchdown solution u = u(x, t) of (5.1.1) at finitetime T , we use the associated similarity variables

y =x− a√T − t , s = −log(T − t) , u(x, t) = (T − t) 1

3wa(y, s) , (1.3.8)

where a is any interior point of Ω. Then wa(y, s) is defined in Wa := (y, s) : a + ye−s/2 ∈Ω, s > s′ = −logT, and it solves

ρ(wa)s −∇ · (ρ∇wa)− 13ρwa +

λρf(a+ ye−s2 )

w2a

= 0 ,

where ρ(y) = e−|y|2/4. Here wa(y, s) is always strictly positive in Wa. The slice of Wa at agiven time s1 is denoted by Ωa(s1) := Wa ∩ s = s1 = es

1/2(Ω − a). Then for any interiorpoint a of Ω, there exists s0 = s0(a) > 0 such that Bs := y : |y| < s ⊂ Ωa(s) for s ≥ s0. Weintroduce the frozen energy functional

Es[wa](s) =12

∫

Bs

ρ|∇wa|2dy − 16

∫

Bs

ρw2ady −

∫

Bs

λρf(a)wa

dy . (1.3.9)

12


By estimating the energy Es[wa](s) in Bs, in §5.2.3 we shall prove the upper bound estimateof wa, see Theorem 5.2.10.

Applying certain a priori estimates of §5.2, we establish refined touchdown profiles in §5.3,using self-similarity methods and center manifold analysis. We note that for N ≥ 2, we areonly able to apply Theorem 5.3.5 to study the refined touchdown profiles for special touchdownpoints –such as x = 0 in the radial case– and the situation is widely open for more generalcases.

Adapting various analytical and numerical techniques, we focus in §5.4 on the set of touch-down points. In §5.4.1 we discuss the radially symmetric case of (5.1.1), and we prove therethat suppose f(r) = f(|x|) satisfies (5.1.2) and f ′(r) ≤ 0 in a bounded ball BR(0) ⊂ RN withN ≥ 1, then r = 0 is the unique touchdown point of u, which is the maximum value point off(r) = f(|x|), see Theorem 5.1.4 and Remark 5.1.1.

For the one dimensional case, Theorem 5.1.4 already implies that touchdown points mustbe unique when the permittivity profile f(x) is uniform. In §5.4.2 we further discuss the onedimensional case of (5.1.1) for varying profile f(x), where numerical simulations show that thetouchdown set may consist of a discrete set or a finite compact subsets of the domain.

Finally, a summary of this thesis is given in Chapter 6.

13

Chapter 2

Pull-In Voltage and Steady-States

2.1 Introduction

In this Chapter we study pull-in voltage and stationary deflection of the elastic membranesatisfying (1.2.30), such that our discussion is centered on the following elliptic problem

−∆u =λf(x)

(1− u)2in Ω,

0 0 characterizes the applied voltage, while nonnegative f(x) describes the varyingpermittivity profile of the elastic membrane shown in Figure 1.1. We focus on the stable andsemi-stable stationary deflections of the membrane, while the unstable case is considered inChapter 3, and the dynamic case in Chapters 4 and 5. Throughout this Chapter and unlessmentioned otherwise, solutions for (S)λ are taken in the classical sense. The permittivityprofile f(x) will be allowed to vanish somewhere, and will be assumed to satisfy

f ∈ Cα(Ω) for some α ∈ (0, 1], 0 ≤ f ≤ 1 andf > 0 on a subset of Ω of positive measure.

(2.1.1)

This Chapter is organized as follows. In §2.2 we mainly show the existence of a specificpull-in voltage in the sense

λ∗(Ω, f) = supλ > 0 | (S)λ possesses at least one solution ,

and we also study its dependence on the size and shape of the domain, as well as on thepermittivity profile. These monotonicity properties will help us establish in §2.3 new lowerand upper bound estimates on the pull-in voltage. We shall write |Ω| for the volume of adomain Ω in RN and P (Ω) :=

∫∂Ω ds for its “perimeter”, with ωN referring to the volume of

the unit ball B1(0) in RN . We denote by µΩ the first eigenvalue of −∆ on H10 (Ω) and by φΩ

the corresponding positive eigenfunction normalized with∫

Ω φΩdx = 1.

Theorem 2.1.1. Assume f is a function satisfying (2.1.1) on a bounded domain Ω in RN ,then there exists a finite pull-in voltage λ∗ := λ∗(Ω, f) > 0 such that

1. If 0 ≤ λ < λ∗, there exists at least one solution for (S)λ.

2. If λ > λ∗, there is no solution for (S)λ.

14

Chapter 2. Pull-In Voltage and Steady-States

3. The following bounds on λ∗ hold for any bounded domain Ω:

λ := max

8N27

,6N − 8

9

1

supΩ f

(ωN|Ω|) 2N ≤ λ∗(Ω) , (2.1.2a)

minλ1 :=

4µΩ

27 infx∈Ω

f(x), λ2 :=

µΩ

3∫

Ω fφΩ dx

≥ λ∗(Ω) . (2.1.2b)

4. If Ω is a strictly star-shaped domain, that is if x · ν(x) ≥ a > 0 for all x ∈ ∂Ω, whereν(x) is the unit outer normal at x ∈ ∂Ω, and if f ≡ 1, then

λ∗(Ω) ≤ λ3 =(N + 2)2P (Ω)

8aN |Ω| . (2.1.3)

In particular, if Ω = B1(0) ⊂ RN then we have the bound

λ∗(B1(0)) ≤ (N + 2)2

8.

5. If f(x) ≡ |x|α with α ≥ 0 and Ω is a ball of radius R, then we have

λ∗(BR, |x|α) ≥ λc(α) := max4(2 + α)(N + α)27

,(2 + α)(3N + α− 4)

9R−(2+α). (2.1.4)

Moreover, if N ≥ 8 and 0 ≤ α ≤ α∗∗(N) := 4−6N+3√

6(N−2)4 , we have

λ∗(B1, |x|α) =(2 + α)(3N + α− 4)

9. (2.1.5)

In §2.3.3 we give some numerical estimates on λ∗ to compare them with the analytic boundsgiven in Theorem 1.1 above. Note that the upper bound λ1 is relevant only when f is boundedaway from 0, while the upper bound λ2 is valid for all permittivity profiles. However, the orderbetween these two upper bounds can vary in general. For example, in the case of exponentialpermittivity profiles of the form f(x) = eα(|x|2−1) on the unit disc, one can see that λ1 is abetter upper bound than λ2 for small α, while the reverse holds true for larger values of α.The lower bounds in (2.1.2) and (2.1.4) can be improved in small dimensions, but they areoptimal –at least for the ball– in dimension larger than 8.

We also consider issues of uniqueness and multiplicity of solutions for (S)λ with 0 < λ ≤ λ∗.The bifurcation diagrams in Figure 2.1 show the complexity of the situation, even in theradially symmetric case. One can see that the number of branches –and of solutions– is closelyconnected to the space dimension, a fact that we establish analytically in §4, by focussing onthe very first branch of solutions considered to be “minimal” in the following way.

Definition 2.1.1. A solution uλ(x) of (S)λ is said to be minimal if for any other solution uof (S)λ we have uλ(x) ≤ u(x) for all x ∈ Ω.

15

00

1f(x) ≡ 1 with different ranges of N

u(0)

N = 1 2 ≤ N ≤ 7

N ≥ 8

λ* λ* λ* λ* = (6N−8)/9

λ

Figure 2.1: Plots of u(0) versus λ for the constant permittivity profile f(x) ≡ 1 defined in theunit ball B1(0) ⊂ RN with different ranges of N . In the case of N ≥ 8, we have λ∗ = (6N−8)/9.

One can also consider for any solution u of (S)λ, the linearized operator at u defined byLu,λ = −∆ − 2λf(x)

(1−u)3 and its eigenvalues µk,λ(u); k = 1, 2, .... The first eigenvalue is thensimple and is given by

µ1,λ(u) = inf〈Lu,λφ, φ〉H1

0 (Ω) ; φ ∈ C∞0 (Ω),∫

Ω|φ(x)|2dx = 1

.

Stable solutions (resp., semi-stable solutions) of (S)λ are those solutions u such that µ1,λ(u) > 0(resp., µ1,λ(u) ≥ 0). Our main results in this direction can be stated as follows.

Theorem 2.1.2. Assume f is a function satisfying (2.1.1) on a bounded domain Ω in RN ,and consider λ∗ := λ∗(Ω, f) as defined in Theorem 2.1.1. Then the following hold:

1. For any 0 ≤ λ < λ∗, there exists a unique minimal solution uλ of (S)λ such thatµ1,λ(uλ) > 0. Moreover, for each x ∈ Ω, the function λ → uλ(x) is strictly increas-ing and differentiable on (0, λ∗).

2. If 1 ≤ N ≤ 7 then –by means of energy estimates– one has supλ∈(0,λ∗) ‖ uλ ‖∞< 1 andconsequently, u∗ = lim

λ↑λ∗uλ exists in C2,α(Ω) with 0 < α < 1 and is a solution for (S)λ∗

such that µ1,λ∗(u∗) = 0. In particular, u∗ –often referred to as the extremal solution ofproblem (S)λ– is unique.

3. On the other hand, if N ≥ 8, f(x) = |x|α with 0 ≤ α ≤ α∗∗(N) := 4−6N+3√

6(N−2)4 and

Ω is the unit ball, then the extremal solution is necessarily u∗(x) = 1 − |x| 2+α3 and is

therefore singular.

We note that, in general, the function u∗ exists in any dimension, does solve (S)λ∗ in a suitableweak sense, and is the unique solution in an appropriate class. The above theorem says thatit is, however, a classical solution in dimensions 1 ≤ N ≤ 7, and this will allow us to startanother branch of non-minimal (unstable) solutions. Indeed, we show in §2.5 –following ideas

16

of Crandall-Rabinowitz [20]– that, for 1 ≤ N ≤ 7, and for λ close enough to λ∗, there exists aunique second branch Uλ of solutions for (S)λ, bifurcating from u∗, with

µ1,λ(Uλ) < 0 while µ2,λ(Uλ) > 0. (2.1.6)

In Chapter 3, we shall provide a variational (mountain pass) characterization of these unstablesolutions and more importantly, we establish –under the same dimension restriction as above–a compactness result along the second branch of unstable solutions leading to a –nonzero–second bifurcation point.

Issues of uniqueness, multiplicity and other qualitative properties of the solutions for (S)λare still far from being well understood, even in the radially symmetric case which we considerin §2.6. Some of the classical work of Joseph-Lundgren [43] and many that followed can beadapted to this situation when the permittivity profile is constant. However, the case of apower-law permittivity profile f(x) = |x|α defined in a unit ball already presents a muchricher situation. In §2.6 we present some numerical evidence for various conjectures relatingto this case, some of which will be tackled in Chapter 3. A detailed and involved analysis ofcompactness along the unstable branches will be discussed there, as well as some informationabout the second bifurcation point.

2.2 The pull-in voltage λ∗

In this section, we first establish the existence and some monotonicity properties for the pull-involtage λ∗, which is defined as

λ∗(Ω, f) = supλ > 0 | (S)λ possesses at least one solution . (2.2.1)

In other words, λ∗ is called pull-in voltage if there exist uncollapsed states for 0 < λ < λ∗

while there are none for λ > λ∗. We then study how λ∗(Ω, f) varies with the domain Ω, thedimension N and the permittivity profile f .

2.2.1 Existence of the pull-in voltage

For any bounded domain Γ in RN , we denote by µΓ the first eigenvalue of −∆ on H10 (Γ) and by

ψΓ the corresponding positive eigenfunction normalized with supx∈Γ ψΓ = 1. We also associatewith any domain Γ in RN the following parameter:

νΩ = supµΓH(inf

ΩψΓ); Γ domain of RN , Γ ⊃ Ω

, (2.2.2)

where H is the function H(t) = t(t+1+2√t)

(t+1+√t)3 .

Theorem 2.2.1. Assume f is a function satisfying (2.1.1) on a bounded domain Ω in RN ,then there exists a finite pull-in voltage λ∗ := λ∗(Ω, f) > 0 such that

1. if λ < λ∗, there exists at least one solution for (S)λ;

2. if λ > λ∗, there is no solution for (S)λ.

17

Moreover, we have the lower bound

λ∗(Ω, f) ≥ νΩ

supx∈Ω f(x). (2.2.3)

Proof: We need to show that (S)λ has at least one solution when λ < νΩ(supΩ f(x))−1. Indeed,it is clear that u ≡ 0 is a subsolution of (S)λ for all λ > 0. To construct a supersolution of(S)λ, we consider a bounded domain Γ ⊃ Ω with smooth boundary, and let (µΓ , ψΓ) be its firsteigenpair normalized in such a way that

supx∈Γ

ψΓ(x) = 1 and infx∈Ω

ψΓ(x) := s1 > 0.

We construct a supersolution in the form ψ = AψΓ where A is a scalar to be chosen later.First, we must have AψΓ ≥ 0 on ∂Ω and 0 < 1−AψΓ < 1 in Ω, which requires that 0 < A < 1.We also require

−∆ψ − λf(x)(1−Aψ)2

≥ 0 in Ω , (2.2.4)

which can be satisfied as long as:

µΓA ψΓ ≥λ supΩ f(x)(1−A ψΓ)2

in Ω , (2.2.5)

orλ sup

Ωf(x) < β(A,Γ) := µΓ inf

g(sA); s ∈ [s1(Γ), 1]

, (2.2.6)

where g(s) = s(1 − s)2. In other words, λ∗ supΩf(x) ≥ supβ(A,Γ); 0 < a < 1,Γ ⊃ Ω, and

therefore it remains to show that

νΩ = supβ(A,Γ); 0 < a < 1,Γ ⊃ Ω

. (2.2.7)

For that, we note first that

infs∈[s1,1]

g(As) = ming(As1), g(A)

.

We also have that g(As1) ≤ g(A) if and only if A2(s31 − 1)− 2A(s2

1 − 1) + (s1 − 1) ≤ 0 whichhappens if and only if A2(s2

1 + s1 + 1)− 2A(s1 + 1) + 1 ≥ 0 or if and only if either A ≤ A− orA ≥ A+ where

A+ =s1 + 1 +

√s1

s21 + 1 + s1

=1

s1 + 1−√s1, A− =

s1 + 1−√s1

s21 + 1 + s1

=1

s1 + 1 +√s1.

Since A− < 1 < a+, we get that

G(A) = infs∈[s1,1]

g(As) =

g(As1) if 0 ≤ A ≤ A− ,g(A) if A− ≤ A ≤ 1 .

(2.2.8)

18

We now have that dGdA = g′(As1)s1 ≥ 0 for all 0 ≤ A ≤ A−. And since A− ≥ 1

3 , we havedGdA = g′(A) ≤ 0 for all A− ≤ A ≤ 1. It follows that

sup0<a<1

infs∈[s1,1]

g(As) = sup0<a<1

G(A) = G(A−) = g(A−)

=1

s1 + 1 +√s1

(1− 1

s1 + 1 +√s1

)2

=s1(s1 + 1 + 2

√s1)

(s1 + 1 +√s1)3

= H(

infΩ ψΓ)),

which proves our lower estimate.Now that we know that λ∗ > 0, pick λ ∈ (0, λ∗) and use the definition of λ∗ to find a

λ ∈ (λ, λ∗) such that (S)λ has a solution uλ,

−∆uλ =λf(x)

(1− uλ)2, x ∈ Ω ; uλ = 0 , x ∈ ∂Ω ,

and in particular −∆uλ ≥ λf(x)(1−uλ)2 for x ∈ Ω which then implies that uλ is a supersolution of

(S)λ. Since u ≡ 0 is a subsolution of (S)λ, then we can conclude again that there is a solutionuλ of (S)λ for every λ ∈ (0, λ∗).

It is also easy to show that λ∗ is finite, since if (S)λ has at least one solution 0 µΩ ≥ µΩ

∫

ΩuψΩ = −

∫

Ωu∆ψΩ = −

∫

ΩψΩ∆u = λ

∫

Ω

ψΩf

(1− u)2dx ≥ λ

∫

ΩψΩfdx

(2.2.9)and therefore λ∗ < +∞. The definition of λ∗ implies that there is no solution of (S)λ for anyλ > λ∗.

2.2.2 Monotonicity results for the pull-in voltage

In this subsection, we give a more precise characterization of λ∗, namely as the endpoint for thebranch of minimal solutions. This will allow us to establish various monotonicity propertiesfor λ∗ that will help in the estimates given in the next subsections. First we give a recursivescheme for the construction of minimal solutions.

Theorem 2.2.2. Assume f is a function satisfying (2.1.1) on a bounded domain Ω in RN ,then for any 0 < λ < λ∗(Ω, f) there exists a unique minimal positive solution uλ for (S)λ. Itis obtained as the limit of the sequence un(λ;x) constructed recursively as follows: u0 ≡ 0 inΩ and, for each n ≥ 1,

−∆un =λf(x)

(1− un−1)2, x ∈ Ω ;

0 ≤ un < 1 , x ∈ Ω ; un = 0 , x ∈ ∂Ω .

(2.2.10)

19

Proof: Let u be any positive solution for (S)λ, and consider the sequence un(λ;x) definedin (2.2.10). Clearly u(x) > u0 ≡ 0 in Ω, and whenever u(x) ≥ un−1 in Ω, then

−∆(u− un) = λf(x)[ 1(1− u)2

− 1(1− un−1)2

] ≥ 0 , x ∈ Ω ,

u− un = 0 , x ∈ ∂Ω .

The maximum principle and an immediate induction yield that 1 > u(x) ≥ un in Ω for alln ≥ 0. In a similar way, the maximum principle implies that the sequence un(λ;x) ismonotone increasing. Therefore, un(λ;x) converges uniformly to a positive solution uλ(x),satisfying u(x) ≥ uλ(x) in Ω, which is a minimal positive solution of (S)λ. It is also clear thatuλ is unique in this class of solutions.

Remark 2.2.1. Let g(x, ξ,Ω) be the Green’s function of the Laplace operator, with g(x, ξ,Ω) = 0on ∂Ω. Then the iteration in (2.2.10) can be replaced by u0 ≡ 0 in Ω, and for each n ≥ 1,

un(λ;x) = λ

∫

Ω

f(ξ)g(x, ξ,Ω)(1− un−1(λ; ξ))2

dξ , x ∈ Ω ;

un(λ;x) = 0 , x ∈ ∂Ω .(2.2.11)

The same reasoning as above yields that limn→∞ un(λ;x) = uλ(x) for all x ∈ Ω.

The above construction of solutions yields the following monotonicity result for the pull-involtage.

Proposition 2.2.3. If Ω1 ⊂ Ω2 and if f is a function satisfying (2.1.1) on Ω2, then λ∗(Ω1) ≥λ∗(Ω2) and the corresponding minimal solutions satisfy uΩ1

(λ, x) ≤ uΩ2(λ, x) on Ω1 for every

0 < λ < λ∗(Ω2).

Proof: Again the method of sub/supersolutions immediately yields that λ∗(Ω1) ≥ λ∗(Ω2).Now consider, for i = 1, 2, the sequences un(λ, x,Ωi) on Ωi defined by (2.2.11) whereg(x, ξ,Ωi) are the corresponding Green’s functions on Ωi. Since Ω1 ⊂ Ω2, we have thatg(x, ξ,Ω1) ≤ g(x, ξ,Ω2) on Ω1. Hence, it follows that

u1(λ, x,Ω2) = λ

∫

Ω2

f(ξ)g(x, ξ,Ω2)dξ ≥ λ∫

Ω1

f(ξ)g(x, ξ,Ω1)dξ = u1(λ, x,Ω1)

on Ω1. By induction we conclude that un(λ, x,Ω2) ≥ un(λ, x,Ω1) on Ω1 for all n. On the otherhand, since un(λ, x,Ω2) ≤ un+1(λ, x,Ω2) on Ω2 for n, we get that un(λ, x,Ω1) ≤ uΩ2

(λ, x) onΩ1, and we are done.

We also note the following easy comparison results, and we omit the details.

Corollary 2.2.4. Suppose f1, f2 : Ω→ R are two functions satisfying (2.1.1) such that f1(x) ≤f2(x) on Ω, then λ∗(Ω, f1) ≥ λ∗(Ω, f2), and for 0 < λ < λ∗(Ω, f2) we have u1(λ, x) ≤ u2(λ, x)on Ω, where u1(λ, x) (resp., u2(λ, x)) are the unique minimal positive solution of

−∆u = λf1(x)(1−u)2 (resp., −∆u = λf2(x)

(1−u)2 ) on Ω and u = 0 on ∂Ω.

Moreover, if f2(x) > f1(x) on a subset of positive measure, then u1(λ, x) < u2(λ, x) for allx ∈ Ω.

20

We shall also need the following result which is adapted from [6] (Theorem 4.10) where itis proved for regular nonlinearities.

Proposition 2.2.5. For any bounded domain Γ in RN and any function f satisfying (2.1.1)on Γ, we have

λ∗(Γ, f) ≥ λ∗(BR, f∗)where BR = BR(0) is the Euclidean ball in RN with radius R > 0 and with volume |BR| = |Γ|,and where f∗ is the Schwarz symmetrization of f .

Proof: For any bounded Γ ⊂ RN , define its symmetrized domain Γ∗ = BR to be the ballx : |x| < R with |Γ| = |BR|. If u is a real-valued function on Γ, we define its symmetrizedfunction u∗ : Γ∗ = BR → R by u∗(x) = supµ : x ∈ BR(µ) where BR(µ) is the symmetrizationof the superlevel set Γ(µ) = x ∈ Ω : µ ≤ u(x) (i.e., BR(µ) = Γ(µ)∗). If h and g are continuousfunctions on Γ, then the following inequality holds (See Lemma 2.4 of [6])

∫

Γhgdx ≤

∫

BR

h∗g∗dx . (2.2.12)

As in Theorem 4.10 of [6], we consider for any λ ∈ (0, λ∗(BR)) the minimal sequence un for(Sλ) in Γ as defined in (2.2.10), and let vn be the minimal sequence for the correspondingSchwarz symmetrized problem:

−∆v =λf∗(x)(1− v)2

x ∈ BR , (2.2.13a)

v = 0 x ∈ ∂BR (2.2.13b)

with 0 < v < 1 on BR = Γ∗. Since λ ∈ (0, λ∗(BR)), we can consider the corresponding minimalsolution v

λfor (2.2.13b). As in Theorem 2.2.2 we have 0 ≤ vn ≤ v

λ< 1 on BR for all n ≥ 1.

We shall show that un also satisfies 0 ≤ u∗n ≤ vλ < 1 on BR for all n ≥ 1.We now write f(a = ωN |x|N ) for f∗(x), and un(a = ωN |x|N ) for u∗n(x). Applying (2.2.12)

and the argument for (4.9) in [6], we obtain that u0 = v0 = 0 in (0, R), and for each n ≥ 1,

dunda

+λ

q(a)

∫ a

0

f

(1− un−1)2dr ≥ 0 in (0, R) , (2.2.14)

anddvnda

+λ

q(a)

∫ a

0

f

(1− vn−1)2dr = 0 in (0, R) (2.2.15)

with q(a) = [Nω1/NN

a(N−1)/N ]2 > 0. We claim that for any n ≥ 1, we have

un(a) ≤ vn(a) a ∈ (0, R) . (2.2.16)

In fact, for n = 1 we have du1/da ≥ dv1/da, and integration yields that

−u1(a) = u1(R)− u1(a) ≥ v1(R)− v1(a) = −v1(a) ,

and hence u1(a) ≤ v1(a) on [0, R]. (2.2.16) is now proved by induction. Suppose it holds forn ≤ k − 1, then one gets from (2.2.14) and (2.2.15) that duk/da ≥ dvk/da, which establishes(2.2.16) for all n ≥ 1.

21

Therefore, the minimal sequence un(x) on Γ is bounded by maxx∈BR vλ(x) < 1, andagain as in the proof of Theorem 2.2.2, there exists a minimal solution u

λfor (S)λ on Γ. This

means λ∗(Γ, f) ≥ λ∗(BR, f∗).

2.3 Estimates for the pull-in voltage

In this section, analytically and numerically we shall discuss estimates of pull-in voltage λ∗.For that we shall write |Ω| for the volume of a domain Ω in RN and P (Ω) :=

∫∂Ω dS for its

“perimeter”, with ωN referring to the volume of the unit ball B1(0) in RN . We denote byµΩ the first eigenvalue of −∆ on H1

0 (Ω) and by φΩ the corresponding positive eigenfunctionnormalized with

∫Ω φΩdx = 1.

2.3.1 Lower bounds for λ∗

While the lower bound in (2.2.3) is useful to prove existence, it is not easy to compute. Thefollowing proposition gives more computationally accessible lower estimates for λ∗.

Proposition 2.3.1. Assume f is a function satisfying (2.1.1) on a bounded domain Ω in RN ,then we have the following lower bound:

λ∗(Ω, f) ≥ max8N

27,

6N − 89

1supΩ f

(ωN|Ω|) 2N. (2.3.1)

Moreover, if f(x) ≡ |x|α with α ≥ 0 and Ω is a ball of radius R, then we have

λ∗(BR, |x|α) ≥ max4(2 + α)(N + α)

27,(2 + α)(3N + α− 4)

9

R−(2+α). (2.3.2)

Finally, if N ≥ 8 and 0 ≤ α ≤ α∗∗(N) := 4−6N+3√

6(N−2)4 , we have

λ∗(B1, |x|α) =(2 + α)(3N + α− 4)

9. (2.3.3)

Proof: Setting R =( |Ω|ωN

) 1N , it suffices –in view of Proposition 2.2.5– and since sup

BR

f∗ = supΩf ,

to show thatλ∗(BR, f∗) ≥ max

8N27R2 supΩ f

∗ ,6N − 8

9R2 supΩ f∗

(2.3.4)

for the case where Ω = BR. In fact, the function w(x) = 13(1− |x|2

R2 ) satisfies on BR

−∆w =2N3R2

=2N(1− 1

3)2

3R2

1(1− 1

3)2

≥ 8N27R2 supΩ f

f(x)

[1− 13(1− |x|2

R2 )]2

=8N

27R2 supΩ f

f(x)(1− w)2

.

22


So for λ ≤ 8N27R2 supΩ f

, w is a supersolution of (S)λ in BR. Since on the other hand w0 ≡ 0is a subsolution of (S)λ and w0 ≤ w in BR, then there exists a solution of (S)λ in BR whichproves a part of (2.3.4).

A similar computation applied to the function v(x) = 1 − ( |x|R )23 shows that v is also a

supersolution as long as λ ≤ 6N−89R2 supΩ f

.

In order to prove (2.3.2), it suffices to note that w(x) = 13

(1− |x|2+α

R2+α

)is a supersolution for

(S)λ on BR provided λ ≤ 4(2+α)(N+α)27R2+α , and that v(x) = 1− ( |x|R )

2+α3 is a supersolution for (S)λ

on BR, provided λ ≤ (2+α)(3N+α−4)9R2+α .

In order to complete the proof of Proposition 2.3.1, we need to establish that the functionu∗(x) = 1 − |x| 2+α

3 is the extremal function as long as N ≥ 8 and 0 ≤ α ≤ α∗∗(N) =4−6N+3

√6(N−2)

4 . This will then yield that for such dimensions and these values of α, thevoltage λ = (2+α)(3N+α−4)

9 is exactly the pull-in voltage λ∗.First, it is easy to check that u∗ is a H1

0 (Ω)-weak solution of (S)λ∗ . Since ‖u∗‖∞ = 1, andby the characterization of Theorem 2.5.1 below, we need only to prove that

∫

Ω|∇φ|2 ≥

∫

Ω

2λ|x|α(1− u∗)3

φ2 ∀φ ∈ H10 (Ω). (2.3.5)

However, Hardy’s inequality gives for N ≥ 2:∫

B1

|∇φ|2 ≥ (N − 2)2

4

∫

B1

φ2

|x|2

for any φ ∈ H10 (B1), which means that (2.3.5) holds whenever 2λ∗ ≤ (N−2)2

4 or, equivalently,

if N ≥ 8 and 0 ≤ α ≤ α∗∗ = 4−6N+3√

6(N−2)4 .

Remark 2.3.1. The above lower bounds can be improved at least in low dimensions. Firstnote from (2.3.2) that if N > 12+α

5 , then λ2 = (2+α)(3N+α−4)9 is the better lower bound and

is actually sharp on the ball as soon as N ≥ 8 and α ≤ α∗∗. For lower dimensions, thebest lower bounds are more complicated even when one considers supersolutions of the formv(x) = a(1 − ( |x|R )k) and optimize λ(a, k,R) over a and k. For example, in the case whereα = 0, N = 2 and R = 1, one can see that a better lower bound can be obtained via thesupersolution v(x) = 1

2.4(1− |x|1.6).

2.3.2 Upper bounds for λ∗

We note that (2.2.9) already yields a finite upper bound for λ∗. However, Pohozaev-typearguments can be used to establish better and more computable upper bounds. For a generaldomain Ω, the following upper bounds on λ∗(Ω) were established in [50] and [32], respectively.

Proposition 2.3.2. (1) Assume f is a function satisfying (2.1.1) on a bounded domain Ω inRN such that infΩ f > 0, then

λ∗(Ω, f) ≤ λ1 ≡ 4µΩ

27(

infΩf)−1

. (2.3.6)

23

(2) If we only suppose that f > 0 on a set of positive measure, then

λ∗(Ω, f) ≤ λ2 ≡ µΩ

3( ∫

ΩfφΩ dx

)−1. (2.3.7)

Proof: (1). We multiply the equation (S)λ by φΩ, integrate the resulting equation over Ω,and use Green’s identity to obtain

∫

Ω

(− µΩu+λf(x)

(1− u)2

)φΩ dx = 0 . (2.3.8)

Since C := infΩ f > 0 and φΩ > 0, the equality in (2.3.8) is impossible when

− µΩu+λC

(1− u)2> 0 , for all x ∈ Ω. (2.3.9)

A simple calculation using (2.3.9) shows that (2.3.9) holds when λ > λ1, where λ1 is given in(2.3.6). This completes the proof of Proposition 2.3.2(1).

As shown below, the bound (2.3.6) on λ∗ is rather good when applied to the constantpermittivity profile f(x) ≡ 1. However, this bound is useless when the minimum of f(x) on Ωis zero, and cannot be used to estimate λ∗ for the power-law permittivity profile f(x) = |x|αwith α > 0. Therefore, it is desirable to obtain a bound on λ∗ that depends more on the globalproperties of f . Such a bound was established in [32] and here is a sketch of its proof.

(2). Multiply now (S)λ by φΩ(1− u)2, and integrate the resulting equation over Ω to get∫

ΩλfφΩ dx =

∫

ΩφΩ(1− u)2∆u dx . (2.3.10)

Using the identity ∇ · (Hg) = g∇ ·H + H · ∇g for any smooth scalar field g and vector fieldH, together with the Divergence theorem, we calculate∫

ΩλfφΩ dx =

∫

∂Ω(1− u)2φΩ∇u · ν dS +

∫

Ω∇u · ∇ [φΩ(1− u)2

]dx , (2.3.11)

where ν is the unit outward normal to ∂Ω. Since φΩ = 0 on ∂Ω, the first term on the right-handside of (2.3.11) vanishes. By calculating the second term on the right-hand side of (2.3.11) weget: ∫

ΩλfφΩ dx = −

∫

Ω2(1− u)φΩ|∇u|2 dx+

∫

Ω(1− u)2∇u · ∇φΩ dx (2.3.12a)

≤ −∫

Ω

13∇φΩ · ∇

[(1− u)3

]dx . (2.3.12b)

The right-hand side of (2.3.12b) is evaluated explicitly by∫

ΩλfφΩ dx ≤ −1

3

∫

∂Ω(1− u)3∇φΩ · ν dS − µΩ

3

∫

Ω(1− u)3φΩ dx . (2.3.13)

For 0 ≤ u < 1, the last term on the right-hand side of (2.3.13) is positive. Moreover, u = 0 on∂Ω so that

∫∂Ω∇φΩ · ν dS = −µΩ since

∫Ω φΩ dx = 1. Therefore, if (Sλ) has a solution, then

(2.3.13) yields

λ

∫

ΩfφΩ dx ≤ µΩ

3. (2.3.14)

This proves that there is no solution for λ > λ2, which gives (2.3.7).

24

Remark 2.3.2. The above estimate is not sharp, at least in dimensions 1 ≤ N ≤ 7, as one canshow that there exists 1 > α(Ω, N) > 0 such that

λ ≤ µΩ

3(1− α(Ω, N))

( ∫

ΩfφΩdx

)−1. (2.3.15)

Indeed, this follows from inequality (2.3.13) above and Theorem 2.4.5 below where it will beshown that in these dimensions, there exists 0 < C(Ω, N) < 1 independent of λ such that‖uλ‖∞ ≤ C(Ω, N) for any minimal solution uλ. It is now easy to see that α(Ω, N) can betaken to be

α(Ω, N) :=(1− C(Ω, N)

)3 ∫

ΩφΩdx.

We now consider problem (S)λ in the case where Ω ⊂ RN is a strictly star-shaped domaincontaining 0, meaning that Ω satisfies the additional property that there exists a positiveconstant a such that

x · ν ≥ a > 0 for all x ∈ ∂Ω , (2.3.16)

where ν is the unit outer normal to ∂Ω.

Proposition 2.3.3. Suppose f ≡ 1 and that the strictly star-shaped smooth domain Ω ⊂ RNsatisfies (2.3.16). Then the pull-in voltage λ∗(Ω) satisfies:

λ∗(Ω) ≤ λ3 =(N + 2)2P (Ω)

8aN |Ω| . (2.3.17)

where |Ω| is the volume and P (Ω) is the perimeter of Ω.In particular, if Ω is the Euclidean unit ball in RN , then we have the bound

λ∗(B1(0)) ≤ (N + 2)2

8.

Proof: Recall Pohozaev’s identity: If u is a solution of

∆u+ λg(u) = 0 for x ∈ Ω ,u = 0 for x ∈ ∂Ω ,

thenNλ

∫

ΩG(u)dx− N − 2

2λ

∫

Ωug(u)dx =

12

∫

∂Ω(x · ν)

(∂u∂ν

)2dS , (2.3.18)

where G(u) =∫ u

0 g(s)ds. Applying this with g(u) = 1(1−u)2 and G(u) = u

1−u yields

λ

2

∫

Ω

u(N + 2− 2Nu)(1− u)2

dx =12

∫

∂Ω(x · ν)

(∂u∂ν

)2dS

≥ a

2P (Ω)

(∫

∂Ω

∂u

∂νdS)2

=a

2P (Ω)

(−∫

Ω∆udx

)2

=aλ2

2P (Ω)

(∫

Ω

dx

(1− u)2

)2,

(2.3.19)

25


where we have used the Divergence Theorem and Holder’s inequality∫

∂Ω

∂u

∂νdS ≤

(∫

∂Ω

(− ∂u

∂ν

)2dS)1/2(∫

∂ΩdS)1/2

.

Since∫

Ω

u(N + 2− 2Nu)(1− u)2

dx =∫

Ω

[− 2N

(u− N + 2

4N)2 +

(N + 2)2

8N

] 1(1− u)2

dx

≤ (N + 2)2

8N

∫

Ω

dx

(1− u)2,

we deduce from (2.3.19) that

(N + 2)2

8N≥ aλ

P (Ω)

∫

Ω

dx

(1− u)2≥ aλ|Ω|P (Ω)

,

which implies the upper bound (2.3.17) for λ∗.Finally, for the special case where Ω = B1(0) ⊂ RN , we have a = 1 and P (B1(0))

ωN

= N and

hence the bound λ∗(B1(0)) ≤ λ3 = (N+2)2

8 .

2.3.3 Numerical estimates for λ∗

In this subsection, we apply numerical methods to discussing the bounds of λ∗. In the com-putations below we shall consider two choices for the domain Ω,

Ω : [−1/2, 1/2] (slab) ; Ω : x2 + y2 ≤ 1 (unit disk) . (2.3.20)

(a). λ∗ versus α (slab) (b). λ∗ versus α (unit disk)

05101520

0 3 6 9 12 15

04812

16

0 1 2 3 4 5

Figure 2.2: Plots of λ∗ versus α for a power-law profile (heavy solid curve) and the exponentialprofile (solid curve). The left figure corresponds to the slab domain, while the right figurecorresponds to the unit disk.

26


Exponential Profiles:

Ω α λ λ∗ λ1 λ2

slab 0 1.185 1.401 1.462 3.290slab 1.0 1.185 1.733 1.878 4.023slab 3.0 1.185 2.637 3.095 5.965slab 6.0 1.185 4.848 6.553 10.50

unit disk 0 0.593 0.789 0.857 1.928unit disk 0.5 0.593 1.153 1.413 2.706unit disk 1.0 0.593 1.661 2.329 3.746unit disk 3.0 0.593 6.091 17.21 11.86

Table 2.1: Numerical values for pull-in voltage λ∗ with the bounds given in Theorem 2.1.1.Here the exponential permittivity profile is chosen as (2.3.21).

For the permittivity profile, following [32] we consider

slab : f(x) = |2x|α (power-law) ; f(x) = eα(x2−1/4) (exponential) , (2.3.21a)

unit disk : f(x) = |x|α (power-law) ; f(x) = eα(|x|2−1) (exponential) , (2.3.21b)

with α ≥ 0. To compute the bounds λ1 and λ2, we must calculate the first eigenpair µΩ andφΩ of −∆ on Ω, normalized by

∫Ω φΩ dx = 1, for each of these domains. A simple calculation

yields that

µΩ = π2 , φΩ =π

2sin[π(x+

12

)](Slab) ; (2.3.22a)

µΩ = z20 ≈ 5.783 , φΩ =

z0

J1(z0)J0(z0|x|) (Unit Disk) . (2.3.22b)

Here J0 and J1 are Bessel functions of the first kind, and z0 ≈ 2.4048 is the first zero of J0(z).The bounds λ1 and λ2 can be evaluated by substituting (2.3.22) into (2.1.2b). Notice that λ2

is, in general, determined only up to a numerical quadrature.In Figure 2.2(a) we plot the saddle-node value λ∗ versus α for the slab domain. A similar

plot is shown in Figure 2.2(b) for the unit disk. The numerical computations are done usingBVP solver COLSYS [2] to solve the boundary value problem (S)λ and Newton’s method todetermine the saddle-node point. Theorem 2.1.1 guarantees a finite pull-in voltage for anyα > 0, while λ∗ is seen to increase rapidly with α. Therefore, by increasing α, or equivalentlyby increasing the spatial extent where f(x) 1, one can increase the stable operating range ofthe MEMS capacitor. In Table 2.1 we give numerical results for λ∗, together with the boundsgiven by Theorem 2.1.1, in the case of exponential permittivity profiles, while Table 2.2 dealswith power-law profiles. From Table 2.1, we observe that the bound λ1 for λ∗ is better thanλ2 for small values of α. However, for α 1, we can use Laplace’s method on the integraldefining λ2 to obtain for the exponential permittivity profile that

λ1 =4b2127ec1α , λ2 ∼ c2α

2 . (2.3.23)

27

Power-Law Profiles:Ω α λc(α) λ∗ λ1 λ2

slab 0 1.185 1.401 1.462 3.290slab 1.0 3.556 4.388 ∞ 9.044slab 3.0 11.851 15.189 ∞ 28.247slab 6.0 33.185 43.087 ∞ 76.608

unit disk 0 0.593 0.789 0.857 1.928unit disk 1.0 1.333 1.775 ∞ 3.019unit disk 5.0 7.259 9.676 ∞ 15.82unit disk 20 71.70 95.66 ∞ 161.54

Table 2.2: Numerical values for pull-in voltage λ∗ with the bounds given in Theorem 2.1.1.Here the power-law permittivity profile is chosen as (2.3.21).

Here b1 = π2, c1 = 1/4, c2 = 1/3 for the slab domain, and b1 = z20 , c1 = 1, c2 = 4/3 for the

unit disk, where z0 is the first zero of J0(z) = 0. Therefore, for α 1, the bound λ2 is betterthan λ1. A similar calculation can be done for the power-law profile, see Table 2.2. For thiscase, it is clear that the lower bound λc(α) in (2.1.4) is better than λ in (2.1.2a), and the upperbound λ1 is undefined. However, by using Laplace’s method, we readily obtain for α 1 thatλ2 ∼ α2/3 for the unit disk and λ2 ∼ 4α2/3 for the slab domain.

Therefore, what is remarkable is that λ1 and λ2 are not comparable even when f is boundedaway from 0 and that neither one of them provides the optimal value for λ∗. This leads us toconjecture that there should be a better estimate for λ∗, one involving the distribution of f inΩ, as opposed to the infimum or its average against the first eigenfunction φΩ .

2.4 The branch of minimal solutions

In the rest of this Chapter, we consider issues of uniqueness and multiplicity of solutions for(S)λ with 0 < λ ≤ λ∗. The bifurcation diagrams in Figure 2.1 show the complexity of thesituation, even in the radially symmetric case. One can see that the number of branches –andof solutions– is closely connected to the space dimension. In this section, we focus on the veryfirst branch of solutions considered to be “minimal”.

The branch of minimal solutions corresponds to the lowest branch in the bifurcation dia-gram, the one connecting the origin point λ = 0 to the first fold at λ = λ∗. To analyze furtherthe properties of this branch, we consider for each solution u of (S)λ, the operator

Lu,λ = −∆− 2λf(1− u)3

(2.4.1)

associated with the linearized problem around u. We denote by µ1(λ, u) the smallest eigenvalueof Lu,λ, that is, the one corresponding to the following Dirichlet eigenvalue problem

−∆φ− 2λf(x)(1− u)3

φ = µφ x ∈ Ω , φ = 0 x ∈ ∂Ω . (2.4.2a)

28

In other words,

µ1(λ, u) = infφ∈H1

0 (Ω)

∫Ω

|∇φ|2 − 2λf(1− u)−3φ2dx∫

Ω φ2dx

.

A solution u for (S)λ is said to be stable (resp., semi-stable) if µ1(λ, u) > 0 (resp., µ1(λ, u) ≥ 0).

2.4.1 Spectral properties of minimal solutions

We start with the following crucial lemma, which shows among other things that semi-stablesolutions are necessarily minimal solutions.

Lemma 2.4.1. Let f be a function satisfying (2.1.1) on a bounded domain Ω in RN , and letλ∗ := λ∗(Ω, f) be as in Theorem (2.1.1). Suppose u is a positive solution of (S)λ, and considerany –classical– supersolution v of (S)λ, that is,

−∆v ≥ λf(x)(1− v)2

x ∈ Ω , (2.4.3a)

0 ≤ v(x) < 1 x ∈ Ω (2.4.3b)v = 0 x ∈ ∂Ω. (2.4.3c)

If µ1(λ, u) > 0, then v ≥ u on Ω, and if µ1(λ, u) = 0, then v = u on Ω.

Proof: For a given λ and x ∈ Ω, use the fact that f(x) ≥ 0 and that t→ λf(x)(1−t)2 is convex on

(0, 1), to obtain

−∆(u+ τ(v − u))− λf(x)[1− (u+ τ(v − u))]2

≥ 0 x ∈ Ω , (2.4.4)

for τ ∈ [0, 1]. Note that (2.4.4) is an identity at τ = 0, which means that the first derivativeof the left-hand side for (2.4.4) with respect to τ is nonnegative at τ = 0,

−∆(v − u)− 2λf(x)(1− u)3

(v − u) ≥ 0 , x ∈ Ω , (2.4.5a)

v − u = 0 , x ∈ ∂Ω . (2.4.5b)

Thus, the maximal principle implies that if µ1(λ, u) > 0, we have v ≥ u on Ω, while ifµ1(λ, u) = 0, then Lemma 2.16 of [20] gives

−∆(v − u)− 2λf(x)(1− u)3

(v − u) = 0 x ∈ Ω . (2.4.6)

In the latter case the second derivative of the left-hand side for (2.4.4) with respect to τ isnonnegative a τ = 0 again,

− 6λf(x)(1− u)4

(v − u)2 ≥ 0 x ∈ Ω , (2.4.7)

From (2.4.7) we deduce that v ≡ u in Ω \ Ω0, where

Ω0 = x ∈ Ω : f(x) = 0 for x ∈ Ω . (2.4.8)

29

On the other hand, (2.4.6) reduces to

−∆(v − u) = 0 x ∈ Ω0 ; v − u = 0 x ∈ ∂Ω0 ,

which implies v ≡ u on Ω0. Hence if µ1(λ, u) = 0 then v ≡ u on Ω, which completes the proofof Lemma 2.4.1.

Theorem 2.4.2. Assume f is a function satisfying (2.1.1) on a bounded domain Ω in RN ,and consider the branch λ→ uλ of minimal solutions on (0, λ∗). Then the following hold:

1. For each x ∈ Ω, the function λ→ uλ(x) is differentiable and strictly increasing on (0, λ∗).

2. For each λ ∈ (0, λ∗), the minimal solution uλ is stable and the function λ → µ1,λ :=µ1(λ, uλ) is decreasing on (0, λ∗).

Proof: Consider λ1 < λ2 < λ∗, their corresponding minimal positive solutions uλ1 and uλ2

and let u∗ be a positive solution for (S)λ2 . For the monotone increasing series un(λ1;x)defined in (2.2.10), we then have u∗ > u0(λ1;x) ≡ 0, and if un−1(λ1;x) ≤ u∗ in Ω, then

−∆(u∗ − un) = f(x)[ λ2

(1− u∗)2− λ1

(1− un−1)2

] ≥ 0 , x ∈ Ω

u∗ − un = 0 , x ∈ ∂Ω .

So we have un(λ1;x) ≤ u∗ in Ω. Therefore, uλ1 = limn→∞ un(λ1;x) ≤ u∗ in Ω, and inparticular uλ1 ≤ uλ2 in Ω. Therefore, duλ(x)

dλ ≥ 0 for all x ∈ Ω.That λ→ µ1,λ is decreasing follows easily from the variational characterization of µ1,λ, the

monotonicity of λ→ uλ, as well as the monotonicity of (1− u)−3 with respect to u.Now we define

λ∗∗ = supλ; uλ is a stable solution for (S)λ

.

It is clear that λ∗∗ ≤ λ∗, and to show the equality, it suffices to prove that there is no minimalsolution for (S)µ with µ > λ∗∗. For that, suppose w is a minimal solution of (S)λ∗∗+δ withδ > 0, then we would have for λ ≤ λ∗∗,

−∆w =(λ∗∗ + δ)f(x)

(1− w)2≥ λf(x)

(1− w)2x ∈ Ω .

Since for 0 < λ < λ∗∗ the minimal solutions uλ are stable, it follows from Lemma 2.4.1 that1 > w ≥ u

λfor all 0 < λ < λ∗∗. Consequently, u

¯= limλλ∗∗ uλ exists in C1(Ω) and is a

solution for (S)λ∗∗ . Now from the definition of λ∗∗, we necessarily have µ1,λ∗∗ = 0, hence byagain applying Lemma 2.4.1, we obtain that w ≡ u

¯and δ = 0 on Ω, which is a contradiction,

and hence λ∗∗ = λ∗.Since each uλ is stable, then by setting F (λ, uλ) := −∆ − λf

(1−uλ)2 , we get that Fuλ(λ, uλ)

is invertible for 0 < λ < λ∗. It then follows from the implicit function theorem that uλ(x) is

differentiable with respect to λ.Finally, by differentiating (S)λ with respect to λ, and since λ → uλ(x) is non-decreasing,

we get

−∆duλdλ− 2λf(x)

(1− uλ)3

duλdλ

=f(x)

(1− uλ)2≥ 0 , x ∈ Ω

duλdλ≥ 0 , x ∈ ∂Ω .

30

Applying the strong maximum principle, we conclude that duλdλ > 0 on Ω for all 0 < λ < λ∗,

and the theorem is proved.

Remark 2.4.1. Lemma 3 of [20] yields µ1(1, 0) as an upper bound for λ∗∗ – at least in the casewhere infΩ f > 0 on Ω. Since λ∗∗ = λ∗, this gives another upper bound for λ∗ in our setting.It is worth noting that the upper bound in Theorem 2.1.1 gives a better estimate, since in thecase where f ≡ 1, we have µ(1, 0) = µΩ/2, while the estimate in Theorem 2.1.1 gives 4µΩ

27 foran upper bound.

2.4.2 Energy estimates and regularity

We start with the following easy observations.

Lemma 2.4.3. Let f be a function satisfying (2.1.1) on a bounded domain Ω in RN . Then,

1. Any (weak) solution u in H10 (Ω) of (S)λ then satisfies

∫Ω

f(1−u)2dx <∞.

2. If infΩ f > 0 and N ≥ 3, then any solution u such that f/(1−u) ∈ L3N/2(Ω) is a classicalsolution.

Proof: (1) Since u ∈ H10 (Ω) is a positive solution of (S)λ, we have

∫

Ω

λf

(1− u)2−∫

Ω

λf

1− u =∫

Ω

λuf

(1− u)2=∫

Ω|∇u|2 =: C < +∞ ,

which implies that∫

Ω

λf

(1− u)2≤ C +

∫

Ω

λf

1− u ≤ C +∫

Ω

[Cε

λf

(1− u)2+C

εf] ≤ C + Cε

∫

Ω

λf

(1− u)2

with ε > 0. Therefore, by choosing ε > 0 small enough, we conclude that∫

Ωf

(1−u)2 <∞ .

(2) Suppose u is a weak solution such that f(x)(1−u)3 ∈ Lp(Ω), which means that f(x)

(1−u)2 ∈L3p/2(Ω). By Sobolev’s theorem we can already deduce that u ∈ C0,α(Ω) with α = 2 − 2N

3p .To get more regularity, it suffices to show that u < 1 on Ω, but then if not we consider x0 ∈ Ωsuch that u(x0) = ‖u‖C(Ω) = 1, then we have

|1− u(x)| = |u(x0)− u(x)| ≤ C|x0 − x|α on Ω .

This inequality shows that if p ≥ N2 then we have

∞ >

∫

Ω

( f(x)(1− u)3

)pdx ≥ C ′

∫

Ω|x− x0|−3pαdx = C ′

∫

Ω|x− x0|−Ndx =∞ ,

a contradiction, which implies that we must have ‖u‖C(Ω) < 1. Note that the above argument cannot be applied to the case where f(x) ≥ 0 vanishes on

Ω, and therefore we have to use the iterative scheme outlined in the next theorem.

31

Theorem 2.4.4. Let f be a function satisfying (2.1.1) on a bounded domain Ω in RN . Thenfor any constant C > 0 there exists 0 < K(C,N) < 1 such that a positive weak solution u of(S)λ (0 < λ < λ∗) is a classical solution and ‖ u ‖

C(Ω)≤ K(C,N) provided

1. N = 1 and ‖ f(1−u)3 ‖L1(Ω)

≤ C,

2. N ≥ 2 and ‖ f(1−u)3 ‖

LN/2(Ω)≤ C.

Proof: We prove this theorem by considering the following three cases separately:

(1) If N = 1, then for any I > 0 we write using the Sobolev inequality with constant K(1) > 0,

K(1) ‖ (1− u)−1 − 1 ‖2L∞

≤∫

Ω

∣∣∇[(1− u)−1 − 1]∣∣2

=13

∫

Ω∇u · ∇[(1− u)−3 − 1

]

=λ

3

∫

Ωf(1− u)−2[(1− u)−3 − 1]

≤ CI + C

∫

(1−u)−3≥I8f(1− u)−2

+C∫

(1−u)−3≥If[(1− u)−3 + 2(1− u)−2 + 4(1− u)−1

][(1− u)−1 − 1

]2

≤ CI + C + C ‖ (1− u)−1 − 1 ‖2L∞((1−u)−3≥I)

∫

(1−u)−3≥I

f

(1− u)3

≤ CI + C + Cε(I) ‖ (1− u)−1 − 1 ‖2L∞ ,

(2.4.9)

with ε(I) =∫

(1−u)−3≥I

f

(1− u)3, where Lemma 2.4.3(1) is applied in the second inequality.

From the assumption f/(1−u)3 ∈ L1(Ω), we have ε(I)→ 0 as I →∞. We now choose I suchthat ε(I) ≤ K(1)

2C , so that the above estimates imply that ‖ (1−u)−1−1 ‖L∞< K(C) . Standardregularity theory for elliptic problems now implies that 1/(1 − u) ∈ C2,α(Ω). Therefore, u isclassical, and there exists a constant K(C,N) which can be taken strictly less than 1 such that‖ u ‖

C(Ω)≤ K(C,N) < 1.

(2) Assuming N = 2, we need to show that

(1− u)−1 ∈ Lp(Ω) for any p > 1. (2.4.10)

Fix p > 1 and let us introduce Tku = minu, 1− k, the truncated function of u at level 1− k,0 < k < 1. For k small, we take (1− Tku)−1 − 1 ∈ H1

0 (Ω) as a test function for (S)λ:∫

Ω

|∇Tku|2(1− Tku)2

=∫

Ω

λf(x)(1− u)2

((1− Tku)−1 − 1

)≤∫

Ω

λf(x)|1− u|3 ≤ C < +∞ (2.4.11)

32

Note that the classical consequence of the Moser-Trudinger inequality gives: there exists C > 0so that ∫

Ωepv ≤ C exp

( p2

16π‖v‖2H1

0 (Ω)

)∀ v ∈ H1

0 (Ω) , p > 1. (2.4.12)

Since now log(

11−Tku

) ∈ H10 (Ω), (2.4.12) and (2.4.11) now yield that for any p > 1:

∫

Ω(1− Tku)−p ≤ C1exp

( p2

16π

∫

Ω|∇ log(

11− Tku)|2) ≤ C2

where C1, C2 denote positive constants depending only on p and C. Taking the limit as k → 0and using that u ≤ 1, we get the validity of (2.4.10).

(3) The case when N > 2 is more elaborate, and we first show that (1− u)−1 ∈ Lq(Ω) for allq ∈ (1,∞). Since u ∈ H1

0 (Ω) is a solution of (S)λ, we already have∫

Ωf

(1−u)2 < C. Now we

proceed by iteration to show that if∫

Ωf

(1−u)2+2θ < C for some θ ≥ 0, then∫

Ω1

(1−u)2∗(1+θ) < C

with 2∗ = 2NN−2 .

Indeed, for any constant θ ≥ 0 and ` > 0 we choose a test function φ = [(1 − u)−3 −1] min(1− u)−2θ, `2. By applying this test function to both sides of (S)λ, we have

λ

∫

Ωf(1− u)−2[(1− u)−3 − 1] min(1− u)−2θ, `2

=∫

Ω∇u · ∇[((1− u)−3 − 1

)min(1− u)−2θ, `2]

= 3∫

Ω|∇u|2(1− u)−4 min(1− u)−2θ, `2

+2θ∫

(1−u)−θ≤`|∇u|2(1− u)−2θ−1[(1− u)−3 − 1] .

(2.4.13)

We now suppose∫

Ωf

(1−u)2+2θ < C. We then obtain from (2.4.13) and the fact that 1(1−u)5 ≤

CI1

(1−u)3 ( 11−u − 1)2 whenever (1− u)−3 ≥ I > 1 that:

33


∫

Ω

∣∣∇[((1− u)−1 − 1

)min(1− u)−θ, `]∣∣2

≤ 2∫

Ω|∇u|2(1− u)−4 min(1− u)−2θ, `2

+2θ2

∫

(1−u)−θ≤`|∇u|2(1− u)−2θ−2

[(1− u)−1 − 1

]2

= 2∫

Ω|∇u|2(1− u)−4 min(1− u)−2θ, `2

+2θ2

∫

(1−u)−θ≤`|∇u|2(1− u)−2θ−1

[ 1(1− u)3

+1

1− u −2

(1− u)2

]

≤ Cλ∫

Ωf(1− u)−2[(1− u)−3 − 1] min(1− u)−2θ, `2

≤ Cλ∫

Ωf(1− u)−5 min(1− u)−2θ, `2

≤ CI + C

∫

(1−u)−3≥If(1− u)−5 min(1− u)−2θ, `2

≤ CI + C

∫

(1−u)−3≥If(1− u)−3

[(1− u)−1 − 1

]2 min(1− u)−2θ, `2

≤ CI + C[ ∫

(1−u)−3≥I

( f

(1− u)3

)N2] 2N

×[ ∫

(1−u)−3≥I

([(1− u)−1 − 1

]min(1− u)−θ, `)

2NN−2

]N−2N

≤ CI + Cε(I)∫

Ω

∣∣∇[((1− u)−1 − 1

)min(1− u)−θ, `]∣∣2

(2.4.14)

with

ε(I) =[ ∫

(1−u)−3≥I

( f

(1− u)3

)N2] 2N.

From the assumption f/(1−u)3 ∈ LN2 (Ω) we have ε(I)→ 0 as I →∞. We now choose I such

that ε(I) = 12C , and the above estimates imply that

∫

(1−u)−θ≤`

∣∣∇[(1− u)−θ−1 − (1− u)−θ]∣∣2 ≤ CI ,

34

where the bound is uniform with respect to `. This estimate leads to

1(θ+1)2

∫(1−u)−θ≤`

∣∣∇[(1− u)−θ−1]∣∣2 =

∫

(1−u)−θ≤`(1− u)−2θ−4

∣∣∇u∣∣2

≤ CI + C

∫

(1−u)−θ≤`(1− u)−2θ−3

∣∣∇u∣∣2

≤ CI +∫

(1−u)−θ≤`

[Cε(1− u)−2θ−4 + C/ε

]∣∣∇u∣∣2

≤ CI + C/ε+ Cε

∫

(1−u)−θ≤`(1− u)−2θ−4

∣∣∇u∣∣2

with ε > 0. This means that for ε > 0 sufficiently small∫

(1−u)−θ≤`

∣∣∇(1− u)−θ−1∣∣2 =

∫

(1−u)−θ≤`(θ + 1)2(1− u)−2θ−4

∣∣∇u∣∣2 < C .

Thus we can let ` → ∞ and we get that (1 − u)−θ−1 ∈ H1(Ω) → L2∗(Ω), which means that∫Ω

1(1−u)2∗(1+θ) < C.

By iterating the above argument for θi + 1 = NN−2(θi−1 + 1) for i ≥ 1 and starting with

θ0 = 0, we find that 1/(1− u) ∈ Lq(Ω) for all q ∈ (1,∞).Standard regularity theory for elliptic problems applies again to give that 1/(1 − u) ∈

C2,α(Ω). Therefore, u is a classical solution and there exists a constant 0 < K(C,N) < 1 suchthat ‖ u ‖

C(Ω)≤ K(C,N) < 1. This completes the proof of Theorem 2.4.4.

Theorem 2.4.5. For any dimension 1 ≤ N ≤ 7 there exists a constant 0 < C(N) < 1independent of λ such that for any 0 < λ < λ∗ the minimal solution uλ satisfies ‖ u

λ‖C(Ω)≤

C(N).Consequently, u∗ = lim

λ↑λ∗uλ exists in the topology of C2,α(Ω) with 0 < α < 1. It is the

unique classical solution for (S)λ∗ and satisfies µ1,λ∗(u∗) = 0.

This result will follow from the following uniform energy estimate on the minimal solutionsuλ.

Proposition 2.4.6. There exists a constant C(p) > 0 such that for each λ ∈ (0, λ∗), the

minimal solution uλ

satisfies ‖ f(1−u

λ)3 ‖Lp(Ω) ≤ C(p) as long as p < 1 + 4

3 + 2√

23 .

Proof: Since minimal solutions are stable, we have

λ

∫

Ω

2f(x)(1− u

λ)3w2dx ≤ −

∫

Ωw∆wdx =

∫

Ω|∇w|2dx (2.4.15)

for all 0 < λ < λ∗ and nonnegative w ∈ H10 (Ω). Setting

w = (1− uλ)i − 1 > 0 , where − 2−

√6 < i < 0 , (2.4.16)

then (2.4.15) becomes

i2∫

Ω(1− u

λ)2i−2|∇u

λ|2dx ≥ λ

∫

Ω

2[1− (1− uλ)i]2f(x)

(1− uλ)3

dx . (2.4.17)

35

On the other hand, multiplying (S)λ by i2

1−2i [(1 − uλ)2i−1 − 1] and applying integration byparts yield that

i2∫

Ω(1− u

λ)2i−2|∇u

λ|2dx = λ

i2

2i− 1

∫

Ω

[1− (1− uλ)2i−1]f(x)

(1− uλ)2

dx . (2.4.18)

Hence (2.4.17) and (2.4.18) reduce to

λ i2

2i− 1

∫

Ω

f(x)(1− u

λ)2dx− 2λ

∫

Ω

f(x)(1− u

λ)3dx+ 4λ

∫

Ω

f(x)(1− u

λ)3−idx

≥ λ(2 +i2

2i− 1)∫

Ω

f(x)(1− u

λ)3−2i

dx .

(2.4.19)

From the choice of i in (2.4.16) we have 2 + i2

2i−1 > 0. So (2.4.19) implies that∫

Ω

f(x)(1− u

λ)3−2i

dx ≤ C∫

Ω

f(x)(1− u

λ)3−idx

≤ C(∫

Ω

∣∣∣ f3−i3−2i

(1− uλ)3−i

∣∣∣3−2i3−i

dx) 3−i

3−2i ·(∫

Ω

∣∣∣f −i3−2i

∣∣∣3−2i−idx) −i

3−2i

≤ C(∫

Ω

f(x)(1− u

λ)3−2i

dx) 3−i

3−2i,

(2.4.20)

where Holder’s inequality is applied. From the above we deduce that∫

Ω

f(x)(1− u

λ)3−2i

dx ≤ C . (2.4.21)

Further we have ∫

Ω

∣∣∣ f(x)(1− u

λ)3

∣∣∣3−2i

3dx =

∫

Ωf−2i

3 · f

(1− uλ)3−2i

dx

≤ C∫

Ω

f

(1− uλ)3−2i

dx ≤ C .(2.4.22)

Therefore, we get that

‖ f(x)(1− u

λ)3‖Lp≤ C , (2.4.23a)

where –in view of (2.4.16)–

p =3− 2i

3≤ 1 +

43

+ 2

√23. (2.4.23b)

Proof of Theorem 2.4.5: The existence of u∗ as a classical solution follows from Proposition

2.4.6 and Theorem 2.4.4, as long as N2 < 1 + 4

3 + 2√

23 , which happens when N ≤ 7.

Since µ1,λ > 0 on the minimal branch for any λ < λ∗, we have the limit µ1,λ∗ ≥ 0. If nowµ1,λ∗ > 0 the implicit function theorem could be applied to the operator Lu

λ∗ ,λ∗ , and would

allow the continuation of the minimal branch λ 7→ uλ

of classical solutions beyond λ∗, whichis a contradiction, and hence µ1,λ∗ = 0. The uniqueness in the class of classical solutions thenfollows from Lemma 2.4.1.

36

2.5 Uniqueness and multiplicity of solutions

The purpose of this section is to discuss uniqueness and multiplicity of solutions for (S)λ.

2.5.1 Uniqueness of the solution at λ = λ∗

We first note that in view of the monotonicity in λ and the uniform boundedness of the firstbranch of solutions, the extremal function defined by u∗(x) = lim

λ↑λ∗uλ(x) always exists, and

can always be considered as a solution for (S)λ∗ in a generalized sense. Now if there exists0 < C < 1 such that ‖ u

λ‖C(Ω)≤ C for each λ < λ∗ –just like in the case where 1 ≤ N ≤ 7–

then we have seen in Theorem 2.4.5 that u∗ is unique among the classical solutions. In thesequel, we tackle the important case when u∗ is a weak solution (i.e., in H1

0 (Ω)) of (S)λ∗ butwith the possibility that ‖u∗‖∞ = 1. Here and in the sequel, u will be called a H1

0 (Ω)-weaksolution of (S)λ if 0 ≤ u ≤ 1 a.e. while u solves (S)λ in the weak sense of H1

0 (Ω).We shall borrow ideas from [10, 14], where the authors deal with the case of regular nonlin-

earities. However, unlike those papers where solutions are considered in a very weak sense, weconsider here a more focussed and much simpler situation. We establish the following usefulcharacterization of the extremal solution.

Theorem 2.5.1. Assume f is a function satisfying (2.1.1) on a bounded domain Ω in RN .For λ > 0, consider u ∈ H1

0 (Ω) to be a weak solution of (S)λ (in the H10 (Ω) sense) such that

‖ u ‖L∞(Ω)= 1. Then the following assertions are equivalent:

1. µ1,λ ≥ 0, that is u satisfies∫

Ω|∇φ|2 ≥

∫

Ω

2λf(x)(1− u)3

φ2 ∀φ ∈ H10 (Ω) , (2.5.1)

2. λ = λ∗ and u = u∗.

We need the following uniqueness result.

Proposition 2.5.2. Assume f is a function satisfying (2.1.1) on a bounded domain Ω in RN .Let u1, u2 be two H1

0 (Ω)-weak solutions of (S)λ so that µ1,λ(ui) ≥ 0 for i = 1, 2. Then u1 = u2

almost everywhere in Ω.

Proof: For any θ ∈ [0, 1] and φ ∈ H10 (Ω), φ ≥ 0, we have that

Iθ,φ : =∫

Ω∇(θu1 + (1− θ)u2

)∇φ−∫

Ω

λf(x)(1− θu1 − (1− θ)u2

)2φ

= λ

∫

Ωf(x)

( θ

(1− u1)2+

1− θ(1− u2)2

− 1(1− θu1 − (1− θ)u2

)2)φ ≥ 0

due to the convexity of 1/(1 − u)2 with respect to u. Since I0,φ = I1,φ = 0, the derivative ofIθ,φ at θ = 0, 1 provides

∫

Ω∇(u1 − u2

)∇φ−∫

Ω

2λf(x)(1− u2

)3 (u1 − u2)φ ≥ 0 ,

∫

Ω∇(u1 − u2

)∇φ−∫

Ω

2λf(x)(1− u1

)3 (u1 − u2)φ ≤ 0

37

for any φ ∈ H10 (Ω), φ ≥ 0. Testing the first inequality on φ = (u1 − u2)− and the second one

on (u1 − u2)+, we get that∫

Ω

[|∇(u1 − u2)−|2 − 2λf(x)

(1− u2)3

((u1 − u2)−

)2] ≤ 0 ,∫

Ω

[|∇(u1 − u2)+|2 − 2λf(x)

(1− u1)3

((u1 − u2)+

)2] ≤ 0 .

Since µ1,λ(u1) ≥ 0, we have1) either µ1,λ(u1) > 0 and then u1 ≤ u2 a.e.,2) or µ1,λ(u1) = 0, which then gives

∫

Ω∇(u1 − u2

)∇φ−∫

Ω

2λf(x)(1− u1

)3 (u1 − u2)φ = 0 (2.5.2)

where φ = (u1 − u2)+. Since Iθ,φ ≥ 0 for any θ ∈ [0, 1] and I1,φ = ∂θI1,φ = 0, we get that:

∂2θθI1,φ = −

∫

Ω

6λf(x)(1− u1)4

((u1 − u2)+

)3 ≥ 0 .

Let Z0 = x ∈ Ω : f(x) = 0. Clearly, (u1 − u2)+ = 0 a.e. in Ω \ Z0 and, by (2.5.2) we get:∫

Ω|∇(u1 − u2

)+|2 = 0.

Hence, u1 ≤ u2 a.e. in Ω. The same argument applies to prove the reversed inequality: u2 ≤ u1

a.e. in Ω. Therefore, u1 = u2 a.e. in Ω, and the proof is complete.

Since ‖ uλ ‖< 1 for any λ ∈ (0, λ∗), we need –in order to prove Theorem 2.5.1– only to showthat (S)λ does not have any H1

0 (Ω)-weak solution for λ > λ∗. By the definition of λ∗, thisis already true for classical solutions. We shall now extend this property to the class of weaksolutions by means of the following result.

Proposition 2.5.3. If w is a H10 (Ω)-weak solution of (S)λ, then for any ε ∈ (0, 1) there exists

a classic solution wε of (S)λ(1−ε).

Proof: First we prove that for any ψ ∈ C2([0, 1]) concave function so that ψ(0) = 0, we havethat ∫

Ω∇ψ(w)∇ϕ ≥

∫

Ω

λf

(1− w)2ψ(w)ϕ (2.5.3)

for any ϕ ∈ H10 (Ω), ϕ ≥ 0. Indeed, by concavity of ψ we get:

∫

Ω∇ψ(w)∇ϕ =

∫

Ωψ(w)∇w∇ϕ =

∫

Ω∇w∇

(ψ(w)ϕ

)−∫

Ωψ(w)ϕ|∇w|2

≥∫

Ω

λf(x)(1− w)2

ψ(w)ϕ

for any ϕ ∈ C∞0 (Ω), ϕ ≥ 0. By density, we get (2.5.3).

38

Now let ε ∈ (0, 1), and define

ψε(w) := 1−(ε+ (1− ε)(1− w)3

) 13, 0 ≤ w ≤ 1 .

Since ψε ∈ C2([0, 1]) is a concave function, ψε(0) = 0 and

ψε(w) = (1− ε)g(ψε(w)

)

g(w), g(s) := (1− s)−2,

by (2.5.3) we obtain that for any ϕ ∈ H10 (Ω), ϕ ≥ 0:

∫

Ω∇ψε(w)∇ϕ ≥

∫

Ω

λf(x)(1− w)2

ψε(w)ϕ = λ(1− ε)∫

Ωf(x)g

(ψε(w)

)ϕ

=∫

Ω

λ(1− ε)f(x)(1− ψε(w))2

ϕ .

Hence, ψε(w) is a H10 (Ω)-weak supersolution of (S)λ(1−ε) so that 0 ≤ ψε(w) ≤ 1 − ε 1

3 < 1.Since 0 is a subsolution for any λ > 0, we get the existence of a H1

0 (Ω)-weak solution wε of(S)λ(1−ε) so that 0 ≤ wε ≤ 1 − ε 1

3 . By standard elliptic regularity theory, wε is a classicalsolution of (S)λ(1−ε).

2.5.2 Uniqueness of low energy solutions for small voltage

In the following we focus on the uniqueness when λ is small enough. We first define nonminimalsolutions for (S)λ as follows.

Definition 2.5.1. A solution 0 ≤ u < 1 is said to be a nonminimal positive solution of (S)λ,if there exists another positive solution v of (S)λ and a point x ∈ Ω such that u(x) > v(x).

Lemma 2.5.4. Suppose u is a nonminimal solution of (S)λ with λ ∈ (0, λ∗). Then µ1(λ, u) <0, and the function w = u− uλ is in the negative space of Lu,λ = −∆− 2λf(x)

(1−u)3 .

Proof: For a fixed λ ∈ (0, λ∗), let uλ

be the minimal solution of (S)λ. We have w = u−uλ≥ 0

in Ω, and

−∆w − λ(2− u− uλ)f

(1− u)2(1− uλ)2w = 0 in Ω .

Hence the strong maximum principle yields that uλ 0. Direct calculations give that

−∆(u− uλ)− 2λf

(1− u)3(u− u

λ)

= λf[ 1(1− u)2

− 1(1− u

λ)2− 2

(1− u)3(u− u

λ)]

=

0 , x ∈ Ω0 ;

< 0 , x ∈ Ω/Ω0 .

(2.5.4)

From this we get

〈Lu,λw,w〉 = λ

∫

Ω/Ω0

f[ 1(1− u)2

− 1(1− u

λ)2− 2

(1− u)3(u− u

λ)](u− u

λ) < 0 . (2.5.5)

39

Now we are able to prove the following uniqueness result.

Theorem 2.5.5. For every M > 0 there exists 0 < λ∗1(M) < λ∗ such that for λ ∈ (0, λ∗1(M))the equation (S)λ has a unique solution v satisfying

1. ‖ f(1−v)3 ‖1 ≤M and the dimension N = 1,

2. ‖ f(1−v)3 ‖1+ε ≤M for some ε > 0 and N = 2,

3. ‖ f(1−v)3 ‖N/2 ≤M and N > 2.

Proof: For any fixed λ ∈ (0, λ∗), let uλ

be the minimal solution of (S)λ, and suppose (S)λhas a nonminimal solution u. The preceding lemma then gives

∫

Ω|∇(u− u

λ)|2dx <

∫

Ω

2λ(u− uλ)2f(x)

(1− u)3dx .

This implies in the case where N > 2 that

C(N)(∫

Ω(u− u

λ)

2NN−2dx

)N−2N

< λ

∫

Ω

2f(x)(1− u)3

(u− uλ)2dx

≤ 2λ(∫

Ω

∣∣ f

(1− u)3

∣∣N2) 2N(∫

Ω(u− u

λ)

2NN−2dx

)N−2N

≤ 2λM2N

(∫

Ω(u− u

λ)

2NN−2dx

)N−2N

,

which is a contradiction if λ < C(N)

2M2N

unless u ≡ uλ. If N = 1, then we write

C(1)‖(u− uλ)‖2∞ < λ

∫

Ω

2f(x)(1− u)3

(u− uλ)2dx ≤ 2λ‖(u− u

λ)‖2∞

∫

Ω

f

(1− u)3dx ,

and the proof follows. A similar proof holds for dimension N = 2.

Remark 2.5.1. The above gives uniqueness for small λ among all solutions that either stayaway from 1 or those that approach it slowly. We do not know whether, if λ is small enough,any positive solution v of (S)λ satisfies

∫Ω(1 − v)−

3N2 dx ≤ M for some uniform bound M

independent of λ. Numerical computations do show that we may have uniqueness for small λ–at least for radially symmetric solutions– as long as N ≥ 2.

2.5.3 Second solutions around the bifurcation point

Our next result is quite standard.

Lemma 2.5.6. Suppose there exists 0 < C < 1 such that ‖ uλ‖C(Ω)≤ C for each λ < λ∗.

Then there exists δ > 0 such that the solutions of (S)λ near (λ∗, uλ∗ ) form a curve ρ(s) =

(λ(s), v(s)) : |s| < δ, and the pair (λ(s), v(s)) satisfies

λ(0) = λ∗, λ′(0) = 0, λ′′(0) < 0 , and v(0) = uλ∗ , v

′(0)(x) > 0 in Ω . (2.5.6)

40

In particular, if 1 ≤ N ≤ 7, then for λ close enough to λ∗ there exists a unique second branchUλ of solutions for (S)λ, bifurcating from u∗, such that

µ1,λ(Uλ) < 0 while µ2,λ(Uλ) > 0. (2.5.7)

Proof: The proof is similar to a related result of Crandall and Rabinowitz (cf. [19, 20]), sowe will be brief. First, the assumed upper bound on u

λin C1 and standard regularity theory

show that if f ∈ C(Ω) then ‖ uλ‖C2,α(Ω)

≤ C < 1 for some 0 < α < 1 (while if f ∈ L∞, then‖ u

λ‖C1,α(Ω)

≤ C < 1). It follows that (λ, uλ) is precompact in the space R × C2,α, and

hence we have a limiting point (λ∗, uλ∗ ) as desired. Since λ∗f(x)

(1−uλ∗ )2 is nonnegative, Theorem 3.2

of [19] characterizes the solution set of (S)λ near (λ∗, uλ∗ ): λ(0) = λ∗, λ′(0) = 0, v(0) = u

λ∗and v′(0) > 0 in Ω. The same computation as in Theorem 4.8 in [19] gives that λ′′(0) < 0.In particular, if 1 ≤ N ≤ 7 then our Theorem 4.5 gives the compactness of u∗ = u

λ∗ , andthe theory of Crandall and Rabinowitz in [20] then implies that, for λ close enough to λ∗,there exists a unique second branch Uλ of solutions for (S)λ, bifurcating from u∗, such thatµ1,λ(Uλ) < 0 while µ2,λ(Uλ) > 0.

Remark 2.5.2. A version of these results will be established variationally in next Chapter.Indeed, we shall give there a variational characterization for both the stable and unstablesolutions uλ, Uλ in the following sense: For 1 ≤ N ≤ 7, there exists δ > 0 such that for anyλ ∈ (λ∗ − δ, λ∗), the minimal solution uλ is a local minimum for some regularized energyfunctional Jε,λ on the space H1

0 (Ω), while the second solution Uλ is a mountain pass for thefunctional Jε,λ.

2.6 Radially symmetric case and power-law profiles

In this section, we discuss issues of uniqueness and multiplicity of solutions for (S)λ when Ωis a symmetric domain and when f is a radially symmetric permittivity profile. Here, one canagain define the corresponding pull-in voltage λ∗r(Ω, f) requiring the solutions to be radiallysymmetric, that is

λ∗r(Ω, f) = supλ; (S)λ has a radially symmetric solution

.

Proposition 2.6.1. Let Ω be a symmetric domain and let f be a nonnegative bounded radiallysymmetric permittivity profile on Ω. Then the minimal solutions of (S)λ are necessarily radiallysymmetric and consequently λ∗r(Ω, f) = λ∗(Ω, f). Moreover, if Ω is a ball, then any radialsolution of (S)λ attains its maximum at 0.

Proof:: It is clear that λ∗r(Ω, f) ≤ λ∗(Ω, f), and the reverse will be proved if we establishthat every minimal solution of (S)λ with 0 < λ < λ∗(Ω, f) is radially symmetric. But this isa straightforward application of the recursive scheme defined in Theorem 2.2.2 which gives aradially symmetric function at each step and therefore the resulting limiting function –whichis the minimal solution– is radially symmetric.

For any radially symmetric u(r) of (S)λ defined in the ball of radius R, we have ur(0) = 0and

−urr − N − 1r

ur =λf

(1− u)2in (0, R) ,

41

Multiplying by rN−1, we get that −d(rN−1ur)dr = λfrN−1

(1−u)2 ≥ 0, and therefore ur < 0 in (0, R)since ur(0) = 0. This shows that u(r) attains its maximum at 0.

The bifurcation diagrams shown in the introduction actually reflect the radially symmetricsituation, and our emphasis in this section is on whether there is a better chance to analyzemathematically the higher branches of solutions in this case. Now some of the classical workof Joseph-Lundgren [43] and many that followed can be adapted to this situation when thepermittivity profile is constant. However, the case of a power-law permittivity profile f(x) =|x|α defined in a unit ball already presents a much richer situation. We now present variousanalytical and numerical evidences for various conjectures relating to this case, some of whichwill be further discussed in Chapter 3.

Power-law permittivity profiles

Consider the domain Ω to be a unit ball B1(0) ⊂ RN (N ≥ 1), and let f(x) = |x|α (α ≥ 0).We analyze in this case the branches of radially symmetric solutions of (S)λ for λ ∈ (0, λ∗]. Inthis case, (S)λ reduces to

−urr − N − 1

rur =

λrα

(1− u)2, 0 < r ≤ 1,

u′(0) = 0 , u(1) = 0 .(2.6.1)

Here r = |x| and 0 0, equation (2.6.1) implies that

γ2β(w′′ +

N − 1P

w′)

=λPα

β2γα1w2

.

We set w(0) = 1 and λ = γ2+αβ3. This yields the following initial value problem:

w′′ +N − 1P

w′ =Pα

w2, P > 0 ,

w′(0) = 0 , w(0) = 1 .(2.6.2)

Since u(1) = 0 we have β = 1/w(γ). Therefore, we conclude that

u(0) = 1− 1w(γ)

,

λ =γ2+α

w3(γ),

(2.6.3)

where w(γ) is a solution of (2.6.2).As was done in [50], one can numerically integrate the initial value problem (2.6.2) and

use the results to compute the complete bifurcation diagram for (2.6.1). We show such acomputation of u(0) versus λ defined in (2.6.3) for the slab domain (N = 1) in Figure 2.3. In

42

this case, one observes from the numerical results that when N = 1, and 0 ≤ α ≤ 1, thereexist exactly two solutions for (S)λ whenever λ ∈ (0, λ∗). On the other hand, the situationbecomes more complex for α > 1 as u(0) → 1. This leads to the question of determining theasymptotic behavior of w(P ) as P →∞. Towards this end, we proceed as follows.

00

1N = 1 and f(x) = |x|α with different ranges of α

λ

|u(0)|

λ* λ* λ* λ * λ

*

α >α*

1<α<=α*

α<=1

Figure 2.3: Plots of u(0) versus λ for profile f(x) = |x|α (α ≥ 0) defined in the slab domain(N = 1). The numerical experiments point to a constant α∗ > 1 (analytically given in (2.6.10))such that the bifurcation diagrams are greatly different for different ranges of α: 0 ≤ α ≤ 1,1 < α ≤ α∗ and α > α∗.

Setting η = logP and w(P ) = PBV (η) > 0 for some positive constant B, we obtain from(2.6.2) that

PB−2V ′′ + (2B +N − 2)PB−2V ′ +B(B +N − 2)PB−2V =Pα−2B

V 2. (2.6.4)

Choosing B − 2 = α− 2B so that B = (2 + α)/3, we get that

V ′′ +3N + 2α− 2

3V ′ +

(2 + α)(3N + α− 4)9

V =1V 2

. (2.6.5)

We can already identify from this equation the following regimes.

Case 1. Assume thatN = 1 and 0 ≤ α ≤ 1. (2.6.6)

In this case, there is no equilibrium point for (2.6.5), which means that the bifurcationdiagram vanishes at λ = 0, from which one infers that in this case, there exist exactly twosolutions for λ ∈ (0, λ∗) and just one for λ = λ∗.

Case 2. N and α satisfy either one of the following conditions:

N = 1 and α > 1, (2.6.7a)

N ≥ 2. (2.6.7b)

43

There exists then an equilibrium point Ve of (2.6.5) which must be positive and satisfies

V 3e =

9(2 + α)(3N + α− 4)

> 0 . (2.6.8)

Linearizing around this equilibrium point by writing

V = Ve + Ceση , 0 < C << 1,

we obtain thatσ2 +

3N + 2α− 23

σ +(2 + α)(3N + α− 4)

3= 0 .

This reduces to

σ± = −3N + 2α− 26

±√4

6, (2.6.9a)

with4 = −8α2 − (24N − 16)α+ (9N2 − 84N + 100) . (2.6.9b)

We note that σ± < 0 whenever 4 ≥ 0. Now define

α∗ = −12

+12

√272, α∗∗ =

4− 6N + 3√

6(N − 2)4

(N ≥ 8) . (2.6.10)

Next, we discuss the ranges of N and α such that 4 ≥ 0 or 4 < 0.

Case 2.A. N and α satisfy either one of the following:

N = 1 with 1 < α ≤ α∗ , (2.6.11a)

N ≥ 8 with 0 ≤ α ≤ α∗∗ . (2.6.11b)

In this case, we have 4 ≥ 0 and

V ∼( 9

(2 + α)(3N + α− 4)

) 13 + C1e

− 3N+2α−2−√46

η + · · · , as η → +∞ .

Further, we conclude that

w ∼ P 2+α3

( 9(2 + α)(3N + α− 4)

) 13 + C1P

−N−22

+√4

6 + · · · , as P → +∞ .

In both cases, the branch monotonically approaches the value 1 as η → +∞. Moreover, sinceλ = γ2+α/w3(γ), we have

λ ∼ λ∗ =(2 + α)(3N + α− 4)

9as γ →∞ , (2.6.12)

which is another important critical threshold for the voltage.In the case (2.6.11a) illustrated by Figure 2.3, we have λ∗ < λ∗, and the number of solutions

increase but remains finite as λ approaches λ∗. On the other hand, in the case of (2.6.11b)

44

illustrated by Figure 2.4, we have λ∗ = λ∗, and there seems to be only one branch of solutions.

Case 2.B. N and α satisfy any one of the following three:

N = 1 with α > α∗ , (2.6.13a)

2 ≤ N ≤ 7 with α ≥ 0 , (2.6.13b)

N ≥ 8 with α > α∗∗ . (2.6.13c)

00

1(a). 2 <= N <= 7 and f(x) = |x|α for any α >= 0

λ

|u(0)|

λ * λ *

00

1(b). N >= 8 and f(x) = |x|α with different ranges of α

|u(0)|

λ λ * λ * λ *

0 <= α <= α **

α > α **

Figure 2.4: Top figure: Plots of u(0) versus λ for 2 ≤ N ≤ 7, where u(0) oscillates around thevalue λ∗ defined in (2.6.12) and u∗ is regular. Bottom figure: Plots of u(0) versus λ for N ≥ 8:when 0 ≤ α ≤ α∗∗, there exists a unique solution for (S)λ with λ ∈ (0, λ∗) and u∗ is singular;when α > α∗∗, u(0) oscillates around the value λ∗ defined in (2.6.12) and u∗ is regular.

In this case, we have 4 < 0 and

V ∼( 9

(2 + α)(3N + α− 4)

) 13 + C1e

− 3N+2α−26

ηcos(√−4

6η + C2

)+ · · · as η → +∞ .

45

We also have for P → +∞,

w ∼ P 2+α3

( 9(2 + α)(3N + α− 4)

) 13 + C1P

−N−22 cos

(√−46

logP + C2

)+ · · · , (2.6.14)

and from the fact that λ = γ2+α/w3(γ) we get again that

λ ∼ λ∗ =(2 + α)(3N + α− 4)

9as γ →∞ .

Note the oscillatory behavior of w(P ) in (2.6.14) for large P , which means that u(0) is expectedto oscillate around the value λ∗ = (2+α)(3N+α−4)

9 as P →∞. The diagrams below point to theexistence of a sequence λi satisfying

λ0 = 0 , λ2k λ∗ as k →∞ ;

λ1 = λ∗ , λ2k−1 λ∗ as k →∞and such that exactly 2k + 1 solutions for (S)λ exist when λ ∈ (λ2k, λ2k+2), while thereare exactly 2k solutions when λ ∈ (λ2k+1, λ2k−1). Furthermore, (S)λ has infinitely multiplesolutions at λ = λ∗.

The three cases (2.6.13a), (2.6.13b) and (2.6.13c) considered here for N and α are illustratedby the diagrams in Figure 2.3, Figure 2.4(a) and Figure 2.4(b), respectively.

We now conclude from above that the bifurcation diagrams show four possible regimes –atleast if the domain is a ball:

A. There is exactly one branch of solution for 0 < λ < λ∗. This regime occurs when N ≥ 8,and if 0 ≤ α ≤ α∗∗(N) := 4−6N+3

√6(N−2)

4 . The results of this section actually show thatin this range, the first branch of solutions “disappears” at λ∗ which happens to be equalto λ∗(α,N) = (2+α)(3N+α−4)

9 .

B. There exists an infinite number of branches of solutions. This regime occurs when

• N = 1 and α ≥ α∗ := −12 + 1

2

√27/2.

• 2 ≤ N ≤ 7 and α ≥ 0;

• N ≥ 8 and α > α∗∗(N) := 4−6N+3√

6(N−2)4 .

In this case, λ∗(α,N) < λ∗ and the multiplicity becomes arbitrarily large as λ approaches–from either side– λ∗(α,N) at which there is a touchdown solution u (i.e., ‖ u ‖∞= 1).

C. There exists a finite number of branches of solutions. In this case, we have again thatλ∗(α,N) < λ∗, but now the branch approaches the value 1 monotonically, and thenumber of solutions increase but remains finite as λ approaches λ∗(α,N). This regimeoccurs when N = 1 and 1 < α ≤ α∗ := −1

2 + 12

√27/2.

D. There exist exactly two branches of solutions for 0 < λ < λ∗ and one solution for λ = λ∗.The bifurcation diagram vanishes when it returns to λ = 0. This regime occurs whenN = 1 and 0 ≤ α ≤ 1.

Some of these questions will be considered in Chapter 3. A detailed and involved analysis ofcompactness along the unstable branches will be discussed there, as well as some informationabout the second bifurcation point.

46

Chapter 3

Compactness along Lower Branches

3.1 Introduction

In this Chapter we continue the analysis of the problem

−∆u =λf(x)

(1− u)2in Ω,

0 0 is a parameter, Ω ⊂ RN is a bounded smooth domain and f satisfies (2.1.1).Following the notations and terminology of Chapter 2, the solutions of (S)λ are considered tobe in the classical sense, and the minimal solution uλ of (S)λ is the classical solution of (S)λsatisfying uλ(x) ≤ u(x) in Ω for any solution u of (S)λ.

There already exist in the literature many interesting results concerning the properties ofthe branch of semi-stable solutions for Dirichlet boundary value problems of the form −∆u =λh(u) where h is a regular nonlinearity (for example of the form eu or (1 + u)p for p > 1). Seefor example the seminal papers [20, 43, 44] and also [15] for a survey on the subject and anexhaustive list of related references. The singular situation was considered in a very generalcontext in [48], and this analysis is completed in Chapter 2 to allow for a general continuouspermittivity profile f(x) ≥ 0. Fine properties of steady states –such as regularity, stability,uniqueness, multiplicity, energy estimates and comparison results– are shown there to dependon the dimension of the ambient space and on the permittivity profile.

Now for any solution u of (S)λ, one can introduce the linearized operator at u defined by:

Lu,λ = −∆− 2λf(x)(1− u)3

,

and its corresponding eigenvalues µk,λ(u); k = 1, 2, .... Note that the first eigenvalue is simpleand is given by:


0 (Ω) ; φ ∈ C∞0 (Ω),∫

Ω|φ(x)|2dx = 1

with the infimum being attained at a first eigenfunction φ1, while the second eigenvalue isgiven by the formula:


0 (Ω) ; φ ∈ C∞0 (Ω),∫

Ω|φ(x)|2dx = 1 ,

∫

Ωφ(x)φ1(x)dx = 0

.

This construction can then be iterated to obtain the k-th eigenvalue µk,λ(u) with the conventionthat eigenvalues are repeated according to their multiplicities.

47

Chapter 3. Compactness along Lower Branches

The usual analysis of the minimal branch (composed of semi-stable solutions) is extendedin Chapter 2 to cover the singular situation (S)λ above – best illustrated by the followingbifurcation diagram– is obtained.

00

1f(x) ≡ 1 with different ranges of N

u(0)

N = 1 2 ≤ N ≤ 7

N ≥ 8

λ* λ* λ* λ* = (6N−8)/9

λ

00

1f(x) ≡ 1 and 2 ≤ N ≤ 7

λ

u(0)

λ* λ*2

Figure 3.1: Top figure: plots of u(0) versus λ for the case where f(x) ≡ 1 is defined in theunit ball B1(0) ⊂ RN with different ranges of dimension N , where we have λ∗ = (6N − 8)/9for dimension N ≥ 8. Bottom figure: plots of u(0) versus λ for the case where f(x) ≡ 1 isdefined in the unit ball B1(0) ⊂ RN with dimension 2 ≤ N ≤ 7, where λ∗ (resp. λ∗2) is the first(resp. second) turning point.

One can recognize from Chapter 2 a clear distinction –in techniques and in the availableresults– between the case where the permittivity profile f is bounded away from zero, and whereit is allowed to vanish somewhere. A test case for the latter situation –that has generated muchinterest among both mathematicians and engineers– is when we have a power-law permittivityprofile f(x) = |x|α (α ≥ 0) on a ball. Our first goal of this Chapter is the study of the effect ofpower-like permittivity profiles f(x) ' |x|α for the problem (S)λ on the unit ball B = B1(0).We extend Theorem 2.1.2 to higher dimensions:

Theorem 3.1.1. Assume N ≥ 8 and α > αN = 3N−14−4√

64+2√

6. Let f ∈ C(B) be such that:

f(x) = |x|αg(x) , g(x) ≥ C > 0 in B. (3.1.1)

48

Let (λn)n be such that λn → λ ∈ [0, λ∗] and un be a solution of (S)λn so that µ1,n := µ1,λn(un) ≥0. Then,

supn∈N‖ un ‖∞< 1.

In particular, the extremal solution u∗ = limλ↑λ∗

uλ is a solution of (S)λ∗ satisfying µ1,λ∗(u∗) = 0.

As to non-minimal solutions, it is also shown in Chapter 2 –following ideas of Crandall-Rabinowitz [20]– that, for 1 ≤ N ≤ 7, and for λ close enough to λ∗, there exists a uniquesecond branch Uλ of solutions for (S)λ, bifurcating from u∗, such that

µ1,λ(Uλ) < 0 while µ2,λ(Uλ) > 0. (3.1.2)

For N ≥ 8 and α > αN , the same remains true for problem (S)λ on the unit ball with f(x) asin (3.3.1) and Uλ is a radial function.

In the sequel, we try to provide a rigorous analysis for other features of the bifurcationdiagram, in particular the second branch of unstable solutions, as well as the second bifurcationpoint. But first, and for the sake of completeness, we shall give a variational characterizationfor the unstable solution Uλ in the following sense:

Theorem 3.1.2. Assume f is a non-negative function in C(Ω) where Ω is a bounded domainin RN . If 1 ≤ N ≤ 7, then there exists δ > 0 such that for any λ ∈ (λ∗ − δ, λ∗), the secondsolution Uλ is a Mountain Pass solution for some regularized energy functional Jε,λ on thespace H1

0 (Ω).Moreover, the same result is still true for N ≥ 8 provided Ω is a ball, and f(x) is as in (3.3.1)with α > αN .

We are now interested in continuing the second branch till the second bifurcation point, bymeans of the implicit function theorem. For that, we have the following compactness result:

Theorem 3.1.3. Assume 2 ≤ N ≤ 7. Let f ∈ C(Ω) be such that:

f(x) =( k∏

i=1

|x− pi|αi)g(x) , g(x) ≥ C > 0 in Ω, (3.1.3)

for some points pi ∈ Ω and exponents αi ≥ 0. Let (λn)n be a sequence such that λn → λ ∈ [0, λ∗]and let un be an associated solution such that

µ2,n := µ2,λn(un) ≥ 0. (3.1.4)

Then, supn∈N‖ un ‖∞< 1. Moreover, if in addition µ1,n := µ1,λn(un) < 0, then necessarily λ > 0.

Let us remark that the case α1 = · · · = αk = 0 in (3.1.3) corresponds to a function f(x)bounded away from zero.We also mention that Theorem 3.1.3 yields another proof –based ona blow-up argument– of the compactness result for minimal solutions established in Chapter2 by means of some energy estimates, though under the more stringent assumption (3.1.3) onf(x). We expect that the same result should be true for radial solutions on the unit ball forN ≥ 8, α > αN , and f ∈ C(Ω) as in (3.3.1).

49

As far as we know, there are no compactness results of this type in the case of regularnonlinearities, marking a substantial difference with the singular situation. Theorem 3.1.3 isbased on a blow up argument and the knowledge of linear instability for solutions of a limitproblem on RN , a result which is interesting in itself (see for example [16]) and which somehowexplains the special role of dimension 7 and α = αN for this problem.

Theorem 3.1.4. Assume that either 1 ≤ N ≤ 7 and α ≥ 0 or that N ≥ 8 and α > αN . LetU be a solution of

∆U =|y|αU2

in RN ,U(y) ≥ C > 0 in RN .

(3.1.5)

Then, U is linearly unstable in the following sense:

µ1(U) = inf∫

RN(|∇φ|2 − 2|y|α

U3φ2)dx; φ ∈ C∞0 (RN ) ,

∫

RNφ2 = 1

< 0 . (3.1.6)

Moreover, if N ≥ 8 and 0 ≤ α ≤ αN , then there exists at least a solution U of (3.1.5) suchthat µ1(U) ≥ 0, given by the limit as λ → λ∗ of suitable rescaling of the minimal solution uλof (S)λ on the unit ball with f(x) = |x|α.

Theorem 3.1.4 is the main tool to control the blow up behavior of a possible non compactsequence of solutions. The usual asymptotic analysis for equations with Sobolev critical non-linearity, based on some energy bounds (usually L

2NN−2 (Ω)-bounds), does not work in our con-

text. In view of Chapter 2, a possible loss of compactness can be related to the L3N2 (Ω)-norm

along the sequence. Essentially, the blow up associated to a sequence un (in the sense of theblowing up of (1−un)−1) corresponds exactly to the blow up of the L

3N2 (Ω)-norm. We replace

these energy bounds by some spectral information and, based on Theorem 3.1.4, we providean estimate of the number of blow up points (counted with their “multiplicities”) in terms ofthe Morse index along the sequence.

We now define the second bifurcation point in the following way for (S)λ:

λ∗2 = infβ > 0 : ∃ a curveVλ ∈ C([β, λ∗];C2(Ω)) of solutions for (S)λs.t. µ2,λ(Vλ) ≥ 0, Vλ ≡ Uλ ∀λ ∈ (λ∗ − δ, λ∗).

We then have the following multiplicity result.

Theorem 3.1.5. Assume f ∈ C(Ω) to be of the form (3.1.3). Then, for 2 ≤ N ≤ 7 we havethat λ∗2 ∈ (0, λ∗) and for any λ ∈ (λ∗2, λ

∗) there exist at least two solutions uλ and Vλ for (S)λ,so that

µ1,λ(Vλ) < 0 while µ2,λ(Vλ) ≥ 0.

In particular, for λ = λ∗2, there exists a second solution, namely V ∗ := limλ↓λ∗2

Vλ so that

µ1,λ∗2(V ∗) < 0 and µ2,λ∗2(V ∗) = 0.

One can compare Theorem 3.1.5 with the multiplicity result of [1] for nonlinearities of the formλuq + up (0 < q < 1 < p), where the authors show that for p subcritical, there exists a second

50

–Mountain Pass– solution for any λ ∈ [0, λ∗). On the other hand, when p is critical, the secondbranch blows up as λ→ 0 (see also [4] for a related problem). We note that in our situation,the second branch cannot approach the value λ = 0 as illustrated by the bifurcation diagramabove.

Now let Vλ, λ ∈ (β, λ∗) be one of the curves appearing in the definition of λ∗2. By (3.1.2),we have that LVλ,λ is invertible for λ ∈ (λ∗ − δ, λ∗) and, as long as it remains invertible, wecan use the implicit function theorem to find Vλ as the unique smooth extension of the curveUλ (in principle Uλ exists only for λ close to λ∗). We now define λ∗∗ in the following way

λ∗∗ = infβ > 0 : ∀λ ∈ (β, λ∗) ∃ Vλ solution of (S)λ so thatµ2,λ(Vλ) > 0, Vλ ≡ Uλ forλ ∈ (λ∗ − δ, λ∗).

Then, λ∗2 ≤ λ∗∗ and there exists a smooth curve Vλ for λ ∈ (λ∗∗, λ∗) so that Vλ is the uniquemaximal extension of the curve Uλ. This is what the second branch is supposed to be. If nowλ∗2 < λ∗∗, then for λ ∈ (λ∗2, λ

∗∗) there is no longer uniqueness for the extension and the “secondbranch” is defined only as one of potentially many continuous extensions of Uλ.

It remains open the problem whether λ∗2 is the second turning point for the solution diagramof (S)λ or if the “second branch” simply disappears at λ = λ∗2. Note that if the “second branch”does not disappear, then it can continue for λ less than λ∗2 but only along solutions whose firsttwo eigenvalues are negative.

In dimension 1, we have a stronger but somewhat different compactness result. Recall thatµk,λn(un) is the k−th eigenvalue of Lun,λn counted with their multiplicity.

Theorem 3.1.6. Let I be a bounded interval in R and f ∈ C1(I) be such that f ≥ C > 0 inI. Let (un)n be a solution sequence for (S)λn on I, where λn → λ ∈ [0, λ∗]. Assume that forany n ∈ N and k large enough, we have:

µk,n := µk,λn(un) ≥ 0. (3.1.7)

If λ > 0, then again supn∈N‖ un ‖∞< 1 and compactness holds.

Even in dimension 1, we can still define λ∗2 but we don’t know when λ∗2 = 0 (this is indeedthe case when f(x) = 1, see [56]) or when λ∗2 > 0. In the latter situation, there would exist asolution V ∗ for (S)λ∗2 which could be –in some cases– the second turning point. Let us remarkthat the multiplicity result of Theorem 3.1.5 holds also in dimension 1 for any λ ∈ (λ∗2, λ

∗).This Chapter is organized as follows. In §3.2 we provide the Mountain Pass variational

characterization of Uλ for λ close to λ∗ as stated in Theorem 3.1.2. The compactness result ofTheorem 3.1.1 on the unit ball is proved in §3.3. §3.4 is concerned with the compactness of thesecond branch of (S)λ as stated in Theorem 3.1.3. In §3.5 we give the proof of the multiplicityresult in Theorem 3.1.5. §3.6 deals with the dimension 1 of Theorem 3.1.6.

3.2 Mountain Pass solutions

This section is devoted to the variational characterization of the second solution Uλ of (S)λ forλ ↑ λ∗ and in dimension 1 ≤ N ≤ 7. Let us stress that the argument works also for problem(S)λ on the unit ball with f(x) in the form (3.3.1) provided N ≥ 8, α > αN .

51

Since the nonlinearity g(u) = 1(1−u)2 is singular at u = 1, we need to consider a regularized

C1 nonlinearity gε(u), 0 < ε < 1, of the following form:

gε(u) =

1(1− u)2

u ≤ 1− ε ,1ε2− 2(1− ε)

pε3+

2pε3(1− ε)p−1

up u ≥ 1− ε ,(3.2.1)

where p > 1 if N = 1, 2 and 1 < p < N+2N−2 if 3 ≤ N ≤ 7. For λ ∈ (0, λ∗), we study the

regularized semilinear elliptic problem: −∆u = λf(x)gε(u) in Ω,

u = 0 on ∂Ω.(3.2.2)

From a variational viewpoint, the action functional associated to (3.2.2) is

Jε,λ(u) =12

∫

Ω|∇u|2dx− λ

∫

Ωf(x)Gε(u)dx , u ∈ H1

0 (Ω) , (3.2.3)

where Gε(u) =∫ u

−∞gε(s)ds.

In view of Theorem 2.4.5, we now fix 0 < ε < 1−‖u∗‖∞2 . For λ ↑ λ∗, the minimal solution

uλ of (S)λ is still a solution of (3.2.2) so that µ1

( − ∆ − λf(x)g′ε(uλ))> 0. The proof of

existence of second solutions for (3.2.2) relies on the standard Mountain Pass Theorem [3]. Forselfcontainedness, we include this theorem as follows:Mountain Pass Theorem: Suppose C1 functional Jε,λ(u) defines on a Banach space Esatisfying (P.S.) conditions and

1. there exists a neighborhood U of uλ in E and a constant σ > 0 such that Jε,λ(v1) ≥Jε,λ(uλ) + σ for all v1 ∈ ∂U ;

2. ∃v2 6∈ U such that Jε,λ(v2) ≤ Jε,λ(uλ).

DefineΓ =

γ ∈ C([0, 1],H) : γ(0) = uλ , γ(1) = v2

,

thencε,λ = inf

γ∈Γmax0≤t≤1

Jε,λ(γ(t)) : t ∈ (0, 1)

is a critical value of Jε,λ.We next briefly sketch the proof of Theorem 3.1.2 as follows. First, we prove that uλ is a

local minimum for Jε,λ(u) for λ ↑ λ∗. Then, by Mountain Pass Theorem, we show the existenceof a second solution Uε,λ for (3.2.2). Using subcritical growth:

0 ≤ gε(u) ≤ Cε(1 + |u|p) (3.2.4)

and applying the inequality:

θGε(u) ≤ ugε(u) for u ≥Mε, (3.2.5)

52


for some Cε, Mε > 0 large and θ = p+32 > 2, we obtain that Jε,λ satisfies the Palais-Smale

condition and, by means of a bootstrap argument, we get the uniform convergence of Uε,λ.On the other hand, a similar proof as in §2.4.1 shows that the convexity of gε(u) ensures thatproblem (3.2.2) has the unique solution u∗ at λ = λ∗, which then allows us to deduce thatUε,λ → u∗ in C(Ω) as λ ↑ λ∗, and it implies Uε,λ ≤ 1− ε. Therefore, Uε,λ is a second solutionfor (S)λ bifurcating from u∗. But since Uε,λ is a MP solution and since (S)λ has exactly twosolutions (cf. Lemma 2.5.6) uλ, Uλ for λ ↑ λ∗, it finally yields that Uε,λ = Uλ.

In order to complete the details for the proof of Theorem 3.1.2, we first need to show thefollowing:

Lemma 3.2.1. For λ ↑ λ∗, the minimal solution uλ of (S)λ is a local minimum of Jε,λ onH1

0 (Ω).

Proof: First, we show that uλ is a local minimum of Jε,λ in C1(Ω). Indeed, since

µ1,λ := µ1

(−∆− λf(x)g′ε(uλ))> 0,

we have the following inequality:∫

Ω|∇φ|2dx− 2λ

∫

Ω

f(x)(1− uλ)3

φ2dx ≥ µ1,λ

∫

Ωφ2 (3.2.6)

for any φ ∈ H10 (Ω), since uλ ≤ 1− ε. Now, take any φ ∈ H1

0 (Ω)∩C1(Ω) such that ‖φ‖C1 ≤ δλ.Since uλ ≤ 1− 3

2ε, if δλ ≤ ε2 , then uλ + φ ≤ 1− ε and we have that:

Jε,λ(uλ + φ)− Jε,λ(uλ)

=12

∫

Ω|∇φ|2dx+

∫

Ω∇uλ · ∇φdx− λ

∫

Ωf(x)

( 11− uλ − φ −

11− uλ

)

≥ µ1,λ

2

∫

Ωφ2 − λ

∫

Ωf(x)

( 11− uλ − φ −

11− uλ −

φ

(1− uλ)2− φ2

(1− uλ)3

),

(3.2.7)

where we have applied (3.2.6). Since now∣∣∣ 11− uλ − φ −

11− uλ −

φ

(1− uλ)2− φ2

(1− uλ)3

∣∣∣ ≤ C|φ|3

for some C > 0, (3.2.7) gives that

Jε,λ(uλ + φ)− Jε,λ(uλ) ≥ (µ1,λ

2− Cλ ‖ f ‖∞ δλ

) ∫

Ωφ2 > 0

provided δλ is small enough. This proves that uλ is a local minimum of Jε,λ in the C1 topology.Since (3.2.4) is satisfied, we can then directly apply Theorem 1 in [12] to get that uλ is a localminimum of Jε,λ in H1

0 (Ω). Since now f 6≡ 0, fix some small ball B2r ⊂ Ω of radius 2r, r > 0, so that

∫Brf(x)dx > 0.

Take a cut-off function χ so that χ = 1 on Br and χ = 0 outside B2r. Let wε = (1 − ε)χ ∈H1

0 (Ω). We have that:

Jε,λ(wε) ≤ (1− ε)2

2

∫

Ω|∇χ|2dx− λ

ε2

∫

Br

f(x)→ −∞

53

as ε→ 0, and uniformly for λ far away from zero. Since

Jε,λ(uλ) =12

∫

Ω|∇uλ|2dx− λ

∫

Ω

f(x)1− uλdx→

12

∫

Ω|∇u∗|2dx− λ∗

∫

Ω

f(x)1− u∗dx

as λ→ λ∗, we can find that for ε > 0 small, the inequality

Jε,λ(wε) < Jε,λ(uλ) (3.2.8)

holds for any λ close to λ∗.Fix now ε > 0 small enough so that (3.2.8) holds for λ close to λ∗, and define

cε,λ = infγ∈Γ

maxu∈γ Jε,λ(u),

where Γ = γ : [0, 1] → H10 (Ω); γ continuous and γ(0) = uλ, γ(1) = wε. We can then use

the Mountain Pass Theorem to get a solution Uε,λ of (3.2.2) for λ close to λ∗, provided thePalais-Smale condition holds at level c. We next prove this (PS)-condition in the followingform:

Lemma 3.2.2. Assume that wn ⊂ H10 (Ω) satisfies

Jε,λn(wn) ≤ C , J ′ε,λn(wn)→ 0 in H−1 (3.2.9)

for λn → λ > 0. Then the sequence (wn)n is uniformly bounded in H10 (Ω) and therefore admits

a convergent subsequence in H10 (Ω).

Proof: By (3.2.9) we have that:∫

Ω|∇wn|2dx− λn

∫

Ωf(x)gε(wn)wndx = o(‖ wn ‖H1

0)

as n→ +∞ and then,

C ≥ 12

∫

Ω|∇wn|2dx− λn

∫

Ωf(x)Gε(wn)dx

=(1

2− 1θ

) ∫

Ω|∇wn|2dx+ λn

∫

Ωf(x)

(1θwngε(wn)−Gε(wn)

)dx+ o(‖ wn ‖H1

0)

≥ (12− 1θ

) ∫

Ω|∇wn|2dx+ o(‖ wn ‖H1

0)− Cε

+λn∫

wn≥Mεf(x)

(1θwngε(wn)−Gε(wn)

)dx

≥ (12− 1θ

) ∫

Ω|∇wn|2dx+ o(‖ wn ‖H1

0)− Cε

in view of (3.2.5). Hence, supn∈N‖ wn ‖H1

0< +∞.

Since p is subcritical, the compactness of the embedding H10 (Ω) → Lp+1(Ω) provides that,

up to a subsequence, wn → w weakly in H10 (Ω) and strongly in Lp+1(Ω), for some w ∈ H1

0 (Ω).By (3.2.9) we get that

∫Ω |∇w|2 = λ

∫Ω f(x)gε(w)w, and then, by (3.2.4), we deduce that

∫

Ω|∇(wn − w)|2 =

∫

Ω|∇wn|2 −

∫

Ω|∇w|2 + o(1)

= λn

∫

Ωf(x)gε(wn)wn − λ

∫

Ωf(x)gε(w)w + o(1)→ 0

54

as n→ +∞. Completion of Theorem 3.1.2: Consider for any λ ∈ (λ∗−δ, λ∗) the Mountain Pass solutionUε,λ of (3.2.2) at energy level cε,λ, where δ > 0 is small enough. Since cε,λ ≤ cε,λ∗−δ for anyλ ∈ (λ∗ − δ, λ∗), and applying again Lemma 3.2.2, we get that ‖ Uε,λ ‖H1

0≤ C, for any λ close

to λ∗. Then, by (3.2.4) and elliptic regularity theory, we get that Uε,λ is uniformly boundedin C2,α(Ω) for λ ↑ λ∗, for α ∈ (0, 1). Hence, we can extract a subsequence Uε,λn , λn ↑ λ∗,converging in C2(Ω) to some function U∗, where U∗ is a solution for problem (3.2.2) at λ = λ∗.Also u∗ is a solution for (3.2.2) at λ = λ∗ so that µ1

(−∆−λ∗f(x)g′ε(u∗))

= 0. By convexity ofgε(u), similar to §2.4.1 it is classical to show that u∗ is the unique solution of this equation andtherefore U∗ = u∗. Since along any convergent sequence of Uε,λ as λ ↑ λ∗ the limit is alwaysu∗, we get that limλ↑λ∗ Uε,λ = u∗ in C2(Ω). Therefore, since u∗ ≤ 1 − 2ε, there exists δ > 0so that for any λ ∈ (λ∗ − δ, λ∗) Uε,λ ≤ u∗ + ε ≤ 1 − ε and hence, Uε,λ is a solution of (S)λ.Since the Mountain Pass energy level cε,λ satisfies cε,λ > Jε,λ(uλ), we have that Uε,λ 6= uλ andthen Uε,λ = Uλ for any λ ∈ (λ∗ − δ, λ∗). Note that by [20], we know that uλ, Uλ are the onlysolutions of (S)λ as λ ↑ λ∗.

Applying the compactness of Theorem 3.3.1 proved in next section, one can note that theargument of Theorem 3.1.2 works also for problem (S)λ on the unit ball with f(x) in the form(3.3.1) provided N ≥ 8, α > αN . This leads to the following proposition for higher dimensionalcase.

Proposition 3.2.3. Theorem 3.1.2 is still true for N ≥ 8 provided Ω is a ball, and f(x) is asin (3.3.1) with α > αN .

3.3 Minimal branch for power-law profiles

Our goal of this section is to study the effect of power-like permittivity profiles f(x) ' |x|α onthe problem (S)λ defined in the unit ball B = B1(0). The following main results of this sectionextend Theorem 2.1.2 to higher dimensions N ≥ 8, which give Theorem 3.1.1 concerning thecompactness of minimal branch.

Theorem 3.3.1. Assume N ≥ 8 and α > αN = 3N−14−4√

64+2√

6. Let f ∈ C(B) be such that:

f(x) = |x|αg(x) , g(x) ≥ C > 0 in B. (3.3.1)

Let (λn)n be such that λn → λ ∈ [0, λ∗] and un be a solution of (S)λn so that µ1,n := µ1,λn(un) ≥0. Then,

supn∈N‖ un ‖∞< 1.

In particular, the extremal solution u∗ = limλ↑λ∗

uλ is a solution of (S)λ∗ such that µ1,λ∗(u∗) = 0.

Proof: Let B be the unit ball, and let (λn)n be such that λn → λ ∈ [0, λ∗] and un be asolution of (S)λn on B so that

µ1,n := µ1,λn(un) ≥ 0 . (3.3.2)

By Proposition 2.5.2 un coincides with the minimal solution uλn and, by some symmetrizationarguments, it is shown in Proposition 2.6.1 that the minimal solution un is radial and achievesits absolute maximum only at zero.

55

Given a permittivity profile f(x) as in (3.3.1), in order to get Theorem 3.3.1, we want toshow:

supn∈N‖ un ‖∞< 1, (3.3.3)

provided N ≥ 8 and α > αN = 3N−14−4√

64+2√

6. In particular, since uλ is non decreasing in λ and

supλ∈[0,λ∗)

‖ uλ ‖∞< 1,

the extremal solution u∗ = limλ↑λ∗

uλ would be a solution of (S)λ∗ so that µ1,λ∗(u∗) ≥ 0. Prop-

erty µ1,λ∗(u∗) = 0 must hold because otherwise, by implicit function theorem, we could findsolutions of (S)λ for λ > λ∗, which contradicts the definition of λ∗.

In order to prove (3.3.3), let us argue by contradiction. Up to a subsequence, assume thatun(0) = max

Bun → 1 as n → +∞. Since λ = 0 implies un → 0 in C2(B), we can assume

that λn → λ > 0. Let εn := 1 − un(0) → 0 as n → +∞ and introduce the following rescaledfunction:

Un(y) =1− un

(ε

32+αn λ

− 12+α

n y)

εn, y ∈ Bn := B

ε− 3

2+αn λ

12+αn

(0). (3.3.4)

The function Un satisfies:

∆Un =|y|αg(ε

32+αn λ

− 12+α

n y)

U2n

in Bn,

Un(y) ≥ Un(0) = 1,(3.3.5)

and Bn → RN as n → +∞. This would reduce to a contradiction between (3.3.2) and thefollowing Proposition 3.3.2.

Proposition 3.3.2. There exists a subsequence Unn defined in (3.3.4) such that Un → Uin C1

loc(RN ), where U is a solution of the problem

∆U = g(0)|y|αU2

in RN ,

U(y) ≥ U(0) = 1 in RN .(3.3.6)

Moreover, there exists φn ∈ C∞0 (B) such that:∫

B

(|∇φn|2 − 2λn|x|αg(x)(1− un)3

φ2n

)< 0.

The rest of this section is devoted to the proof of Proposition 3.3.2. First, we establish thefollowing theorem, which characterizes the instability for solutions of a limit problem (3.3.7)on RN .

Theorem 3.3.3. Assume either 1 ≤ N ≤ 7 or N ≥ 8 and α > αN . Let U be a solution of theproblem

∆U =|y|αU2

in RN ,

U(y) ≥ C > 0 in RN .(3.3.7)

56

Then,

µ1(U) = inf∫

RN(|∇φ|2 − 2|y|α

U3φ2); φ ∈ C∞0 (RN ) and

∫

RNφ2 = 1

< 0 . (3.3.8)

Moreover, if N ≥ 8 and 0 ≤ α ≤ αN , then there exists at least one solution U of (3.3.7)satisfying µ1(U) ≥ 0.

Proof: By contradiction, assume that

µ1(U) = inf∫

RN(|∇φ|2 − 2|y|α

U3φ2)

; φ ∈ C∞0 (RN ) and∫

RNφ2 dx = 1

≥ 0.

By the density of C∞0 (RN ) in D1,2(RN ), we have∫|∇φ|2 ≥ 2

∫ |y|αU3

φ2 , ∀ φ ∈ D1,2(RN ) . (3.3.9)

In particular, the test function φ = 1

(1+|y|2)N−2

4 + δ2

∈ D1,2(RN ) applied in (3.3.9) gives that

∫ |y|α(1 + |y|2)

N−22

+δU3≤ C

∫1

(1 + |y|2)N2

+δ< +∞

for any δ > 0. Therefore, we have

∫1

(1 + |y|2)N−2−α

2+δU3

=∫

B1

(1 + |y|2)α2

(1 + |y|2)N−2

2+δU3

+∫

Bc1

(1 + |y|2)α2

(1 + |y|2)N−2

2+δU3

≤ C∫

B1

1U3

+ C

∫

Bc1

|y|α(1 + |y|2)

N−22

+δU3

≤ C + C

∫ |y|α(1 + |y|2)

N−22

+δU3,

(3.3.10)

which gives ∫1

(1 + |y|2)N−2−α

2+δU3

≤ C + C

∫1

(1 + |y|2)N2

+δ< +∞ . (3.3.11)

Step 1. We want to show that (3.3.9) allows us to perform the following Moser-type iterationscheme: for any 0 < q < 4 + 2

√6 and β there holds

∫1

(1 + |y|2)β−1−α2 U q+3

≤ Cq(1 +

∫1

(1 + |y|2)βU q)

(3.3.12)

(provided the second integral is finite).Indeed, let R > 0 and consider a smooth radial cut-off function η so that: 0 ≤ η ≤ 1,

η = 1 in BR(0), η = 0 in RN \ B2R(0). Multiplying (3.3.7) byη2

(1 + |y|2)β−1U q+1, q > 0, and

57

integrating by parts we get:∫ |y|αη2

(1 + |y|2)β−1U q+3

=4(q + 1)q2

∫ ∣∣∣∇( η

(1 + |y|2)β−1

2 Uq2

)∣∣∣2− 4(q + 1)

q2

∫1U q

∣∣∣∇( η

(1 + |y|2)β−1

2

)∣∣∣2

−q + 2q2

∫∇(

1U q

)∇( η2

(1 + |y|2)β−1

)

=4(q + 1)q2

∫ ∣∣∣∇( η

(1 + |y|2)β−1

2 Uq2

)∣∣∣2− 2q

∫1U q

∣∣∣∇( η

(1 + |y|2)β−1

2

)∣∣∣2

+2(q + 2)q2

∫1U q

η

(1 + |y|2)β−1

2

∆( η

(1 + |y|2)β−1

2

),

where the relation ∆(ψ)2 = 2|∇ψ|2 + 2ψ∆ψ is used in the second equality. Then, by (3.3.9)we deduce that

(8q + 8− q2)∫ |y|αη2

(1 + |y|2)β−1U q+3

≤ C ′q

∫1U q

(∣∣∇( η

(1 + |y|2)β−1

2

)∣∣2 +η

(1 + |y|2)β−1

2

∣∣∆( η

(1 + |y|2)β−1

2

)∣∣).

Assuming |∇η| ≤ CR and |∆η| ≤ C

R2 , it is straightforward to see that:

∣∣∇( η

(1 + |y|2)β−1

2

)∣∣2 +η

(1 + |y|2)β−1

2

∣∣∆( η

(1 + |y|2)β−1

2

)∣∣

≤ C[ 1

(1 + |y|2)β+

1R2(1 + |y|2)β−1

χB2R(0)\BR(0)

]

for some constant C independent of R. Then,

(8q + 8− q2)∫ |y|αη2

(1 + |y|2)β−1U q+3≤ C ′′q

∫1

(1 + |y|2)βU q.

Let q+ = 4 + 2√

6. For any 0 < q < q+, we have 8q + 8− q2 > 0 and therefore:∫ |y|αη2

(1 + |y|2)β−1U q+3≤ Cq

∫1

(1 + |y|2)βU q,

where Cq does not depend on R > 0. Taking the limit as R→ +∞, we get that:∫ |y|α

(1 + |y|2)β−1U q+3≤ Cq

∫1

(1 + |y|2)βU q,

and then, the validity of (3.3.12) easily follows from the same argument of (3.3.10).

Step 2. Let either 1 ≤ N ≤ 7 or N ≥ 8 and α > αN . We want to show that∫

1(1 + |y|2)U q

< +∞ (3.3.13)

58

for some 0 < q < q+ = 4 + 2√

6.Indeed, set β0 = N−2−α

2 + δ, δ > 0, and q0 = 3. By (3.3.11) we get that∫

1(1 + |y|2)β0U q0

< +∞.

Let βi = β0 − i(1 + α2 ) and qi = q0 + 3i, i ∈ N. Since q0 < q1 < q+ = 4 + 2

√6 < q2, we can

iterate (3.3.12) exactly two times to get that:∫

1(1 + |y|2)β2U q2

< +∞ (3.3.14)

where β2 = N−6−3α2 + δ, q2 = 9.

Let 0 < q < q+ = 4 + 2√

6 < 9. By (3.3.14) and Holder inequality we get that:∫

1(1 + |y|2)U q

=∫

(1 + |y|2)q9

( 6−N2−δ+ 3

2α)

U q· 1

(1 + |y|2)q9

( 6−N2−δ+ 3

2α)+1

≤(∫ 1

(1 + |y|2)β2U q2

) q9(∫ 1

(1 + |y|2)q

9−q ( 6−N2−δ+ 3

2α)+ 9

9−q

) 9−q9< +∞

provided − 2q9−qβ2 + 18

9−q > N or equivalently

q >9N − 18

6− 2δ + 3α. (3.3.15)

To assure (3.3.15) for some δ > 0 small and q < q+ at the same time, it requires 3N−62+α < q+

or equivalently

1 ≤ N ≤ 7 or N ≥ 8 , α > αN =3N − 14− 4

√6

4 + 2√

6.

Our assumptions then provide the existence of some 0 < q < q+ = 4 + 2√

6 such that (3.3.13)holds.

Step 3. We are ready to obtain a contradiction. Let 0 < q < 4 + 2√

6 be such that (3.3.13)holds, and suppose η is the cut-off function of Step 1. Using equation (3.3.7) we compute:

∫ ∣∣∇( ηU

q2

)∣∣2 −∫

2|y|αU3

( ηU

q2

)

=q2

4

∫η2|∇U |2U q+2

+∫ |∇η|2

U q+

12

∫∇(η2)∇( 1

U q)−

∫2|y|αη2

U q+3

= − q2

4(q + 1)

∫∇U · ∇( η2

U q+1

)+∫ |∇η|2

U q

+ q+24(q+1)

∫∇(η2)∇( 1

U q)−

∫2|y|αη2

U q+3

= −8q + 8− q2

4(q + 1)

∫ |y|αη2

U q+3+∫ |∇η|2

U q− q + 2

4(q + 1)

∫∆η2

U q.

59

Since 0 < q < 4 + 2√

6, we have 8q + 8− q2 > 0 and∫ ∣∣∇( η

Uq2

)∣∣2 −∫

2|y|αU3

( ηU

q2

)2

≤ −8q + 8− q2

4(q + 1)

∫

B1(0)

|y|αη2

U q+3+O

( 1R2

∫

B2R(0)\BR(0)

1U q

)

≤ −8q + 8− q2

4(q + 1)

∫

B1(0)

|y|αη2

U q+3+O

(∫

|y|≥R

1(1 + |y|2)U q

).

Since (3.3.13) implies: limR→+∞∫

|y|≥R

1(1 + |y|2)U q

= 0, we get that for R large

∫ ∣∣∇( η

U q/2)∣∣2 −

∫2|y|αU3

( η

U q/2)2

≤ −8q + 8− q2

4(q + 1)

∫

B1(0)

|y|αU q+3

+O(∫

|y|≥R

1(1 + |y|2)U q

)< 0.

A contradiction to (3.3.9). Hence, (3.3.8) holds and the proof of the first part of Theorem 3.3.3is complete.

We now deal with the second part of Theorem 3.3.3. Consider the minimal solution un of(S)λn as λn → λ∗ on the unit ball with f(x) = |x|α. Using the density of C∞0 (Ω) in H1

0 (Ω),Theorem 2.5.1 gives

∫

B

(|∇φn|2 − 2λn|x|α(1− un)3

φ2n

)dx ≥ 0 ∀φn ∈ C∞0 (Ω) .

In view of the rescaled function (3.3.4), we now define

φn(x) =(ε

32+αn λ

− 12+α

n

)−N−22 φ

(ε− 3

2+αn λ

12+αn x

).

Then we have∫ (|∇φ|2 − 2|y|α

U3φ2)dy = limn→∞

∫ (|∇φ|2 − 2|y|α

U3n

φ2)dy

= limn→∞∫

B

(|∇φn|2 − 2λn|x|α

(1− un)3φ2n

)dx ≥ 0 ,

since φ has compact support and Un → U in C1loc(RN ). This completes the proof of Theorem

3.3.3. Proof of Proposition 3.3.2: Let R > 0. For n large, decompose Un = U1

n + U2n, where U2

n

satisfies: ∆U2

n = ∆Un in BR(0) ,U2n = 0 on ∂BR(0) .

By (3.3.5) we get that on BR(0):

0 ≤ ∆Un ≤ Rα ‖ g ‖∞,

60

and standard elliptic regularity theory gives that U2n is uniformly bounded in C1,β(BR(0)) for

some β ∈ (0, 1). Up to a subsequence, we get that U2n → U2 in C1(BR(0)). Since U1

n = Un ≥ 1on ∂BR(0), by harmonicity U1

n ≥ 1 in BR(0) and then the Harnack inequality gives

supBR/2(0)

U1n ≤ CR inf

BR/2(0)U1n ≤ CRU1

n(0) = CR(1− U2

n(0)) ≤ CR

(1 + sup

n∈N|U2n(0)|) <∞ .

Hence, U1n is uniformly bounded in C1,β(BR/4(0)) for some β ∈ (0, 1). Up to a further subse-

quence, we get that U1n → U1 in C1(BR/4(0)) and then, Un → U1 +U2 in C1(BR/4(0)) for any

R > 0. By a diagonal process and up to a subsequence, we find that Un → U in C1loc(RN ),

where U is a solution of the equation (3.3.6).If either 1 ≤ N ≤ 7 or N ≥ 8 and α > αN , since g(0) > 0 Theorem 3.3.3 shows that

µ1(U) < 0 and then, we find φ ∈ C∞0 (RN ) so that:∫ (|∇φ|2 − 2g(0)

|y|αU3

φ2)< 0.

Defining now

φn(x) =(ε

32+αn λ

− 12+α

n

)−N−22 φ

(ε− 3

2+αn λ

12+αn x

),

then we have∫

B

(|∇φn|2 − 2λn|x|αg(x)(1− un)3

φ2n

)=∫ (|∇φ|2 − 2|y|α

U3n

g(ε3

2+αn λ

− 12+α

n y)φ2)

→∫ (|∇φ|2 − 2g(0)

|y|αU3

φ2)< 0

as n→ +∞, since φ has compact support and Un → U in C1loc(RN ). The proof of Proposition

3.3.2 is now complete.

3.4 Compactness along the second branch

In this section, we are interested in continuing the second branch till the second bifurcationpoint, by means of the implicit function theorem. Our main result of this section is thecompactness of Theorem 3.1.3 for 2 ≤ N ≤ 7.

In order to prove Theorem 3.1.3, we now assume that f ∈ C(Ω) is in the form (3.1.3), andlet (un)n be a solution sequence for (S)λn where λn → λ ∈ [0, λ∗].

3.4.1 Blow-up analysis

Assume that the sequence (un)n is not compact, which means that up to passing to a subse-quence, we may assume that max

Ωun → 1 as n→∞. Let xn be a maximum point of un in Ω

(i.e., un(xn) = maxΩ

un) and set εn = 1− un(xn). Let us assume that xn → p as n→ +∞. We

have three different situations depending on the location of p and the rate of |xn − p|:1) blow up outside the zero set p1, . . . , pk of f(x), i.e. p 6∈ p1, . . . , pk;2) “slow” blow up at some pi in the zero set of f(x), i.e. xn → pi and ε−3

n λn|xn− pi|α+2 →+∞ as n→ +∞;

61

3) “fast” blow at some pi in the zero set of f(x), i.e. xn → pi and limn→+∞ sup(ε−3n λn|xn−

pi|α+2)< +∞.

Accordingly, for 2 ≤ N ≤ 7 we next discuss each one of these situations.1st Case Assume that p /∈ p1, . . . , pk. In general, we are not able to prove that a blow

up point p is always far away from ∂Ω, even though we suspect it to be true. However,the following weaker estimate is available and –as explained later– will be sufficient for ourpurposes.

Lemma 3.4.1. Let hn be a function on a smooth bounded domain An in RN . Let Wn be asolution of the problem

∆Wn =hn(x)W 2n

in An,

Wn(y) ≥ C > 0 in An,Wn(0) = 1,

(3.4.1)

for some C > 0. Assume that supn∈N

‖ hn ‖∞< +∞ and An → Tµ as n → +∞ for some

µ ∈ (0,+∞), where Tµ is a hyperspace so that 0 ∈ Tµ and dist (0, ∂Tµ) = µ. Then forsufficiently large n, either

inf∂An∩B2µ(0)

Wn ≤ C (3.4.2)

orinf

∂An∩B2µ(0)∂νWn ≤ 0, (3.4.3)

where ν is the unit outward normal of An.

Proof: Assume that ∂νWn > 0 on ∂An ∩B2µ(0). Let

G(x) =

− 12π

log|x|2µ

if N = 2

cN( 1|x|N−2

− 1(2µ)N−2

)if N ≥ 3

be the Green function of the operator −∆ in B2µ(0) with homogeneous Dirichlet boundarycondition, where cN = 1

(N−2)|∂B1(0)| and | · | stands for the Lebesgue measure.Here and in the sequel, when there is no ambiguity on the domain, ν and dS will denote

the unit outward normal and the boundary integration element of the corresponding domain.By the representation formula we have that:

Wn(0) = −∫

An∩B2µ(0)∆Wn(x)G(x)dx−

∫

∂An∩B2µ(0)Wn(x)∂νG(x)dS

+∫

∂An∩B2µ(0)∂νWn(x)G(x)dS −

∫

∂B2µ(0)∩AnWn(x)∂νG(x)dS.

(3.4.4)

Note that on ∂Tµ we have

− ∂νG(x) =

1

2πx|x|2 · ν > 0 if N = 2 ;

(N − 2)cN x|x|N · ν > 0 if N ≥ 3 . (3.4.5)

62


Since ∂An → ∂Tµ as n→∞, it yields that for sufficiently large n,

∂νG(x) ≤ 0 on ∂An ∩B2µ(0). (3.4.6)

Hence, by (3.4.4), (3.4.6) and the assumptions on Wn, we then get for sufficiently large n,

1 ≥ −∫

An∩B2µ(0)

hn(x)W 2n(x)

G(x)dx− ( inf∂An∩B2µ(0)

Wn

) ∫

∂An∩B2µ(0)∂νG(x)dS ,

since G(x) ≥ 0 in B2µ(0) and ∂νG(x) ≤ 0 on ∂B2µ(0). On the other hand, we have

∣∣∫

An∩B2µ(0)

hn(x)W 2n(x)

G(x)∣∣ ≤ C ,

and (3.4.5) also implies that for sufficiently large n,

−∫

∂An∩B2µ(0)∂νG(x)dσ(x)→ −

∫

∂Tµ∩B2µ(0)∂νG(x)dS > 0.

Then for sufficiently large n, 1 ≥ −C + C−1(

inf∂An∩B2µ(0)

Wn

)for some C > 0 large enough.

Therefore, we conclude that for sufficiently large n, inf∂An∩B2µ(0)

Wn is bounded and the proof is

complete. We are now ready to completely discuss this first case. Introduce the following rescaled

function:

Un(y) =1− un(ε

32nλ− 1

2n y + xn)

εn, y ∈ Ωn =

Ω− xnε

32nλ− 1

2n

, (3.4.7)

then Un satisfies

∆Un =f(ε

32nλ− 1

2n y + xn)U2n

in Ωn,

Un(0) = 1 in Ωn.

(3.4.8)

In addition, we have that Un ≥ Un(0) = 1 as long as xn is the maximum point of un in Ω.We would like to prove the following:

Proposition 3.4.2. Let xn ∈ Ω and εn := 1− un(xn), and assume

xn → p /∈ p1, . . . , pk , ε3nλ−1n → 0 as n→ +∞. (3.4.9)

Let Un and Ωn be defined as in (3.4.7) satisfying

Un ≥ C > 0 in Ωn ∩BRn(0) (3.4.10)

for some Rn → +∞ as n→ +∞. Then, there exists a subsequence of (Un)n such that Un → Uin C1

loc(RN ), where U is a solution of the problem

∆U =f(p)U2

in RN ,

U(y) ≥ C > 0 in RN .(3.4.11)

63

Moreover, there exists a function φn ∈ C∞0 (Ω) such that∫

Ω

(|∇φn|2 − 2λnf(x)(1− un)3

φ2n

)< 0 , (3.4.12)

and Supp φn ⊂ BMε

32n λ− 1

2n

(xn) for some M > 0.

Proof: We first claim that Lemma 3.4.1 provides us with a stronger estimate:

ε3nλ−1n (dist (xn, ∂Ω))−2 → 0 as n→ +∞. (3.4.13)

Indeed, by contradiction and up to a subsequence, assume that ε3nλ−1n d−2

n → δ > 0 as n→ +∞,where dn := dist (xn, ∂Ω). We get from (3.4.9) that dn → 0 as n → +∞. We introduce thefollowing rescaling Wn:

Wn(y) =1− un(dny + xn)

εn, y ∈ An =

Ω− xndn

.

Since dn → 0, we get that An → Tµ as n → +∞, where Tµ is a hyperspace containing 0 sothat dist (0, ∂Tµ) = µ. The function Wn solves problem (3.4.1) with hn(y) = λnd2

nε3n

f(dny+ xn)and C = Wn(0) = 1. We have that:

‖ hn ‖∞≤ λnd2n

ε3n

‖ f ‖∞≤ 2δ‖ f ‖∞

and Wn = 1εn→ +∞ on ∂An as n → ∞. By Lemma 3.4.1 we get that (3.4.3) must hold, a

contradiction to Hopf Lemma applied to un. This gives the validity of (3.4.13).(3.4.13) now implies that Ωn → RN as n → +∞. Arguing as in the proof of Proposition

3.3.2, we get that Un → U in C1loc(RN ), where U is a solution of (3.4.11) by means of (3.4.8)-

(3.4.10).If 2 ≤ N ≤ 7, since f(p) > 0 by Theorem 3.3.3 we get that µ1(U) < 0 and then, we find

φ ∈ C∞0 (RN ) so that: ∫ (|∇φ|2 − 2f(p)U3

φ2)< 0.

Defining now φn(x) =(ε

32nλ− 1

2n

)−N−22 φ

(ε− 3

2n λ

12n (x− xn)

), then we have

∫

Ω

(|∇φn|2 − 2λnf(x)(1− un)3

φ2n

)=∫ (|∇φ|2 − 2f(ε

32nλ− 1

2n y + xn)U3n

φ2)

→∫ (|∇φ|2 − 2f(p)

U3φ2)< 0

as n→ +∞, since φ has compact support and Un → U in C1loc(R). The proof of Proposition

3.4.2 is now complete. 2nd Case Assume that xn → pi and ε−3

n λn|xn − pi|αi+2 → +∞ as n → +∞. For conve-nience, in the following we denote

α := αi , fi(x) :=( k∏

j=1, j 6=i|x− pj |αj

)g(x) , (3.4.14)

64

and we rescale the function un in a different way:

Un(y) =1− un(ε

32nλ− 1

2n |xn − pi|−α2 y + xn)

εn, y ∈ Ωn =

Ω− xnε

32nλ− 1

2n |xn − pi|−α2

. (3.4.15)

In this situation, Un satisfies:

∆Un =∣∣ε

32nλ− 1

2n |xn − pi|−

α+22 y +

xn − pi|xn − pi|

∣∣α fi(ε32nλ− 1

2n |xn − pi|−α2 y + xn)

U2n

in Ωn,

Un(0) = 1 in Ωn,(3.4.16)

and we have the following


xn → pi , ε−3n λn|xn − pi|α+2 → +∞ as n→ +∞. (3.4.17)

Let Un and Ωn be defined as in (3.4.15) satisfying (3.4.10). Then, up to a subsequence, Un → Uin C1

loc(RN ), where U is a solution of the equation:

∆U =fi(pi)U2

in RN ,

U(y) ≥ C > 0 in RN .(3.4.18)

Moreover, there holds (3.4.12) for some φn ∈ C∞0 (Ω) such that

Supp φn ⊂ BMε

32n λ− 1

2n |xn−pi|−

α2

(xn) for some M > 0 .

Proof: Similar to Proposition 3.4.2 we get from (3.4.17) that Ωn → RN as n → +∞. Asbefore, Un → U in C1

loc(RN ) and U is a solution of (3.4.18) in view of (3.4.10) and (3.4.16)-(3.4.17). Since 2 ≤ N ≤ 7 and fi(pi) > 0, Theorem 3.3.3 implies µ1(U) < 0 and the existenceof some φ ∈ C∞0 (RN ) so that:

∫ (|∇φ|2 − 2fi(pi)U3

φ2)< 0.

Defining now φn(x) =(ε

32nλ− 1

2n |xn − pi|−α2

)−N−22 φ

(ε− 3

2n λ

12n |xn − pi|α2 (x− xn)

), then we have

∫

Ω

(|∇φn|2 − 2λnf(x)(1− un)3

φ2n

)

=∫ (|∇φ|2 − ∣∣ε

32nλ− 1

2n |xn − pi|−

α+22 y +


∣∣α 2fi(ε32nλ− 1

2n |xn − pi|−α2 y + xn)

U3n

φ2)

→∫ (|∇φ|2 − 2fi(pi)

U3φ2)< 0

as n→ +∞, which completes the proof of Proposition 3.4.3.

65


3rd Case Assume that xn → pi as n→ +∞ and ε−3n λn|xn − pi|αi+2 ≤ C. In the following

we denote α := αi, and we rescale the function un in a still different way:

Un(y) =1− un(ε

32+αn λ

− 12+α

n y + xn)εn

, y ∈ Ωn =Ω− xn

ε3

2+αn λ

− 12+α

n

. (3.4.19)

Then Un satisfies

∆Un =∣∣y + ε

− 32+α

n λ1

2+αn (xn − pi)

∣∣α fi(ε3

2+αn λ

− 12+α

n y + xn)U2n

in Ωn,

Un(0) = 1 in Ωn,

(3.4.20)

where fi is defined in (3.4.14).In this situation, the result we have is the following


ε3nλ−1n → 0 , xn → pi , ε

− 3α+2

n λ1

α+2n (xn − pi)→ y0 as n→ +∞. (3.4.21)

Let Un and Ωn be defined as in (3.4.19) satisfying either (3.4.10) or

Un ≥ C(ε− 3α+2

n λ1

α+2n |xn − pi|

)−α3∣∣y + ε

− 3α+2

n λ1

α+2n (xn − pi)

∣∣α3 in Ωn ∩BRn(0) (3.4.22)

for some Rn → +∞ as n→ +∞ and C > 0. Then, up to a subsequence, Un → U in C1loc(RN )

and U satisfies:

∆U = |y + y0|α fi(pi)U2

in RN ,U(y) ≥ C > 0 in RN .

(3.4.23)

Moreover, we have (3.4.12) for some function φn ∈ C∞0 (Ω) such that

Supp φn ⊂ BMε

32+αn λ

− 12+α

n

(xn) for some M > 0 .

Proof: By (3.4.21) we get again that Ωn → RN as n → +∞. If (3.4.10) holds, as before wehave Un → U in C1

loc(RN ) and, by (3.4.10) and (3.4.20)-(3.4.21), U solves (3.4.23).We need to discuss the non trivial case when we have the validity of (3.4.22). Arguing as in

the proof of Proposition 3.3.2, fix R > 2|y0| and decompose Un = U1n +U2

n, where U2n satisfies:

∆U2

n = ∆Un in BR(0) ,U2n = 0 on ∂BR(0) .

By (3.4.20) and (3.4.22) we get that on BR(0):

0 ≤ ∆Un =∣∣y + ε

− 32+α

n λ1

2+αn (xn − pi)

∣∣α fi(ε3

2+αn λ

− 12+α

n y + xn)U2n

≤ C(ε−3

α+2n λ

1α+2n |xn − pi|

) 2α3∣∣y + ε

− 32+α

n λ1

2+αn (xn − pi)

∣∣α3 .

66

Since ε− 3α+2

n λ1

α+2n (xn − pi) is bounded, we get that 0 ≤ ∆Un ≤ CR on BR(0) for n large, and

then, standard elliptic regularity theory gives that U2n is uniformly bounded in C1,β(BR(0)) for

some β ∈ (0, 1). Up to a subsequence, we get that U2n → U2 in C1(BR(0)). Since by (3.4.22)

U1n = Un ≥ C(R − 2|y0|)α3 > 0 on ∂BR(0), by harmonicity U1

n ≥ CR in BR(0) and hence theHarnack inequality gives

supBR/2(0) U1n ≤ CR infBR/2(0) U

1n ≤ CRU1

n(0) = CR(1− U2

n(0))

≤ CR(1 + supn∈N |U2

n(0)|) <∞ .

Therefore, U1n is uniformly bounded in C1,β(BR/4(0)), β ∈ (0, 1). Up to a further subsequence,

we get that U1n → U1 in C1(BR/4(0)) and then, Un → U1 +U2 in C1(BR/4(0)) for any R > 0.

By a diagonal process and up to a subsequence, by (3.4.22) we find that Un → U in C1loc(RN ),

where U ∈ C1(RN ) ∩ C2(RN \ −y0) is a solution of the equation

∆U = |y + y0|α fi(pi)U2

in RN \ −y0 ,U(y) ≥ C|y + y0|α3 in RN ,

for some C > 0. In order to prove that U is a solution of (3.4.23), we still need to provethat U(−y0) > 0. Let B be some ball so that −y0 ∈ ∂B and assume by contradiction thatU(−y0) = 0. Since

−∆U + c(y)U = 0 in B , U ∈ C2(B) ∩ C(B) , U(y) > U(−y0) in B,

and c(y) = fi(pi)|y+y0|αU3 ≥ 0 is a bounded function, Hopf Lemma shows that ∂νU(−y0) < 0,

where ν is the unit outward normal of B at −y0. Hence, U becomes negative in a neighborhoodof −y0, in contradiction with the positivity of U . Hence, U(−y0) > 0 and U satisfies (3.4.23).

Since 2 ≤ N ≤ 7 and fi(pi) > 0, Theorem 3.3.3 implies µ1(U) < 0 and the existence ofsome φ ∈ C∞0 (RN ) so that:

∫ (|∇φ|2 − |y + y0|α 2fi(pi)U3

φ2)< 0.

Let φn(x) =(ε

32+αn λ

− 12+α

n

)−N−22 φ

(ε− 3

2+αn λ

12+αn (x− xn)

), then it reduces to

∫

Ω

(|∇φn|2 − 2λnf(x)(1− un)3

φ2n

)

=∫ (|∇φ|2 − ∣∣y + ε

− 32+α

n λ1

2+αn (xn − pi)

∣∣α 2fi(ε3

2+αn λ

− 12+α

n y + xn)U3n

φ2)

→∫ (|∇φ|2 − |y + y0|α 2fi(pi)

U3φ2)< 0

as n→ +∞, and Proposition 3.4.4 is established.

67


3.4.2 Spectral confinement

Let us now assume the validity of (3.1.4), namely µ2,n := µ2,λn(un) ≥ 0 for any n ∈ N.This information will play a crucial role in controlling the number k of “blow up points”(for (1 − un)−1) in terms of the spectral information on un. Indeed, roughly speaking, wecan estimate k with the number of negative eigenvalues of Lun,λn (with multiplicities). Inparticular, assumption (3.1.4) implies that “blow up” can occur only along the sequence xn ofmaximum points of un in Ω.

Proposition 3.4.5. Assume 2 ≤ N ≤ 7, and suppose f ∈ C(Ω) is as in (3.1.3). Let λn →λ ∈ [0, λ∗] and un be an associated solution. Assume that un(xn) = max

Ωun → 1 as n→ +∞.

Then, there exist constants C > 0 and N0 ∈ N such that

(1− un(x)

) ≥ Cλ13nd(x)

α3 |x− xn|

23 , ∀ x ∈ Ω , n ≥ N0 , (3.4.24)

where d(x)α3 = min|x − pi|

αi3 : i = 1, . . . , k is the distance function from the zero set

p1, . . . , pk of f(x).

Proof: Let εn = 1− un(xn). Then, εn → 0 as n→ +∞ and, even more precisely:

ε2nλ−1n → 0 as n→ +∞. (3.4.25)

Indeed, otherwise we would have along some subsequence:

0 ≤ λnf(x)(1− un)2

≤ λnε2n

‖ f ‖∞≤ C , λn → 0 as n→ +∞.

But if the right hand side of (S)λn is uniformly bounded, then elliptic regularity theory impliesthat un is uniformly bounded in C1,β(Ω) for some β ∈ (0, 1). Hence, up to a further subse-quence, un → u in C1(Ω), where u is a harmonic function such that u = 0 on ∂Ω, and henceu ≡ 0 on Ω. On the other hand, εn → 0 implies that max

Ωu = 1, a contradiction.

By (3.4.25) we get that ε3nλ−1n → 0 as n → +∞, as needed in (3.4.9) and (3.4.21) for

Propositions 3.4.2 and 3.4.4, respectively. Now, depending on the case corresponding to theblow up sequence xn, we can apply one among Propositions 3.4.2 and 3.4.4 to obtaining theexistence of a function φn ∈ C∞0 (Ω) such that (3.4.12) holds, together with a specific controlon Supp φn.

By contradiction, assume now that (3.4.24) is false: up to a subsequence, then there existsa sequence yn ∈ Ω such that

λ− 1

3n d(yn)−

α3 |yn − xn|− 2

3

(1− un(yn)

)

= λ− 1

3n minx∈Ω

(d(x)−

α3 |x− xn|− 2

3

(1− un(x)

))→ 0 as n→ +∞ .

(3.4.26)

Then, µn := 1− un(yn)→ 0 as n→∞ and (3.4.26) can be rewritten as:

µ32nλ− 1

2n

|xn − yn| d(yn)α2

→ 0 as n→ +∞. (3.4.27)

68

We now want to explain the meaning of the crucial choice (3.4.26). Let βn be a sequence ofpositive numbers so that

Rn := β− 1

2n mind(yn)

12 , |xn − yn|

12 → +∞ as n→ +∞. (3.4.28)

Let us introduce the following rescaled function:

Un(y) =1− un(βny + yn)

µn, y ∈ Ωn =

Ω− ynβn

.

Formula (3.4.26) implies:

µn = d(yn)α3 |yn − xn| 23 minx∈Ω

(d(x)−

α3 |x− xn|− 2

3

(1− un(x)

))

≤ µnd(yn)α3 |yn − xn| 23d(βny + yn)−

α3 |βny + yn − xn|− 2

3 Un(y).

Sinced(βny + yn)

d(yn)= min∣∣yn − pi

d(yn)+

βnd(yn)

y∣∣ : i = 1, . . . , k ≥ 1− βn

d(yn)|y|

in view of |yn − pi| ≥ d(yn), by (3.4.28) we get that:

Un(y) ≥ (1− βnRnd(yn)

)α3(1− βnRn|xn − yn|

) 23 ≥ (1

2) 2+α

3

for any y ∈ Ωn ∩ BRn(0). Hence, whenever (3.4.28) holds, we get the validity of (3.4.10) forthe rescaled function Un at yn with respect to βn.

We need to discuss all the possible types of blow up at yn.1st Case Assume that yn → q /∈ p1, . . . , pk. By (3.4.27) we get that µ3

nλ−1n → 0 as

n → +∞. Since d(yn) ≥ C > 0, let βn = µ32nλ− 1

2n and, by (3.4.27) we get that (3.4.28) holds.

Associated to yn, µn, define Un, Ωn as in (3.4.7). We have from above that (3.4.10) holdsby the validity of (3.4.28) for our choice of βn. Hence, Proposition 3.4.2 applied to Un givesthe existence of ψn ∈ C∞0 (Ω) such that (3.4.12) holds and Supp ψn ⊂ B

Mµ32n λ− 1

2n

(yn) for some

M > 0.In the worst case xn → q, given Un be as in (3.4.7) associated to xn, εn, we get by scaling

that for x = ε32nλ− 1

2n y + xn,

λ− 1

3n

(d(x)−

α3 |x− xn|− 2

3

(1− un(x)

))

≥ Cλ−13

n

(|x− xn|− 23

(1− un(x)

))= C|y|− 2

3Un(y) ≥ CR > 0

uniformly in n and y ∈ BR(0) for any R > 0. Then,

ε32nλ− 1

2n

|xn − yn| → 0 as n→ +∞.

Hence, in this situation φn and ψn have disjoint compact supports and obviously, it remainstrue when xn → p 6= q. Hence, µ2,n < 0 in contradiction with (3.1.4).

69


3rd Case Assume that yn → pi in a “fast” way:

µ−3n λn|yn − pi|α+2 ≤ C.

Since d(yn) = |yn − pi|, by (3.4.27) we get that

|yn − pi||xn − yn| =

µ32nλ− 1

2n

|xn − yn| |yn − pi|α2(µ−3n λn|yn − pi|α+2

) 12 → 0 as n→ +∞, (3.4.30)

and then for n large

|xn − pi||yn − pi| ≥

|xn − yn||yn − pi| − 1 ≥ 1 ,

|xn − pi||xn − yn| ≥ 1− |yn − pi||xn − yn| ≥

12. (3.4.31)

Since εn ≤ µn, (3.4.27) and (3.4.31) give that

ε−3n λn|xn − pi|α+2

≥ ( µ32nλ− 1

2n

|xn − yn| |yn − pi|α2)−2( |xn − pi|

|xn − yn|2

α+2 |yn − pi|αα+2

)α+2

≥ C( µ

32nλ− 1

2n


)−2 → +∞ as n→ +∞.

(3.4.32)

The meaning of (3.4.32) is the following: once yn provides a fast blowing up sequence at pi,then no other fast blow up at pi can occur as (3.4.32) states for xn.

Let βn = µ3

2+αn λ

− 12+α

n . By (3.4.27) and (3.4.30) we get that

βn|xn − yn| = µ

32+αn λ

− 12+α

n |xn − yn|−1

=( µ

32nλ− 1

2n


) 22+α( |yn − pi||xn − yn|

) α2+α → 0 as n→ +∞.

(3.4.33)

However, since un blows up fast at pi along yn, we have β−1n d(yn) ≤ C and then, (3.4.28) does

not hold. Letting as before

Un(y) =1− un(βny + yn)

µn, y ∈ Ωn =

Ω− ynβn

,

we need to refine the analysis before in order to get some estimate for Un even when only(3.4.33) does hold. Formula (3.4.26) gives that:

Un(y) ≥ |yn − pi|−α3 |yn − xn|− 23 |βny + yn − pi|α3 |βny + yn − xn| 23

=(µ− 3

2+αn λ

12+αn |yn − pi|

)−α3∣∣ βn|xn − yn|y +

yn − xn|xn − yn|

∣∣ 23

·∣∣y + µ− 3

2+αn λ

12+αn (yn − pi)

∣∣α3

≥ C(µ−3

2+αn λ

12+αn |yn − pi|

)−α3∣∣y + µ

− 32+α

n λ1

2+αn (yn − pi)

∣∣α3

(3.4.34)

71

for |y| ≤ Rn = ( |xn−yn|βn)

12 , and Rn → +∞ as n→ +∞ by (3.4.33). Since (3.4.34) implies that

(3.4.22) holds for µn, yn, Un, Proposition 3.4.4 provides some ψn ∈ C∞0 (Ω) such that (3.4.12)holds and Supp ψn ⊂ B

Mµ3

2+αn λ

− 12+α

n

(yn) for some M > 0.

Since yn cannot lie in any ball centered at xn and radius of order of the scale parameter

(ε32nλ− 1

2n or ε

32nλ− 1

2n |xn − pi|−α2 ), we get from (3.4.33) that φn and ψn have disjoint compact

supports leading to µ2,n < 0, a contradiction to (3.1.4). This completes the proof of Proposition3.4.5.

3.4.3 Compactness issues

We are now in position to give the proof of Theorem 3.1.3. Assume 2 ≤ N ≤ 7, and letf ∈ C(Ω) be as in (3.1.3). Let (λn)n be a sequence such that λn → λ ∈ [0, λ∗] and let un bean associated solution such that (3.1.4) holds, namely

µ2,n := µ2,λn(un) ≥ 0.

The essential ingredient will be the estimate of Proposition 3.4.5 combined with the uniquenessresult of Proposition 2.5.2.

Proof of Theorem 3.1.3: Let xn be the maximum point of un in Ω and, up to asubsequence, assume by contradiction that un(xn) = max

Ωun(x) → 1 as n → ∞. Then

Proposition 3.4.5 gives that for some C > 0 and N0 ∈ N large,

un(x) ≤ 1− Cλ13nd(x)

α3 |x− xn|

23

for any x ∈ Ω and n ≥ N0, where d(x)α3 = min|x−pi|

αi3 : i = 1, . . . , k stands for the distance

function from the zero set of f(x). Thus, we have that:

0 ≤ λnf(x)(1− un)2

≤ C f(x)

d(x)2α3

λ13n

|x− xn| 43(3.4.35)

for any x ∈ Ω and n ≥ N0. Since by (3.1.3)

∣∣ f(x)

d(x)2α3

∣∣ ≤ |x− pi|α3 ‖fi‖∞ ≤ C

for x close to pi, fi as in (3.4.14), we get that f(x)

d(x)2α3

is a bounded function on Ω and then,

(3.4.35) gives that λnf(x)/(1 − un)2 is uniformly bounded in Ls(Ω), for any 1 < s < 3N4 .

Standard elliptic regularity theory now implies that un is uniformly bounded in W 2,s(Ω). BySobolev’s imbedding theorem, un is uniformly bounded in C0,β(Ω) for any 0 < β < 2/3. Up toa subsequence, we get that un → u0 weakly in H1

0 (Ω) and strongly in C0,β(Ω), 0 < β < 2/3,where u0 is a Holderian function solving weakly in H1

0 (Ω) the equation:

−∆u0 =λf(x)

(1− u0)2in Ω ,

0 ≤ u0 ≤ 1 in Ω ,u0 = 0 on ∂Ω.

(3.4.36)

72

Moreover, by uniform convergence

maxΩ

u0 = limn→+∞max

Ωun = 1

and, in particular u0 > 0 in Ω. Clearly, λ > 0 since any weak harmonic function in H10 (Ω)

is identically zero. To reach a contradiction, we shall first show that µ1,λ(u0) ≥ 0 and thendeduce from the uniqueness, stated in Proposition 2.5.2, of the semi-stable solution uλ thatu0 = uλ. But max

Ωuλ < 1 for any λ ∈ [0, λ∗], contradicting max

Ωu0 = 1. Hence, the claimed

compactness must hold.In addition to (3.1.4), assume now that µ1,n < 0, then λ > 0. Indeed, if λn → 0, then

by compactness and standard regularity theory, we get that un → u0 in C2(Ω), where u0 is aharmonic function so that u0 = 0 on ∂Ω. Then, u0 = 0 and un → 0 in C2(Ω). But the onlybranch of solutions for (S)λ bifurcating from 0 for λ small is the branch of minimal solutionsuλ and then, un = uλn for n large contradicting µ1,n < 0.

In order to complete the proof, we need only to show that

µ1,λ(u0) = inf∫

Ω

(|∇φ|2 − 2λf(x)(1− u0)3

φ2); φ ∈ C∞0 (Ω) and

∫

Ωφ2 = 1

≥ 0. (3.4.37)

Indeed, first Propositions 3.4.2 ∼ 3.4.4 imply the existence of a function φn ∈ C∞0 (Ω) so that∫

Ω

(|∇φn|2 − 2λnf(x)(1− un)3

φ2n

)< 0. (3.4.38)

Moreover, Supp φn ⊂ Brn(xn) and rn → 0 as n → +∞. Up to a subsequence, assume thatxn → p ∈ Ω as n→ +∞.

By contradiction, if (3.4.37) were false, then there exist φ0 ∈ C∞0 (Ω) such that∫

Ω

(|∇φ0|2 − 2λf(x)(1− u0)3

φ20

)< 0. (3.4.39)

We will replace φ0 with a truncated function φδ with δ > 0 small enough, so that (3.4.39)is still true while φδ = 0 in Bδ2(p) ∩ Ω. In this way, φn and φδ would have disjoint compactsupports in contradiction to µ2,n ≥ 0.

Let δ > 0 and set φδ = χδφ0, where χδ is a cut-off function defined as:

χδ(x) =

0 |x− p| ≤ δ2 ,

2− log |x− p|log δ

δ2 ≤ |x− p| ≤ δ ,1 |x− p| ≥ δ .

By Lebesgue’s theorem, we have:∫

Ω

2λf(x)(1− u0)3

φ2δ →

∫

Ω

2λf(x)(1− u0)3

φ20 as δ → 0 . (3.4.40)

For the gradient term, we have the expansion:∫

Ω|∇φδ|2 =

∫

Ωφ2

0|∇χδ|2 +∫

Ωχ2δ |∇φ0|2 + 2

∫

Ωχδφ0∇χδ∇φ0 .

73

The following estimates hold:

0 ≤∫

Ωφ2

0|∇χδ|2 ≤ ‖φ0‖2∞∫

δ2≤|x−p|≤δ

1|x− p|2 log2 δ

≤ C

log 1δ

and ∣∣2∫

Ωχδφ0∇χδ∇φ0

∣∣ ≤ 2‖φ0‖∞‖∇φ0‖∞log 1

δ

∫

B1(0)

1|x| ,

which provide ∫

Ω|∇φδ|2 →

∫

Ω|∇φ0|2 as δ → 0. (3.4.41)

Combining (3.4.39)-(3.4.41), we get that∫

Ω

(|∇φδ|2 − 2λf(x)(1− u0)3

φ2δ

)< 0

for δ > 0 sufficiently small. This completes the proof of (3.4.37) and therefore, Theorem 3.1.3is completely established.

3.5 The second bifurcation point

In this section, we discuss the second bifurcation point for (S)λ in the sense

λ∗2 = inf β > 0 : ∃ a curveVλ ∈ C([β, λ∗];C2(Ω)) of solutions for (S)λsuch that µ2,λ(Vλ) ≥ 0, Vλ ≡ Uλ ∀λ ∈ (λ∗ − δ, λ∗) ,

where Uλ and δ are as in Theorem 3.1.2, and we prove the multiplicity result of Theorem 3.1.5.Proof of Theorem 3.1.5: For any λ ∈ (λ∗2, λ

∗), the definition of λ∗2 gives that there existsa solution Vλ such that:

µ1,λ := µ1,λ(Vλ) < 0 ∀ λ ∈ (λ∗2, λ∗). (3.5.1)

In particular, Vλ 6= uλ provides a second solution different from the minimal one.Clearly (3.5.1) is true because first µ1,λ < 0 for λ close to λ∗. Moreover, if µ1,λ = 0 for

some λ ∈ (λ∗2, λ∗), then by Proposition 2.5.2 Vλ = uλ contradicting the fact that µ1,λ(uλ) > 0

for any 0 < λ < λ∗.Since the definition of λ∗2 gives µ2,λ(Vλ) ≥ 0 for any λ ∈ (λ∗2, λ

∗), we can take a sequenceλn ↓ λ∗2 and apply Theorem 3.1.3 that λ∗2 = lim

n→+∞λn > 0, supn∈N‖ Vλn ‖∞< 1. By elliptic

regularity theory, up to a subsequence Vλn → V ∗ in C2(Ω), where V ∗ is a solution for (S)λ∗2 .As before, µ1,λ∗2(V ∗) < 0 and by continuity µ2,λ∗2(V ∗) ≥ 0.

Suppose µ2,λ∗2(V ∗) > 0, let us fix some ε > 0 so small that 0 ≤ V ∗ ≤ 1 − 2ε and considerthe truncated nonlinearity gε(u) as in (3.2.1). Clearly, V ∗ is a solution of (3.2.2) at λ = λ∗2so that −∆ − λ∗2f(x)g′ε(V ∗) has no zero eigenvalues, since µ1,λ∗2(V ∗) < 0 and µ2,λ∗2(V ∗) > 0.Namely, V ∗ solves N(λ∗2, V

∗) = 0, where N is a map from R×C2,α(Ω) into C2,α(Ω), α ∈ (0, 1),defined as:

N : (λ, V ) −→ V + ∆−1(λf(x)gε(V )

).

74

Moreover,

∂VN(λ∗2, V∗) = Id + ∆−1

( 2λ∗2f(x)(1− V ∗)3

)

is an invertible map since −∆− λ∗2f(x)g′ε(V ∗) has no zero eigenvalues. The implicit functiontheorem gives the existence of a curve Wλ, λ ∈ (λ∗2 − δ, λ∗2 + δ), of solution for (3.2.2) so thatlimλ→λ∗2 Wλ = V ∗ in C2,α(Ω). Up to take δ smaller, this convergence implies that µ2,λ(Wλ) > 0and Wλ ≤ 1−ε for any λ ∈ (λ∗2−δ, λ∗2+δ). Hence, Wλ is a solution of (S)λ so that µ2,λ(Wλ) > 0,contradicting the definition of λ∗2. Therefore, µ2,λ∗2(V ∗) = 0, which completes the proof ofTheorem 3.1.5.

3.6 The one dimensional problem

In this section, we discuss the compactness of solutions for (S)λ in one dimensional case. Recallthat µk,λn(un) is the k−th eigenvalue of Lun,λn counted with their multiplicity.

Proof of Theorem 3.1.6: Let I = (a, b) be a bounded interval in R. Assume f ∈ C1(I)so that f ≥ C > 0 in I. We study solutions un of (S)λn in the form

−un =λnf(x)

(1− un)2in I ,

0 < un < 1 in I ,un(a) = un(b) = 0.

(3.6.1)

Assume that un satisfy (3.1.7) and λn → λ ∈ (0, λ∗]. Let xn ∈ I be a maximum point:un(xn) = max

Iun. If (un)n is not compact, then up to a subsequence, we may assume that

un(xn) → 1 with xn → x0 ∈ I as n → +∞. Away from x0, un is uniformly far away from1. Otherwise, by the maximum principle we would have un → 1 on an interval of positivemeasure, and then µk,λn(un) < 0 for any k and n large, a contradiction.

Assume, for example, that a ≤ x0 < b. By elliptic regularity theory, un(x) is uniformlybounded for x far away from x0. Letting ε > 0, we multiply (3.6.1) by un and integrate on(xn, x0 + ε):

u2n(xn)− u2

n(x0 + ε) =∫ x0+ε

xn

2λnf(s)un(s)(1− un(s))2

ds

=2λnf(x0 + ε)

1− un(x0 + ε)− 2λnf(xn)

1− un(xn)−∫ x0+ε

xn

2λnf(s)1− un(s)

ds.

Then, for n large:

u2n(xn) +

Cλ

1− un(xn)≤ u2

n(x0 + ε) + 2λnf(x0 + ε)

1− un(x0 + ε)− 2λn

∫ x0+ε

xn

f(s)1− un(s)

ds

≤ Cε + 4λ ‖ f ‖∞ x0 + ε− xn1− un(xn)

since un(xn) is the maximum value of un in I. Choosing ε > 0 sufficiently small, we get thatfor any n large: 1

1−un(xn) ≤ Cε, contradicting un(xn)→ 1 as n→ +∞.

75

Chapter 4

Dynamic Deflection

4.1 Introduction

The rest of this thesis is devoted to the dynamic deflection of the elastic membrane satisfying(1.2.30). Throughout this Chapter and unless mentioned otherwise, for convenience we studydynamic solutions of (1.2.30) in the form

∂u

∂t−∆u =

λf(x)(1− u)2

for x ∈ Ω , (4.1.1a)

u(x, t) = 0 for x ∈ ∂Ω, u(x, 0) = 0 for x ∈ Ω , (4.1.1b)

where nonnegative f ∈ Cα(Ω) for some α ∈ (0, 1] describes the permittivity profile of the elasticmembrane shown in Figure 1.1, while λ > 0 characterizes the applied voltage, see §1.3.2.

In this Chapter we deal with issues of global convergence, finite and infinite time “touch-down”, and touchdown profiles as well as pull-in distance. Recall that a point x0 ∈ Ω is saidto be a touchdown point for a solution u(x, t) of (4.1.1), if for some T ∈ (0,+∞], we havelimtn→T

u(x0, tn) = 1. T is then said to be a –finite or infinite– touchdown time. For each such

solution, we define its corresponding –possibly infinite– “first touchdown time”:

Tλ(Ω, f, u) = inft ∈ (0,+∞]; sup

x∈Ωu(x, t) = 1

.

We first analyze the relationship between the applied voltage λ, the permittivity profile f , andthe dynamic deflection of the elastic membrane. It is already known that solutions correspond-ing to large voltages λ necessarily touchdown in finite time (see [32]). The following theoremproved in §4.2 completes the picture.

Theorem 4.1.1. Suppose λ∗ := λ∗(Ω, f) is the pull-in voltage defined in Theorem 2.1.1, thenthe following hold:

1. If λ ≤ λ∗, then there exists a unique solution u(x, t) for (4.1.1) which globally convergesas t→ +∞, monotonically and pointwise to its unique minimal steady-state.

2. If λ > λ∗ and infΩ f > 0, then the unique solution u(x, t) of (4.1.1) must touchdown ata finite time.

This “touchdown” phenomenon is referred to sometimes as quenching. Note that in the casewhere the unique minimal steady-state of (4.1.1) at λ = λ∗ is non-regular – which can happenif N ≥ 8 – the above result means that the corresponding dynamic solution must touchdownbut that quenching occurs here in infinite time.

In §4.3 we shall establish that –an isolated– touchdown cannot occur at a point in Ω wherethe permittivity profile is zero, a fact that was observed numerically and conjectured to holdin [32]. More precisely, we prove the following.

76

Chapter 4. Dynamic Deflection

Theorem 4.1.2. Suppose u(x, t) is a touchdown solution of (4.1.1) at a finite time T , thenut > 0 for all 0 < t < T . Furthermore,

1. The permittivity profile f cannot vanish on an isolated set of touchdown points in Ω.

2. On the other hand, zeroes of the permittivity profile can be locations of touchdown ininfinite time.

In §4.4 we shall provide upper and lower estimates for finite touchdown times. Uniquenessconsiderations lead to a first touchdown time Tλ(Ω, f) that only depends on the domain Ωand on the profile f . These touchdown times translate into useful information concerning thespeed of the operation for many MEMS devices, such as Radio Frequency (RF) switches andmicrovalves. Estimates (4.1.4) and (4.1.5) below were already established in [32] for large λ.Considering that λ∗ < minλ1, λ2, the estimate (4.1.3) below gives an upper bound on thefirst touchdown time as soon as we exceed the pull-in voltage λ∗.

Theorem 4.1.3. Suppose f is a non-negative continuous function on a bounded domain Ω,and let Tλ(Ω, f) be the first –possibly infinite– touchdown time corresponding to a voltage λ.

1. The following lower estimate then holds for any λ > 0:

13λ supx∈Ω f(x)

≤ Tλ(Ω, f). (4.1.2)

2. If infx∈Ω f(x) > 0, then the following upper estimate holds for any λ > λ∗:

Tλ(Ω, f) ≤ T0,λ(Ω, f) :=8(λ+ λ∗)2

3 infx∈Ω f(x)(λ− λ∗)2(λ+ 3λ∗)

[1 +

( λ+ 3λ∗

2λ+ 2λ∗)1/2]

. (4.1.3)

3. If infx∈Ω f(x) > 0, and λ > λ1 := 4µΩ

27 infx∈Ω f(x) , then

Tλ(Ω, f) ≤ T1,λ(Ω, f) :=∫ 1

0

[λ infx∈Ω f(x)(1− s)2

− µΩs]−1

ds. (4.1.4)

4. If λ > λ2 := µΩ

3RΩ fφΩ

dx, then

Tλ(Ω, f) ≤ T2,λ(Ω, f) := − 1µΩ

log[1− µΩ

3λ( ∫

ΩfφΩ dx

)−1]. (4.1.5)

Here µΩ and φΩ are the first eigenpair of −∆ on H10 (Ω) with normalized

∫Ω φΩdx = 1.

Note that the upper bounds T0,λ and T1,λ are relevant only when f is bounded away from 0,while the upper bound T2,λ is valid for all permittivity profiles provided of course that λ > λ2.

In §4.5 we discuss touchdown profiles by the method of asymptotic analysis, and our purposeis to provide some information on the refined touchdown rate discussed in next Chapter.§4.6 is devoted to the pull-in distance of MEMS devices, referred to as the maximum stable

deflection of the elastic membrane before touchdown occurs. We provide numerical resultsfor pull-in distance with some explicit examples, from which one can observe that both largerpull-in distance and pull-in voltage can be achieved by properly tailoring the permittivityprofile.

77

4.2 Global convergence or touchdown

In this section, we analyze the relationship between the applied voltage λ, the permittivityprofile f , and the dynamic deflection u of (4.1.1). We first prove in §4.2.1 global convergencein the case λ < λ∗. In §4.2.2 we study finite-time touchdown for the case λ > λ∗. Finally wediscuss the case λ = λ∗ in §4.2.3.

First, we note the following uniqueness result.

Lemma 4.2.1. Suppose u1 and u2 are solutions of (4.1.1) on the interval [0, T ] such that‖ui‖L∞(Ω×[0,T ]) < 1 for i = 1, 2, then u1 ≡ u2.

Proof: Indeed, the difference U = u1 − u2 then satisfies

Ut −∆U = αU in Ω (4.2.1)

with initial data U(x, 0) = 0 and zero boundary condition. Here

α(x, t) =λ(2− u1 − u2)f(x)(1− u1)2(1− u2)2

.

The assumption on u1, u2 implies that α(x, t) ∈ L∞(Ω × [0, T ]). We now fix T1 ∈ [0, T ] andconsider the solution φ of the problem

φt + ∆φ+ αφ = 0 x ∈ Ω, 0 < t < T1,

φ(x, T1) = θ(x) ∈ C0(Ω),φ(x, t) = 0 x ∈ ∂Ω,

(4.2.2)

The standard linear theory (cf. Theorem 8.1 of [47]) gives that the solution of (4.2.2) is uniqueand bounded. Now multiplying (4.2.1) by φ, and integrating it on Ω × [0, T1], together with(4.2.2), yield that ∫

ΩU(x, T1)θ(x)dx = 0

for arbitrary T1 and θ(x), which implies that U ≡ 0, and we are done.

4.2.1 Global convergence when λ < λ∗

Theorem 4.2.2. Suppose λ∗ := λ∗(Ω, f) is the pull-in voltage defined in Theorem 2.1.1, thenfor λ < λ∗ there exists a unique global solution u(x, t) for (4.1.1) which monotonically convergesas t→ +∞ to the unique minimal solution u

λ(x) of (S)λ.

Proof:: This is standard and follows from the maximum principle combined with the existenceof regular minimal steady-state solutions at this range of λ. Indeed, fix 0 < λ < λ∗, and useTheorem 2.1.2 to obtain the existence of a unique minimal solution u

λ(x) of (S)λ. It is clear

that the pair u ≡ 0 and u = uλ(x) are sub- and super-solutions of (4.1.1) for all t > 0. This

implies that the unique global solution u(x, t) of (4.1.1) satisfies 1 > uλ(x) ≥ u(x, t) ≥ 0 in

Ω× (0,∞).By differentiating in time and setting v = ut, we get for any fixed t0 > 0

vt = ∆v +2λf(x)(1− u)3

v x ∈ Ω , 0 < t < t0 , (4.2.3a)

78


v(x, t) = 0 x ∈ ∂Ω , v(x, 0) ≥ 0 x ∈ Ω . (4.2.3b)

Here 2λf(x)(1−u)3 is a locally bounded non-negative function, and by the strong maximum principle,

we get that ut = v > 0 for (x, t) ∈ Ω× (0, t0) or ut ≡ 0. The second case is impossible becauseotherwise u(x, t) ≡ uλ(x) for any t > 0. It follows that ut > 0 holds for all (x, t) ∈ Ω× (0,∞),and since u(x, t) is bounded, this monotonicity property implies that the unique global solutionu(x, t) converges to some function us(x) as t→∞. Hence, 1 > uλ(x) ≥ us(x) > 0 in Ω.

Next we claim that the limit us(x) is a solution of (S)λ. Indeed, consider a solution u1 ofthe linear stationary boundary problem

−∆u1 =λf(x)

(1− us)2x ∈ Ω , u1 = 0 x ∈ ∂Ω . (4.2.4)

Let w(x, t) = u(x, t)− u1(x), then w satisfies

wt −∆w = λf(x)[ 1(1− u)2

− 1(1− us)2

], (x, t) ∈ Ω× (0, T ) ; (4.2.5a)

w(x, t) = 0 x ∈ ∂Ω× (0, T ) ; w(x, 0) = −u1(x) x ∈ Ω . (4.2.5b)

Since the right side of (4.2.5a) converges to zero in L2(Ω) as t→∞, a standard eigenfunctionexpansion implies that the solution w of (4.2.5) also converges to zero in L2(Ω) as t → ∞.This shows that u(x, t)→ u1(x) in L2(Ω) as t→∞. But since u(x, t)→ us(x) pointwise in Ωas t→∞, we deduce that u1(x) ≡ us(x) in L2(Ω), which implies that us(x) is also a solutionfor (S)λ. The minimal property of u

λ(x) then yields that u

λ(x) ≡ us(x) on Ω, which follows

that for every x ∈ Ω, we have u(x, t) ↑ uλ(x) as t→∞.

4.2.2 Touchdown at finite time when λ > λ∗

Recall from Theorem 2.1.1 that there is no solution for (S)λ as soon as λ > λ∗. Since thesolution u(x, t) of (4.1.1) –whenever it exists– is strictly increasing in time t ((see precedingtheorem)), then there must be T ≤ ∞ such that u(x, t) reaches 1 at some point of Ω as t→ T−.Otherwise, a proof similar to Theorem 4.2.2 would imply that u(x, t) would converge to itssteady-state which is then the unique minimal solution uλ of (S)λ, contrary to the hypothesisthat λ > λ∗. Therefore for this case, it only remains to know whether the touchdown time isfinite or infinite. This is exactly what we prove in the following.

Theorem 4.2.3. Suppose λ∗ := λ∗(Ω, f) is the pull-in voltage defined in Theorem 2.1.1. Ifinfx∈Ω f(x) > 0, then for λ > λ∗ there exists a finite time Tλ(Ω, f) at which the unique solutionu(x, t) of (4.1.1) must touchdown. Moreover, we have the bound

Tλ(Ω, f) ≤ T0,λ :=8(λ+ λ∗)2

3 infx∈Ω f(x)(λ− λ∗)2(λ+ 3λ∗)

[1 +

( λ+ 3λ∗

2λ+ 2λ∗)1/2]

. (4.2.6)

We start by transforming the problem from a touchdown situation (i.e. quenching) into ablow-up problem where a concavity method can be used. For that, we set V = 1/(1−u) which

79

reduces (1.1) to the following parabolic problem

Vt = ∆V − 2|∇V |2V

+ λf(x)V 4 for x ∈ Ω , (4.2.7a)

V (x, t) = 1 for x ∈ ∂Ω , (4.2.7b)V (x, 0) = 1 for x ∈ Ω . (4.2.7c)

This transformation implies that when λ > λ∗, the solution of (4.2.7) must blow up (in finiteor infinite time) and that there is no solution for the corresponding stationary equation:

∆V − 2|∇V |2V

+ λf(x)V 4 = 0 , x ∈ Ω ; V = 1 , x ∈ ∂Ω . (4.2.8)

Therefore, proving finite touchdown time of u for (1.1) is equivalent to showing finite blow-uptime of the solution V for (4.2.7).

For the proof, we shall first analyze the following auxiliary parabolic equation

vt = ∆v − 2|∇v|2v

+ λa2t2f(x)v4 for x ∈ Ω , (4.2.9a)

v = 1 for x ∈ ∂Ω , (4.2.9b)v(x, 0) = 1 for x ∈ Ω , (4.2.9c)

where a > 0 is a given constant.

Lemma 4.2.4. Suppose v is a solution of (4.2.9) up to a finite time T , then(vtv4

)t≥ 0 for all

t < T .

Proof: Dividing (4.2.9a) by v4, we obtain

vtv4

=∆vv4− 2|∇v|2

v5+ λa2t2f(x) .

Setting w = v−3, then direct calculations show that

wt −∆w +2|∇w|2

3w+ 3λa2t2f(x) = 0 . (4.2.10)

Differentiate (4.2.10) twice with respect to t, we obtain( |∇w|2

w

)tt

=(2∇w∇wt

w− |∇w|

2wtw2

)t

=2∇w∇wtt

w+

2|∇wt|2w

− 4∇w∇wtwtw2

− |∇w|2wtt

w2+

2|∇w|2w2t

w3,

which means that the functionz = wtt = −3

( vtv4

)t

(4.2.11)

satisfies

L(z) : = zt −∆z +4∇w3w∇z − 2|∇w|2

3w2z

= −6λa2f(x)− 23

[2|∇wt|2w

+2|∇w|2w2

t

w3− 4∇w∇wtwt

w2

]

≤ −6λa2f(x) ,

80

after an application of Cauchy-Schwarz inequality. Hence we have

L(z) ≤ −6λa2f(x) ≤ 0 . (4.2.12)

Now from (4.2.9) and the definition of z, we have z(x, 0) = 0 and z = 0 on ∂Ω. Since thecoefficients of L remain bounded as long as v is bounded, we conclude that z(x, t) ≤ 0 holdsfor all t < T . This completes the proof of Lemma 4.2.4. Proof of Theorem 4.2.3: Let λ > λ∗, and set λ′ = λ− λ∗ > 0, δ = infx∈Ω f(x) > 0, and

a0 =3δλ′(4λ∗ + λ′)

4(2λ∗ + λ′)

[1−

( 4λ∗ + λ′

2(2λ∗ + λ′)

)1/2], (4.2.13a)

and

T0,λ =1a0

=8(λ+ λ∗)2

3δ(λ− λ∗)2(λ+ 3λ∗)

[1 +

( λ+ 3λ∗

2λ+ 2λ∗)1/2]

< +∞ . (4.2.13b)

Consider now a solution v of (4.2.9) corresponding to λ = λ∗+λ′ and a0 as defined in (4.2.13a).We first establish the following

Claim: There exists x0 ∈ Ω such that v(x0, t)→∞ as t T0,λ.

Indeed, let t0 = 1a0

[4λ∗+λ′

2(2λ∗+λ′)

]1/2in such a way that

t0 < T0,λ and a20t

20(λ∗ +

λ′

2) = λ∗ +

λ′

4.

We claim that there exists x0 ∈ Ω such that

∆v(x0, t0)− 2|∇v(x0, t0)|2v(x0, t0)

+ (λ∗ +λ′

4)f(x0)|v(x0, t0)|4 > 0 . (4.2.14)

Essentially, otherwise we get that for all x ∈ Ω

∆v(x, t0)− 2|∇v(x, t0)|2v(x, t0)

+ (λ∗ +λ′

4)f(x0)|v(x, t0)|4 ≤ 0 . (4.2.15)

Since v(x, t0) ≥ 1 on Ω and hence on Ω, this means that the function v(x) = v(x, t0) is asupersolution for the equation

∆V − 2|∇V |2V

+ λf(x)V 4 = 0 , x ∈ Ω ; V = 1 , x ∈ ∂Ω . (4.2.16)

Since v ≡ 1 is obviously a subsolution of (4.2.16), it follows that the latter has a solution whichcontradicts the fact that λ = λ∗ + λ′

4 > λ∗(f,Ω). Hence, assertion (4.2.14) is verified.On the other hand, we do get from (4.2.9) that for t = t0 and every x ∈ Ω,

vt = ∆v − 2|∇v|2v

+ (λ∗ +λ′

4)f(x)v4 +

λ′

2a2

0t20f(x)v4 . (4.2.17)

We then deduce from (4.2.17) and (4.2.14) that at the point (x0, t0), we have

vtv4≥ λ′

2a2

0t20f(x0) > 0.

81

Applying Lemma 4.2.4, we then get for all (x0, t), t0 ≤ t < T0,λ that:

vtv4≥ λ′

2a2

0t20f(x0) > 0. (4.2.18)

Integrating (4.2.18) with respect to t in (t0, T0,λ), we obtain since f(x0) ≥ δ that:

13(1− v−3(x0, T0,λ)

) ≥ λ′

2a2

0t20f(x0)(T0,λ − t0) ≥ λ′

2a2

0t20δ(T0,λ − t0) =

13.

It follows that v(x0, t)→∞ as t T0,λ, and the claim is proved.To complete the proof of Theorem 4.2.3, we note that since a2

0t2 ≤ 1 for all t ≤ T0,λ, we

get from (4.2.9) that

vt ≤ ∆v − 2|∇v|2v

+ λf(x)v4 , (x, t) ∈ Ω× (0, T0,λ).

Setting w = V − v, where V is the solution of (4.2.7), then w satisfies

wt −∆w− 2∇(V + v)V

∇w+[λ(V 2 + v2)(V + v)f(x) +

2|∇v|2V v

]w ≥ 0 , (x, t) ∈ Ω× (0, T0,λ).

Here the coefficients of ∇w and w are bounded functions as long as V and v are both bounded.It is also clear that w = 0 on ∂Ω and w(x, 0) = 0. Applying the maximum principle, we reducethat w ≥ 0 and thus V ≥ v. Consequently, V must also blow up at some finite time T ≤ T0,λ,which means that u must touchdown at some finite time prior to T0,λ.

Remark 4.2.1. A slightly revised proof of Theorem 4.2.3 can be essentially adopted to provefinite-time touchdown for the case where λ > λ∗ and f(x) > 0 is almost everywhere on Ω. Theextension to more general nonnegative profile f(x) is now in progress.

4.2.3 Global convergence or touchdown in infinite time for λ = λ∗

In order to complete the proof of Theorem 4.1.1, the rest is to discuss the dynamic behavior of(4.1.1) at λ = λ∗. For this critical case, there exists a unique steady-state w∗ of (4.1.1) obtainedas a pointwise limit of the minimal solution uλ as λ ↑ λ∗. If w∗ is regular (i.e, if it is a classicalsolution such as in the case when N ≤ 7) a similar proof as in the case where λ < λ∗, yieldsthe existence of a unique solution u∗(x, t) which globally converges to the unique steady-statew∗ as t → ∞. On the other hand, if w∗ is a non-regular steady-state, i.e. if ‖w∗‖∞ = 1, thesituation is complicated as we shall still prove global convergence to the extremal solution,which then amounts to a touchdown in infinite time.

Throughout this subsection, we shall consider the unique solution 0 ≤ u∗ = u∗(x, t) < 1for the problem

u∗t −∆u∗ =λ∗f(x)

(1− u∗)2for (x, t) ∈ Ω× [0, t∗) , (4.2.19a)

u∗(x, t) = 0 for x ∈ ∂Ω× [0, t∗) , (4.2.19b)u∗(x, 0) = 0 for x ∈ Ω , (4.2.19c)

where t∗ is the maximal time for existence. We shall use techniques developed in [10] toestablish the following

82

Theorem 4.2.5. If w∗ is a non-regular minimal steady-state of (4.2.19), then there exists aunique global solution u∗ of (4.2.19) such that u∗(x, t) ≤ w∗(x) for all t <∞, while u∗(x, t)→w∗(x) as t→∞. In particular, lim

t→+∞ ‖u∗(x, t)‖∞ = 1.

The proof of Theorem 4.2.5 needs to use the following lemma.

Lemma 4.2.6. Consider the function δ(x) := dist(x, ∂Ω), then for any 0 < T < ∞, thereexists ε1 = ε1(T ) such that for 0 < ε ≤ ε1 the solution Zε of the problem

Zt −∆Z = −εf(x) in Ω× (0,∞) ,Z(x, t) = 0 on ∂Ω× (0,∞) ,Z(x, 0) = δ(x) in Ω

satisfies Zε ≥ 0 on [0, T ]× Ω.

Proof: Let(T (t)

)t≥0

be the heat semigroup with Dirichlet boundary condition, and considerthe solution ξ0 of

−∆ξ0 = 1 in Ω ; ξ0 = 0 on ∂Ω .

then we have

ξ0 = T (t)ξ0 +∫ t

0T (s)1Ωds

for all t ≥ 0. Since T (t)ξ0 ≥ 0, it follows that∫ t

0T (s)1Ωds ≤ ξ0 ≤ Cδ for all t ≥ 0 . (4.2.20)

On the other hand, we have

Zε(t) = T (t)δ − εf∫ t

0T (s)1Ωds ,

and so we have Zε(t) ≥ T (t)δ − εCδ. Consider now c0, c1 > 0 such that c0φ1 ≤ δ ≤ c1φ1,where φ1 is the first eigenfunction of −∆ in H1

0 (Ω), associated to the eigenvalue µ1. We have

T (t)δ ≥ c0T (t)φ1 = c0e−µ1tφ1 ≥ c0

c1e−µ1tδ .

Therefore, we have Zε(t) ≥ ( c0c1 e−µ1t − Cε)δ. And hence it follows that Zε(t) ≥ 0 on [0, T ]provided ε ≤ c0

c1Ce−µ1T .

Proof of Theorem 4.2.5: We proceed in four steps.

Claim 1. We have that u∗(x, t) ≤ w∗(x) for all (x, t) ∈ Ω× [0, t∗). Indeed, fix any T < t∗ andlet ξ be the solution of the backward heat equation:

ξt −∆ξ = h(x, t) in Ω× (0, T ) ,ξ|∂Ω = 0 , ξ(T ) = 0 ,

83

where h(x, t) ≥ 0 is in Ω× (0, T ). Multiplying (4.2.19) by ξ and integrating on Ω× (0, T ) wefind that ∫ T

0

∫

Ωu∗h dxdt =

∫ T

0

∫

Ω

λ∗ξf(x)(1− u∗)2

dxdt .

On the other hand,

−∫ T

0

∫

Ωw∗ξt dxdt =

∫

Ωw∗ξ(0)dx and −

∫ T

0

∫

Ωw∗∆ξ dxdt =

∫ T

0

∫

Ω

λ∗ξf(x)(1− w∗)2

dxdt.

Therefore, we have∫ T

0

∫

Ω(u∗ − w∗)h dxdt ≤

∫

Ωw∗ξ(0)dx+

∫ T

0

∫

Ω(u∗ − w∗)h dxdt

=∫ T

0

∫

Ω

( 1(1− u∗)2

− 1(1− w∗)2

)λ∗ξf(x) dxdt

≤ C∫ T

0

∫

u∗≥w∗

( 1(1− u∗)2

− 1(1− w∗)2

)ξ dxdt

≤ C∫ T

0

∫

Ω(u∗ − w∗)+ξ dxdt,

since ‖u∗‖∞ < 1 for t ∈ [0, T ). Therefore, we have

∫ T

0

∫

Ω(u∗ − w∗)h dxdt ≤ C

(∫ T

0

∫

Ω[(u∗ − w∗)+]2 dxdt

)1/2(∫ T

0

∫

Ωξ2 dxdt

)1/2.

On the other hand, ξ(x, t) =∫ Tt T (s − t)h(x, s)ds , where T (t) is the heat semigroup with

Dirichlet boundary condition, and hence

‖ξ(x, t)‖2L2 ≤(∫ T

t‖h(x, s)‖L2ds

)2≤ (T − t)

∫ T

0

∫

Ωh2 dxdt .

Therefore, ∫ T

0

∫

Ωξ2 dxdt ≤ T 2

2

∫ T

0

∫

Ωh2 dxdt,

and so,∫ T

0

∫

Ω(u∗ − w∗)h dxdt ≤ CT√

2

(∫ T

0

∫

Ω[(u∗ − w∗)+]2 dxdt

)1/2(∫ T

0

∫

Ωh2 dxdt

)1/2.

Letting h converge to (u∗ − w∗)+ in L2, and since u∗ − w∗ ∈ L1(Ω) we have

∫ T

0

∫

Ω[(u∗ − w∗)+]2 dxdt ≤ CT√

2

∫ T

0

∫

Ω[(u∗ − w∗)+]2 dxdt ,

which gives that u∗ ≤ w∗ provided C2T 2 < 2, and our first claim follows.

84

Claim 2. There exist 0 < τ1 < t∗, and C0, c0 > 0 such that for all x ∈ Ω

u∗(x, τ1) ≤ minC0δ(x); w∗(x)− c0δ(x) . (4.2.21)

Fix 0 < τ < t∗ sufficiently small, and let v be the solution of

vt −∆v =λ∗f(x)(1− v)2

for (x, t) ∈ Ω× [0, T ) , (4.2.22a)

v(x, t) = 0 for x ∈ ∂Ω× [0, T ) , (4.2.22b)v(x, 0) = v0 = u∗(x, τ) for x ∈ Ω , (4.2.22c)

where [0, T ) is the maximal interval of existence for v. Similarly to Claim 1, we can show that0 ≤ v ≤ w∗. Choose now K > 1 sufficiently large such that the path z(x, t) := u∗(x, t) +1KT (t)v0 satisfies ‖z(x, t)‖∞ ≤ 1 for 0 ≤ t < T . We then have

zt −∆z =λ∗f(x)

(1− u∗)2≤ λ∗f(x)

(1− z)2in Ω× (0, T ) ,

z(x, t) = 0 on ∂Ω× (0, T ) ,

z(x, 0) =v0(x)K

in Ω ,

and the maximum principle gives that z ≤ v. Consider now a function γ : [0,∞) → R suchthat γ(t) > 0 and

T (t)v0 ≥ Kγ(t)δ on Ω. (4.2.23)

We then get

u∗ ≤ v − 1KT (t)v0 ≤ w∗ − 1

KT (t)v0 ≤ w∗ − γ(t)δ for 0 ≤ t < T . (4.2.24)

On the other hand, for any 0 ≤ t ≤ T < t∗, u∗ is bounded by some constant M < 1 onΩ× [0, T ] such that

u∗ ≤MT (t)1Ω +C

(1−M)2

∫ t

0T (s)1Ωds .

Consider now a function C : [0,∞)→ R such that T (t)1Ω ≤ C(t)δ for t ≥ 0, which means that

u∗ ≤MC(t)δ + C(M)Cδ

for any 0 ≤ t ≤ T , where (4.2.20) is applied. This combined with (4.2.24) conclude the proofof Claim (4.2.21).

Claim 3. For 0 < ε < 1 there exists wε satisfying ‖wε‖∞ < 1 and∫

Ω∇wε∇ϕ ≥

∫

Ω

( 1(1− wε)2

− ε)λ∗ϕf(x) (4.2.25)

for all ϕ ∈ H10 (Ω) with ϕ ≥ 0 on Ω. Moreover, there exists 0 < ε1 ≤ 1 such that for 0 < ε < ε1,

we also have0 ≤ wε(x)− c0

2δ(x) for x ∈ Ω (4.2.26)

85

c0 being as in (4.2.21).To prove (4.2.25), we set

g(w∗) =1

(1− w∗)2, h(w∗) =

∫ w∗

0

ds

g(s), 0 ≤ w∗ < 1 . (4.2.27)

For any ε ∈ (0, 1) we also set

g(w∗) =1

(1− w∗)2− ε , h(w∗) =

∫ w∗

0

ds

g(s), 0 ≤ w∗ < 1 , (4.2.28)

and φε(w∗) := h−1(h(w∗)

). It is easy to check that φε(0) = 0 and 0 ≤ φε(s) < s for s ≥ 0, and

φε is increasing and concave with

φ′ε(s) =g(φε(s))− ε

g(s)> 0 .

Setting wε = φε(w∗), we have for any ϕ ∈ H10 (Ω) with ϕ ≥ 0 on Ω,

∫

Ω∇wε∇ϕ =

∫

Ωφ′ε(w

∗)∇w∗∇ϕ =∫

Ω∇w∗∇ (φ′ε(w∗)ϕ

)−∫

Ωφ′′ε(w

∗)ϕ|∇w∗|2

≥∫

Ω

λ∗f(x)(1− w∗)2

φ′ε(w∗)ϕ =

∫

Ω

( 1(1− wε)2

− ε)λ∗ϕf(x) ,

which gives (4.2.25) for any ε ∈ (0, ε0).In order to prove (4.2.26), we set

η(x) = minw∗(x), (C0 + c0)δ(x) and ηε = φε η ,

where φε(·) is defined above, and C0 and c0 are as in (4.2.21). Since η ≤ w∗ and φε is increasing,we have ηε ≤ φε(w∗) = wε. Applying (4.2.21) we get that

0 ≤ η(x)− c0δ(x) on Ω . (4.2.29)

We also note that ηε = φε(η) ≤ η ≤ M with M = (C0 + c0)δ(x), and φ′ε(s) → 1 as ε → 0uniformly in [0, 1]. Therefore, for some θ ∈ (0, 1) we have

η − ηε = η − (φε(η)− φε(0))

= η(1− φ′ε(θη)

) ≤ η sup0≤s≤1

(1− φ′ε(s))

≤ (C0 + c0)δ sup0≤s≤1

(1− φ′ε(s)) ≤c0

2δ

provided ε small enough, which gives

η ≤ ηε +c0

2δ . (4.2.30)

We now conclude from (4.2.29) and (4.2.30) that

0 ≤ η − c0δ ≤ ηε − c0

2δ ≤ wε − c0

2δ

86

for small ε > 0, and (4.2.26) is therefore proved.

To complete the proof of Theorem 4.2.5, we assume that t∗ <∞ and we shall work towardsa contradiction. In view of Claim 3), we let ε > 0 be small enough so that 0 ≤ wε − c0

2 δ. UseLemma 4.2.6 and choose K > 2 large enough such that the solution Z of the problem

Zt −∆Z = −ελ∗f(x) in Ω× (0, t∗) ,

Z(x, t) = 0 on ∂Ω× (0, t∗) ,

Z(x, 0) =c0

Kδ in Ω

satisfies 0 ≤ Z < 1− u∗ on Ω× (0, t∗). Let v be the solution of

vt −∆v =( 1

(1− |v|)2− ε)λ∗f(x) in Ω× (0, s∗) ,

v(x, t) = 0 on ∂Ω× (0, s∗) ,v(x, 0) = wε in Ω ,

where [0, s∗) is the maximal interval of existence for v. Setting z(x, t) = Z(x, t) + u∗(x, t) for0 ≤ t < t∗, we then have 0 ≤ u∗ ≤ z < 1 and

zt −∆z =( 1

(1− u∗)2− ε)λ∗f(x) ≤

( 1(1− z)2

− ε)λ∗f(x) in Ω× (0, t∗) ,

z(x, t) = 0 on ∂Ω× (0, t∗) ,

z(x, 0) =c0

Kδ(x) ≤ wε(x) in Ω .

Now the maximum principle gives that z ≤ v on Ω× (0,mins∗, t∗), and in particular we have0 ≤ v on Ω × (0,mins∗, t∗). Furthermore, the maximum principle and (4.2.25) also yieldthat v ≤ wε. Since ‖wε‖∞ < 1 we necessarily have t∗ < s∗ = ∞. Therefore, u∗ ≤ z ≤ v ≤ wεon [0, t∗), which implies that ‖u∗‖∞ < 1 at t = t∗, which contradicts to our initial assumptionthat u∗ is not a regular solution.

4.3 Location of touchdown points

In this section, we first present a couple of numerical simulations for different domains, differentpermittivity profiles, and various values of λ, by applying an implicit Crank-Nicholson scheme(see [32] for details) on (4.1.1). For the connection with the involved figures below, we discuss(4.1.1) in the form of

∂u

∂t−∆u = − λf(x)

(1 + u)2for x ∈ Ω , (4.3.1a)

u(x, t) = 0 for x ∈ ∂Ω ; u(x, 0) = 0 for x ∈ Ω , (4.3.1b)

with the following two choices for the domain Ω

Ω : [−1/2, 1/2] (slab) ; Ω : x2 + y2 ≤ 1 (unit disk) . (4.3.2)

87

−0.5 −0.25 0 0.25 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0 (a). f(x) = |2x| and λ = 4.38

x

u

converge to the maximal negative steady−state

−0.5 −0.25 0 0.25 0.5 −1

−0.8

−0.6

−0.4

−0.2

0 (b). f(x) = |2x| and λ = 4.50

x

u

Figure 4.1: Left Figure: u versus x for λ = 4.38. Right Figure: u versus x for λ = 4.50. Herewe consider (4.3.1) with f(x) = |2x| in the slab domain.

Simulation 1: We consider f(x) = |2x| for a permittivity profile in the slab domain−1/2 ≤ x ≤ 1/2. Here the number of the meshpoints is chosen as N = 2000 for the plots uversus x at different times. Figure 4.1(a) shows, for λ = 4.38, a typical sequence of solutionsu for (4.3.1) approaching to the maximal negative steady-state. In Figure 4.1(b) we takeλ = 4.50 and plot u versus x at different times t = 0, 0.1880, 0.3760, 0.5639, 0.7519, 0.9399,1.1279, 1.3159, 1.5039, 1.6918, 1.879818, and a touchdown behavior is observed at two differentnonzero points x = ±0.14132. These numerical results and Remark 4.2.1 point to a pull-involtage 4.38 ≤ λ∗ < 4.50.

0 0.2 0.4 0.6 0.8 1 −0.4

−0.3

−0.2

−0.1

0(a). f(r) = r and λ = 1.70

u

r = |x|

converge to the maximal negative steady−state

0 0.2 0.4 0.6 0.8 1 −1

−0.8

−0.6

−0.4

−0.2

0 (b). f(r) = r and λ = 1.80

r = |x|

u

Figure 4.2: Left Figure: u versus r for λ = 1.70. Right Figure: u versus r for λ = 1.80. Herewe consider (4.3.1) with f(r) = r in the unit disk domain.

Simulation 2: Here we consider f(r) = r for a permittivity profile in the unit disk domain.The number of meshpoints is again chosen to be N = 2000 for the plots u versus r at differenttimes. Figure 4.2(a) shows how for λ = 1.70, a typical sequence of solutions u for (4.3.1)approach to the maximal negative steady-state. In Figure 4.2(b) we take λ = 1.80 and plot uversus r at different times t = 0, 0.4475, 0.8950, 1.3426, 1.7901, 2.2376, 2.6851, 3.1326, 3.5802,4.0277, 4.4751942, and a touchdown behavior is observed at the nonzero points r = 0.21361.Again these numerical results point to a pull-in voltage 1.70 ≤ λ∗ < 1.80.

88

One can observe from above that touchdown points at finite time are not the zero pointsof the varying permittivity profile f , a fact firstly observed and conjectured in [32]. The mainpurpose of this section is to analyze this phenomena.

Theorem 4.3.1. Suppose u(x, t) is a touchdown solution of (4.1.1) at a finite time T , thenwe have

1. The permittivity profile f cannot vanish on an isolated set of touchdown points in Ω.

2. On the other hand, zeroes of the permittivity profile can be locations of touchdown ininfinite time.

Note that Theorem 4.3.1 holds for any bounded domain. The proof of Theorem 4.3.1 isbased on the following Harnack-type estimate.

Lemma 4.3.2. For any compact subset K of Ω and any m > 0, there exists a constantC = C(K,m) > 0 such that supx∈K |u(x)| ≤ C < 1 whenever u satisfies

∆u ≥ m

(1− u)2x ∈ Ω ; 0 ≤ u < 1 x ∈ Ω . (4.3.3)

Proof: Setting v = 1/(1− u), then (4.3.3) gives that v satisfies

∆vv2− 2|∇v|2

v3≥ mv2 in Ω ,

which means that v is a subsolution of the “linear” equation ∆v = 0 in Ω. In order to applythe Harnack inequality on v, we need to show that for balls Br ⊂ Ω, we have that v ∈ L3(Br)with an L3-norm that only depends on m and the radius r.

Without loss of generality, we may assume 0 ∈ K ⊂ Ω. Let Br = Br(0) ⊂ K be the ballcentered at x = 0 and radius r. For 0 < r1 < r2 ≤ 4r1, let η(x) ∈ C∞0 (Br2

) be such that η ≡ 1in Br1

, 0 ≤ η ≤ 1 in Br2\Br1

and |∇η| ≤ 2/(r2− r1). Multiplying (4.3.3) by φ2/(1−u), whereφ = ηα and α ≥ 1 is to be determined later, and integrating by parts we have

∫

Br2

mφ2

(1− u)3≤∫

Br2

φ2∆u1− u = −

∫

Br2

φ2|∇u|2(1− u)2

−∫

Br2

2φ∇φ · ∇u1− u . (4.3.4)

From the fact,∫

Br2

2φ∇φ · ∇u1− u ≤

∫

Br2

φ2|∇u|2 + 4∫

Br2

|∇φ|2(1− u)2

≤∫

Br2

φ2|∇u|2(1− u)2

+ 4∫

Br2

|∇φ|2(1− u)2

,

(4.3.4) gives that ∫

Br2

mφ2

(1− u)3≤ 4

∫

Br2

|∇φ|2(1− u)2

.

Now choose φ = η2β with β = 32 . Then Holder’s inequality implies that

m

∫

Br2

η4β

(1− u)3≤ 16β2

[ ∫

Br2

|∇η|4β] 1

2β[ ∫

Br2

η4β

(1− u)3

] 2β−12β

.

89

This shows that ∫

Br1

1(1− u)3

≤∫

Br2

η4β

(1− u)3< C(m, r1). (4.3.5)

By virtue of the one-sided Harnack inequality, we have

‖ 11− u ‖L∞(B r1

2)=‖ v ‖L∞(B r1

2)≤ C(r1) ‖ v ‖L3(Br1 )< C(r1,m) .

The rest follows from a standard compactness argument. Proof of Theorem 4.3.1: Set v = ut, then we have for any t1 < T that

vt = ∆v +2λf(x)(1− u)3

v (x, t) ∈ Ω× (0, t1) ; (4.3.6a)

v(x, t) = 0 (x, t) ∈ ∂Ω× (0, t1) ; v(x, 0) ≥ 0 x ∈ Ω. (4.3.6b)

Note that the term 2λf(1−u)3 is locally bounded in Ω × (0, t1), so that by the strong maximum

principle, we may conclude

ut = v > 0 for (x, t) ∈ Ω× (0, t1) (4.3.7)

and therefore, ut > 0 holds for all (x, t) ∈ Ω× (0, T ). Since K is an isolated set of touchdownpoints, there exists an open set U such that K ⊂ U ⊂ U ⊂ Ω with no touchdown points inU \ K. Consider now 0 < t0 < T such that infx∈U ut(x, t0) = C1 > 0. We claim that thereexists ε > 0 such that

J ε(x, t) = ut − ε

(1− u)2≥ 0 for all (x, t) ∈ U × (t0, T ) , (4.3.8)

Indeed, there exists C2 > 0 such that ut(x, T ) ≥ C2 > 0 on U , and since ∂U has no touchdownpoints, there exists ε > 0 such that J ε ≥ 0 on the parabolic boundary of U × (t0, T ). Also,direct calculations imply that

J εt −∆J ε =2λf

(1− u)3J ε +

6ε|∇u|2(1− u)4

≥ 2λf(1− u)3

J ε .

Since ε(1−u)2 is locally bounded on U×(t0, T ), we can apply the maximum principle to obtaining

(4.3.8).If now infx∈K f(x) = 0, then we may combine (4.3.8) and (4.1.1), to deduce that for a

small neighborhood B ⊂ U of some point x0 ∈ K where f(x) ≤ ε/2, we have

∆u ≥ ε

21

(1− u)2for (x, t) ∈ B × (t0, T ) .

In view of Lemma 4.3.2, this contradicts the assumption that x0 is a touchdown point.For the second part, recall from Theorem 2.1.2 that the unique extremal solution for the

stationary problem on the ball in the case N ≥ 8 and for a permittivity profile f(x) = |x|α, isu∗(x) = 1− |x| 2+α

3 as long as α is small enough. Theorem 4.1.1 then implies that the origin 0is a touchdown point of the solution even though it is also a root for the permittivity profile

90

(i.e., f(0) = 0). This complements the statement of Theorem 4.3.1 above. In other words, zeropoints of f in Ω cannot be on the isolated set of touchdown points in finite time (which occurwhen λ > λ∗) but can very well be touchdown points in infinite time of (4.1.1), which can onlyhappen when λ = λ∗. The proof of Theorem 4.3.1 fails for touchdowns in infinite time, simplybecause the maximum principle cannot be applied in the infinite cylinder Ω× (0,∞).

4.4 Estimates for finite touchdown times

In this section we give comparison results and explicit estimates on finite touchdown times ofdynamic deflections u = u(x, t). This often translates into useful information concerning thespeed of the operation for many MEMS devices such as RF switches or micro-valves.

4.4.1 Comparison results for finite touchdown time

We start by comparing the effect on finite touchdown time of two different but comparablepermittivity profiles f(x), at a given voltage λ.

Theorem 4.4.1. Suppose u1 = u1(x, t) (resp., u2 = u2(x, t)) is a touchdown solution for(4.1.1) associated to a fixed voltage λ and permittivity profiles f1 (resp., f2) with a correspondingfinite touchdown time Tλ(Ω, f1) (resp., Tλ(Ω, f2)). If f1(x) ≥ f2(x) on Ω and if f1(x) > f2(x)on a set of positive measure, then necessarily Tλ(Ω, f1) < Tλ(Ω, f2).

Proof: By making a change of variable v = 1−u, we can assume to be working with solutionsof the following equation:

∂v

∂t−∆v = −λf(x)

v2for x ∈ Ω , (4.4.1a)

v(x, t) = 1 for x ∈ ∂Ω , (4.4.1b)v(x, 0) = 1 for x ∈ Ω , (4.4.1c)

where f is either f1 or f2. Suppose now that Tλ(Ω, f1) > Tλ(Ω, f2) and let Ω0 ⊂ Ω be the setof touchdown points of u2 at finite time Tλ(Ω, f2). Setting w = u2 − u1, we get that

wt −∆w − λ(f2u1 + f1u2)u2

1u22

w =λ(f1 − f2)u1u2

≥ 0 (x, t) ∈ Ω× (0, Tλ(Ω, f2)) . (4.4.2)

Since w = 0 at t = 0 as well as on ∂Ω×(0, Tλ(Ω, f2)), we get from the maximum principle that wcannot attain a negative minimum in Ω×(0, Tλ(Ω, f2)), and therefore w ≥ 0 in Ω×(0, Tλ(Ω, f2)).Since u2 → 0 in Ω0 as t → Tλ(Ω, f2), and since our assumption is that Tλ(Ω, f1) > Tλ(Ω, f2),we then have u1 > 0 in Ω0 as t → Tλ(Ω, f2). Therefore, w < 0 in Ω0 as t → Tλ(Ω, f2), whichis a contradiction and therefore Tλ(Ω, f1) ≤ Tλ(Ω, f2).

To prove the strict inequality, we note that the above proof shows that w ≥ 0 in Ω ×(0, Tλ(Ω, f2)), which once combined with (4.4.2) gives that

wt −∆w ≥ 0 , in Ω× (t1, Tλ(Ω, f2)) ,

where t1 > 0 is chosen so that w(x, t1) 6≡ 0 in Ω. Now we compare w with the solution z of

zt −∆z = 0 in Ω× (t1, Tλ(Ω, f2))

91

subject to z(x, t1) = w(x, t1) and z(x, t) = 0 on ∂Ω × (t1, Tλ(Ω, f2)). Clearly, w ≥ z inΩ × (t1, Tλ(Ω, f2)). On the other hand, for any t0 > t1 we have z > 0 in Ω × (t0, Tλ(Ω, f2)).Consequently, w > 0 which means that u2 > u1 in Ω× (t0, Tλ(Ω, f2)) and therefore Tλ(Ω, f1) <Tλ(Ω, f2).

The second comparison result deals with different applied voltages but identical permittivityprofiles.

Theorem 4.4.2. Suppose u1 = u1(x, t) (resp., u2 = u2(x, t)) is a solution for (4.1.1) associatedto a voltage λ1 (resp., λ2) and which has a finite touchdown time Tλ1(Ω, f) (resp., Tλ2(Ω, f)).If λ1 > λ2 then necessarily Tλ1(Ω, f) < Tλ2(Ω, f).

Proof: It is similar to the proof of Theorem 4.4.1, except that for w = u2 − u1, (4.4.2) isreplaced by

wt −∆w − λ1(u1 + u2)fu2

1u22

w =(λ1 − λ2)f

u22

≥ 0 (x, t) ∈ Ω× (0, T ) .

The details are left for the interested reader.

Remark 4.4.1. A reasoning similar to the one found in Proposition 2.5 of [29], gives someinformation on the dependence on the shape of the domain. Indeed, for any bounded domainΓ in RN and any non-negative continuous function f on Γ, we have

λ∗(Γ, f) ≥ λ∗(BR, f∗) and Tλ(Γ, f) ≥ Tλ(BR, f∗),

where BR = BR(0) is the Euclidean ball in RN with radius R > 0 and with volume |BR| = |Γ|,where f∗ is the Schwarz symmetrization of f .

We now present numerical results comparing finite touchdown times in a slab domain.

−0.5 −0.25 0 0.25 0.5−1

−0.8

−0.6

−0.4

−0.2

0(a). λ = 8 with different f(x)

f1(x)

T1

f2(x)

T2

x

u

−0.5 −0.25 0 0.25 0.5−1

−0.8

−0.6

−0.4

−0.2

0(b). f(x) = |2x| with different λ

λ 2 = 8

T2

λ 1 = 10

T1

x

u

Figure 4.3: Left Figure: plots of u versus x for different f(x) at λ = 8 and t = 0.185736. RightFigure: plots of u versus x for different λ with f(x) = |2x| and t = 0.1254864.

Figure 4.3(a): Dependence on the dielectric permittivity profiles fWe consider (4.3.1) for the cases where

f1(x) = |2x| and f2(x) =

|2x| if |x| ≤ 18 ,

1/4 + 2 sin(|x| − 1/8) otherwise.(4.4.3)

92


Using N = 1000 meshpoints, we plot u versus x with λ = 8 at the time t = 0.185736 in Figure4.3(a). The numerical results show that the finite touchdown time Tλ(Ω, f1) for the case f1(x)and Tλ(Ω, f2) for the case f2(x) are 0.185736 and 0.186688, respectively.

Figure 4.3(b): Dependence on the applied voltage λUsing N = 1000 meshpoints and the profile f(x) = |2x|, we plot u of (4.3.1) versus x withdifferent values of λ at the time t = 0.1254864. The numerical results show that finite touch-down time Tλ1(Ω, f) for applied voltage λ1 = 10 and Tλ2(Ω, f) for applied voltage λ2 = 8 are0.1254864 and 0.185736, respectively.

4.4.2 Explicit bounds on touchdown times

We now establish claims 1), 3) and 4) in Theorem 1.3 of the introduction. Note that hereλ1 ≥ λ∗ and λ2 ≥ λ∗ are as in Theorem 2.1.1.

Proposition 4.4.3. Suppose f is a non-negative continuous function on a bounded domainΩ, and let u be a solution of (4.1.1) corresponding to a voltage λ. Then,

1. For any λ > 0, we have Tλ(Ω, f) ≥ T∗ := 13λ supx∈Ω f(x) .

2. If infΩ f > 0, and if λ > λ1 := 4µΩ

27 infx∈Ω f(x) , then

Tλ(Ω, f) ≤ T1,λ :=∫ 1

0

[λ infx∈Ω f(x)(1− s)2

− µΩs]−1

ds. (4.4.4)

3. If f > 0 on a set of positive measure, and if λ > λ2 := µΩ

3RΩ fφΩ

dx, then

Tλ(Ω, f) ≤ T2,λ := − 1µΩ

log[1− µΩ

3λ( ∫

ΩfφΩ dx

)−1. (4.4.5)

Here µΩ and φΩ are the first eigenpair of −∆ on H10 (Ω) with normalized

∫Ω φΩdx = 1.

Proof: 1) Consider the initial value problem:

dη(t)dt

=λM

(1− η(t))2,

η(0) = 0 ,(4.4.6)

where M = supx∈Ω f(x). From (4.4.6) one has 1λM

∫ η(t)0 (1 − s)2ds = t . If T∗ is the time

where limt→T∗ η(t) = 1, then we have T∗ = 1λM

∫ 10 (1 − s)2ds = 1

3λM . Obviously, η(t) is now asuper-function of u(x, t) near touchdown, and thus we have

T ≥ T∗ =1

3λM=

13λ supx∈Ω f(x)

,

which completes the proof of 1).

93


The following analytic upper bounds of finite touchdown time T were established in The-orem 3.1 and 3.2 of [32].

2) Without loss of generality we assume that φΩ > 0 in Ω. Multiplying (4.1.1a) by φΩ ,and integrating over the domain, we obtain

d

dt

∫

ΩφΩu dx =

∫

ΩφΩ∆u dx+

∫

Ω

λφΩf(x)(1− u)2

dx . (4.4.7)

Using Green’s theorem, together with the lower bound C0 of f , we get

d

dt

∫

ΩφΩu dx ≥ −µΩ

∫

ΩφΩu dx+ λC0

∫

Ω

φΩ

(1− u)2dx . (4.4.8)

Next, we define an energy-like variable E(t) by E(t) =∫

Ω φΩu dx so that

E(t) =∫

ΩφΩu dx ≤ sup

Ωu

∫

ΩφΩ dx = sup

Ωu . (4.4.9)

Moreover, E(0) = 0 since u = 0 at t = 0. Then, using Jensen’s inequality on the second termon the right-hand side of (4.4.8), we obtain

dE

dt+ µΩE ≥

λC0

(1− E)2, E(0) = 0 . (4.4.10)

We then compare E(t) with the solution F (t) of

dF

dt+ µΩF =

λC0

(1− F )2, F (0) = 0 . (4.4.11)

Standard comparison principles yield that E(t) ≥ F (t) on their domains of existence. There-fore,

supΩu ≥ E(t) ≥ F (t) . (4.4.12)

Next, we separate variables in (4.4.11) to determine t in terms of F . The touchdown time T1

for F is obtained by setting F = 1 in the resulting formula. In this way, we get

T1 ≡∫ 1

0

[ λC0

(1− s)2− µΩs

]−1ds . (4.4.13)

The touchdown time T1 is finite when the integral in (4.4.13) converges. A simple calculationshows that this occurs when λ > λ1 ≡ 4µ

Ω27C0

. Hence if T1 is finite, then (4.4.12) implies that

the touchdown time T of (4.1.1) must also be finite. Therefore, when λ > λ1 = 4µΩ

27C0, we have

that T satisfies

T ≤ T1 ≡∫ 1

0

[ λC0

(1− s)2− µΩs

]−1ds . (4.4.14)

(3) Multiply now (4.1.1a) by φΩ(1−u)2, and integrate the resulting equation over Ω to get

d

dt

∫

Ω

φΩ

3(1− u)3 dx = −

∫

ΩφΩ(1− u)2∆u dx−

∫

ΩλfφΩ dx . (4.4.15)

94

We calculate the first term on the right-hand side of (4.4.15) to get

d

dt

∫

Ω

φΩ

3(1− u)3 dx (4.4.16a)

=∫

Ω∇u · ∇

[φΩ(1− u)2

]dx+

∫

∂Ω(1− u)2φΩ∇u · n dS −

∫

ΩλfφΩ dx (4.4.16b)

= −∫

Ω2(1− u)φΩ |∇u|2 dx−

∫

Ω

13∇φΩ · ∇

[(1− u)3

]dx−

∫

ΩλfφΩ dx (4.4.16c)

≤ −13

∫

∂Ω∇φΩ · ν dS −

µΩ

3

∫

Ω(1− u)3φΩ dx−

∫

ΩλfφΩ dx , (4.4.16d)

where ν is the unit outward normal to ∂Ω. Since∫∂Ω∇φΩ · ν dS = −µΩ , we further estimate

from (4.4.16d) thatdE

dt+ µΩE ≤ R , R ≡ µΩ

3− λ

∫

ΩfφΩ dx , (4.4.17)

where E(t) is defined by

E(t) ≡ 13

∫

ΩφΩ(1− u)3 dx , E(0) =

13. (4.4.18)

Next, we compare E(t) with the solution F (t) of

dF

dt+ µΩF = R , F (0) =

13. (4.4.19)

Again, comparison principles and the definition of E yield

13

infΩ

(1− u)3 ≤ E(t) ≤ F (t) . (4.4.20)

For λ > λ2 we have that R < 0 in (4.4.17) and (4.4.19). For R < 0, we have that F = 0 atsome finite time t = T2. From (4.4.20), this implies that E = 0 at finite time. Thus, u hastouchdown at some finite-time T < T2. By calculating T2 explicitly, and by using (4.4.20), thetouchdown time T for (4.1.1) is found to satisfy

T ≤ T2 ≡ − 1µΩ

log[1− µΩ

3λ( ∫

ΩfφΩ dx

)−1]. (4.4.21)

Remark 4.4.2. It follows from the above that if λ > maxλ1 , λ2

, then

T ≤ minT0,λ, T1,λ, T2,λ

. (4.4.22)

where T0,λ is given by Theorem 4.2.3. We note that the three estimates on the touchdowntimes are not comparable. Indeed, it is clear that T0,λ is the better estimate when λ∗ < λ <min

λ1 , λ2

since T1,λ and T2,λ are not finite. On the other hand, our numerical simulations

show that T0,λ can be much worse than the others, for λ > maxλ1 , λ2

.

95


Here are now some numerical estimates of touchdown times for several choices of the domainΩ given by (4.3.2) and the exponential profile f(x) satisfying

(slab) : f(x) = eα(x2−1/4) (exponential) , (4.4.23a)

(unit disk) : f(x) = eα(|x|2−1) (exponential) , (4.4.23b)

where α ≥ 0.

[htb]Ω α T∗ T T0,λ T1,λ T2,λ

slab 0 1/60 0.01668 0.2555 0.0175 0.01825slab 1.0 1/60 0.02096 ≤ 0.3383 0.0229 0.02275slab 3.0 1/60 0.03239 ≤ 0.6121 0.0395 0.03588slab 6.0 1/60 0.06312 ≤ 1.7033 0.0973 0.07544

unit disk 0 1/60 0.01667 0.2420 0.0172 0.01745unit disk 0.5 1/60 0.02241 ≤ 0.4103 0.0289 0.02507unit disk 1.0 1/60 0.02927 ≤ 0.7123 0.0492 0.03579unit disk 3.0 1/60 0.09563 ≤ 8.9847 1.1614 0.15544

Table 4.1: Computations for finite touchdown time T with the bounds T∗, T0,λ, T1,λ and T2,λ

given in Proposition 4.4.3. Here the applied voltage λ = 20 and the profile is chosen as (4.4.23).

[htb]Ω T (λ = 5) T (λ = 10) T (λ = 15) (λ = 20)

slab 0.07495 0.03403 0.02239 0.01668unit disk 0.06699 0.03342 0.02235 0.01667

Table 4.2: Numerical values for finite touchdown time T at different applied voltages λ = 5,10, 15 and 20, respectively. Here the constant permittivity profile f(x) ≡ 1 is chosen.

In Table 2.1 of §2.2 we give numerical results for the saddle-node value λ∗ with the boundsλ, λ1 and λ2 given in Theorem 2.1.1, for the exponential permittivity profile chosen as (4.4.23).Following the numerical results of Table 2.1 of §2.2, here we can compute in Table 4.1 the valuesof finite touchdown time T at λ = 20, with the bounds T∗, T0,λ, T1,λ and T2,λ given in Theorem4.2.3 and Proposition 4.4.3. Using the meshpoints N = 800 we compute finite touchdown timeT with error less than 0.00001. The numerical results in Table 4.1 show that the bounds T1,λ

and T2,λ for T are much better than T0,λ. Further the bound T1,λ is better than T2,λ for smallervalues of α, and however the bound T2,λ is better than T1,λ for larger values of α. In fact, forα 1 and λ large enough we can deduce from (2.3.23) that

T1,λ ∼ 13λed1α , T2,λ ∼ d2

λα2 .

Here d1 = 1/4, d2 = 1/3π2 for the slab domain, and d1 = 1, d2 = 4/3z20 for the unit disk, where

z0 is the first zero of J0(z) = 0. Therefore, for α 1 and fixed λ large enough, the bound T2,λ

96

is better than T1,λ. Table 4.1 also shows that for fixed applied voltage λ, the touchdown timeis seen to increase once α is increased or equivalently the spatial extent where f(x) 1 isincreased. However, Theorem 4.4.2 tells us that for fixed permittivity profile f , by increasingthe applied voltage λ within the available power supply, the touchdown time can be decreasedand consequently the operating speed of MEMS devices can be improved. In Table 4.2 we givenumerical values for finite touchdown time T with error less than 0.00001, at different appliedvoltages λ = 5, 10, 15 and 20, respectively. Here the constant permittivity profile f(x) ≡ 1 ischosen and the meshpoints N = 800 again.

4.5 Asymptotic analysis of touchdown profiles

In this section, we discuss touchdown profiles by the method of asymptotic analysis, whichprovide some information on the refined touchdown rate studied in next Chapter.

4.5.1 Touchdown profile: f(x) ≡ 1

We first construct a local expansion of the solution near the touchdown time and touchdownlocation by adapting the method of [45] used for blow-up behavior. In the analysis of thissubsection we assume that f(x) ≡ 1 and touchdown occurs at x = 0 and t = T . In the absenceof diffusion, the time-dependent behavior of (4.1.1) is given by ut = λ(1 − u)−2. Integratingthis differential equation and setting u(T ) = 1, we get (1 − u)3 = −3λ(t − T ). This solutionmotivates the introduction of a new variable v(x, t) defined in terms of u(x, t) by

v =1

3λ(1− u)3 . (4.5.1)

A simple calculation shows that (4.1.1) transforms exactly to the following problem for v:

vt = ∆v − 23v|∇v|2 − 1 , x ∈ Ω , (4.5.2a)

v =1

3λ, x ∈ ∂Ω ; v =

13λ

, t = 0 . (4.5.2b)

Notice that u = 1 maps to v = 0. We will find a formal power series solution to (4.5.2a) nearv = 0.

As in [45] we look for a locally radially symmetric solution to (4.5.2) in the form

v(x, t) = v0(t) +r2

2!v2(t) +

r4

4!v4(t) + · · · , (4.5.3)

where r = |x|. We then substitute (4.5.3) into (4.5.2a) and collect coefficients in r. In thisway, we obtain the following coupled ordinary differential equations for v0 and v2:

v′0 = −1 +Nv2 , v

′2 = − 4

3v0v2

2 +(N + 2)

3v4 . (4.5.4)

We are interested in the solution to this system for which v0(T ) = 0, with v′0 < 0 and v2 > 0

for T − t > 0 with T − t 1. The system (4.5.4) has a closure problem in that v2 depends

97


on v4. However, we will assume that v4 v22/v0 near the singularity. With this assumption,

(4.5.4) reduces to

v′0 = −1 +Nv2 , v

′2 = − 4

3v0v2

2 . (4.5.5)

We now solve the system (4.5.5) asymptotically as t→ T− in a similar manner as was donein [45]. We first assume that Nv2 1 near t = T . This leads to v0 ∼ T − t, and the followingdifferential equation for v2:

v′2 ∼

−43(T − t)v

22 , as t→ T− . (4.5.6)

By integrating (4.5.6), we obtain that

v2 ∼ − 34[

log(T − t)] +B0[

log(T − t)]2+ · · · , as t→ T− , (4.5.7)

for some unknown constant B0. From (4.5.7), we observe that the consistency condition thatNv2 1 as t → T− is indeed satisfied. Substituting (4.5.7) into the equation (4.5.5) for v0,we obtain for t→ T− that

v′0 = −1 +N

(− 3

4[

log(T − t)] +B0[

log(T − t)]2+ · · ·

). (4.5.8)

Using the method of dominant balance, we look for a solution to (4.5.8) as t→ T− in the form

v0 ∼(T − t)+

(T − t)

[ C0[log(T − t)] +

C1[log(T − t)]2

+ · · ·], (4.5.9)

for some C0 and C1 to be found. A simple calculation yields that

v0 ∼(T − t)+

−3N(T − t)4| log(T − t)| +

−N(B0 − 3/4)(T − t)

| log(T − t)|2 + · · · , as t→ T− . (4.5.10)

The local form for v near touchdown is v ∼ v0 + r2v0/2. Using the leading term in v2 from(4.5.7) and the first two terms in v0 from (4.5.10), we obtain the local form

v ∼(T − t

)[1− 3N

4| log(T − t)| +3r2

8(T − t)| log(T − t)| + · · ·], (4.5.11)

for r 1 and t − T 1. Finally, using the nonlinear mapping (4.5.1) relating u and v, weconclude that

u ∼ 1−[3λ(T − t)

]1/3(1− 3N

4| log(T − t)| +3r2

8(T − t)| log(T − t)| + · · ·)1/3

. (4.5.12)

We note, as in [45], that if we use the local behavior v ∼ (T − t) + 3r2/[8| log(T − t)|], weget that

|∇v|2v∼[2

3| log(T − t)|+ 16(T − t)| log(T − t)|2

9r2

]−1. (4.5.13)

98

Hence, the term |∇v|2/v in (4.5.2a) is bounded for any r, even as t → T−. This allows usto use a simple finite-difference scheme to compute numerical solutions to (4.5.2). With thisobservation, we now perform a few numerical experiments on the transformed problem (4.5.2).For the slab domain, we define vmj for j = 1, . . . , N + 2 to be the discrete approximation tov(m∆t,−1/2+(j−1)h), where h = 1/(N + 1) and ∆t are the spatial and temporal mesh sizes,respectively. A second order accurate in space, and first order accurate in time, discretizationof (4.5.2) is

vm+1j = vmj + ∆t

((vmj+1 − 2vmj + vmj−1

)

h2−1−

(vmj+1 − vmj−1

)26vmj h2

), j = 2, . . . , N + 1 , (4.5.14)

with vm1 = vmN+2 =(

3λ)−1

form ≥ 0. The initial condition is v0j =

(3λ)−1

for j = 1, . . . , N+2.

The time-step ∆t is chosen to satisfy ∆t < h2/4 for the stability of the discrete scheme. Usingthis argument, one can compute numerical results of dynamic deflection u, see Figures 4.1-4.3of this Chapter.

4.5.2 Touchdown profile: variable permittivity

In this subsection we obtain some formal asymptotic results for touchdown behavior associatedwith a spatially variable permittivity profile in a slab domain. Suppose u is a touchdownsolution of (4.1.1) at finite time T , and let x = x0 be a touchdown point of u. With thetransformation

v =1

3λ(1− u)3 , (4.5.15)

the problem (4.1.1) for u in the slab domain transforms exactly to

vt = vxx − 23vv2x − f(x) , −1/2 < x < 1/2 , (4.5.16a)

v =1

3λ, x = ±1/2 ; v =

13λ

, t = 0 , (4.5.16b)

where f(x) is the permittivity profile.In order to discuss the touchdown profile of u near (x0, T ), we use the formal power series

method of §4.5.1 to locally construct a power series solution to (4.5.16) near touchdown pointx0 and touchdown time T . For this purpose, we look for a touchdown profile for (4.5.16), nearx = x0, in the form

v(x, t) = v0(t) +(x− x0)2

2!v2(t) +

(x− x0)3

3!v3(t) +

(x− x0)4

4!v4(t) + · · · . (4.5.17)

In order for v to be a touchdown profile, it is clear that we must require that

limt→T−

v0 = 0 , v0 > 0 , for t < T ; v2 > 0 , for t− T 1 . (4.5.18)

We first discuss the case where f(x) is analytic at x = x0 with f(x0) > 0. Therefore, forx− x0 1, f(x) has the convergent power series expansion

f(x) = f0 + f′0(x− x0) +

f′′0 (x− x0)2

2+ · · · , (4.5.19)

99


where f0 ≡ f(x0), f′0 ≡ f

′(x0), and f

′′0 ≡ f

′′(x0). Substituting (4.5.17) and (4.5.19) into

(4.5.16), we equate powers of x− x0 to obtain

v′0 = −f0 + v2 , (4.5.20a)

v′2 = −4v2

2

3v0+ v4 − f ′′0 , (4.5.20b)

v3 = f′0 . (4.5.20c)

We now assume that v2 1 and v4 1 as t→ T−. This yields that v0 ∼ f0(T − t), and

v′2 ∼ −

4v22

3f0(T − t) − f′′0 . (4.5.21)

For t→ T−, we obtain from a simple dominant balance argument that

v2 ∼ − 3f0

4[

log(T − t)] + · · · , as t→ T− . (4.5.22)

Substituting (4.5.22) into (4.5.20a), and integrating, we obtain that

v0 ∼ f0

(T − t)+

−3f0(T − t)4| log(T − t)| + · · · , as t→ T− . (4.5.23)

Next, we substitute (4.5.22), (4.5.23) and (4.5.20c) into (4.5.17), to obtain the local touchdownbehavior

v ∼ f0

(T − t)

[1− 3

4| log(T − t)| +3(x− x0)2

8(T − t)| log(T − t)| +f′0(x− x0)3

6f0(T − t) + · · ·], (4.5.24)

for (x− x0) 1 and t− T 1. Finally, using the nonlinear mapping (4.5.15) relating u andv, we conclude that

u ∼ 1−[3f0λ(T − t)

]1/3(1− 3

4| log(T − t)| +3(x− x0)2

8(T − t)| log(T − t)| +f′0(x− x0)3

6f0(T − t) + · · ·)1/3

.

(4.5.25)Here f0 ≡ f(x0) and f

′0 ≡ f

′(x0).

In the following, we exclude the possibility of f(x0) = 0 by using a formal power seriesanalysis. We discuss the case where f(x) is analytic at x = x0, with f(x0) = 0 and f

′(x0) = 0,

so that f(x) = f0(x− x0)2 + O((x− x0)3

)as x → x0 with f0 > 0. We then look for a power

series solution to (4.5.16) as in (4.5.17). In place of (4.5.20) for v3, we get v3 = 0, and

v′0 = v2 , v

′2 = −4v2

2

3v0+ v4 − 2f0 . (4.5.26)

Assuming that v4 1 as before, we can combine the equations in (4.5.26) to get

v′′0 = −

4(v′0

)2

3v0− 2f0 . (4.5.27)

100

By solving (4.5.27) with v0(T ) = 0, we obtain the exact solution

v0 = −3f0

11(T − t)2 < 0 , v2 =

6f0

11(T − t) . (4.5.28)

Since the criteria (4.5.18) are not satisfied, the form (4.5.28) does not represent a touchdownprofile centered at x = x0. Therefore, the above asymptotical analysis also shows that thepoint x = x0 satisfying f(x0) = 0 is not a touchdown point of u.

4.6 Pull-in distance

One of the primary goals in the design of MEMS devices is to maximize the pull-in distanceover a certain allowable voltage range that is set by the power supply. Here pull-in distancerefers to as the maximum stable deflection of the elastic membrane before touchdown occurs.In this section, we provide numerical results of pull-in distance with some explicit examples,from which one can observe that both larger pull-in distance and pull-in voltage can be achievedby properly tailoring the permittivity profile.

Following from [32], we focus on the dynamic solution u satisfying

∂u

∂t−∆u = − λf(x)

(1 + u)2for x ∈ Ω , (4.6.1a)

u(x, t) = 0 for x ∈ ∂Ω ; u(x, 0) = 0 for x ∈ Ω , (4.6.1b)

One can apply Theorem 4.1.1 that for λ ≤ λ∗, the dynamic solution u(x, t) of (4.6.1) globallyconverges to its unique maximal negative steady-state uλ(x). On the other hand, Theorem2.1.2 implies that the unique maximal negative steady-state uλ(x) is strictly increasing in λ.Therefore, we can deduce that pull-in distance of (4.6.1) is achieved exactly at λ = λ∗. Sincethe space dimension N of MEMS devices is 1 or 2, Theorems 2.1.2 & 4.1.1 give that the pull-indistance D of MEMS devices exactly satisfies

D := limt→∞ ‖ u

∗(x, t) ‖L∞(Ω)=‖ u∗(x) ‖L∞(Ω)≤ C(N) < 1 , N = 1, 2, (4.6.2)

where u∗(x, t) is the unique global solution of (4.6.1) at λ = λ∗, and while u∗(x) is the uniqueextremal steady-state of (4.6.1).

In order to understand the relationship between pull-in distance D and permittivity profilef(x), we first consider the steady-state of (4.6.1) satisfying

∆u =λf(x)

(1 + u)2in Ω,

−1 < u < 0 in Ω,u = 0 on ∂Ω ,

(4.6.3)

where the domain Ω is considered to be a slab or an unit disk defined by (4.3.2). Here we stillchoose the following permittivity profile f(x) as before:

slab : f(x) = |2x|α (power-law) ; f(x) = eα(x2−1/4) (exponential) , (4.6.4a)

unit disk : f(x) = |x|α (power-law) ; f(x) = eα(|x|2−1) (exponential) , (4.6.4b)

101


(a). Exponential Profiles: (b). Power-Law Profiles:Ω α λ∗

slab 0 1.401slab 3 2.637slab 6 4.848slab 10 10.40

unit disk 0 0.789unit disk 3 6.096unit disk 4.8 15.114unit disk 5.6 20.942

Ω α λ∗

slab 0 1.401slab 1 4.388slab 3 15.189slab 6 43.087

unit disk 0 0.789unit disk 1 1.775unit disk 5 9.676unit disk 20 95.66

Table 4.3: Numerical values for pull-in voltage λ∗: Table (a) corresponds to exponential profiles,while Table (b) corresponds to power-law profiles.

0:350:370:390:410:430:45

0:0 2:0 4:0 6:0 8:0 10:0 12:0 14:0 16:0ju(0)j

0:430:440:450:460:470:480:490:50

0:0 2:0 4:0 6:0 8:0 10:0 12:0 14:0 16:0ju(0)j

Figure 4.4: Plots of the pull-in distance |u(0)| = |u∗(0)| versus α for the power-law profile(heavy solid curve) and the exponential profile (solid curve). Left figure: the slab domain.Right figure: the unit disk.

with α ≥ 0. For above choices of domain Ω and profile f(x), since the extremal solution u∗(x)of (4.6.3) is unique, Lemma 2.6.1 shows that u∗(x) must be radially symmetric. Therefore, thepull-in distance D of (4.6.3) satisfies D = |u∗(0)|.

As in §2.2, using Newton’s method and COLSYS [2] to solve the boundary value problem(4.6.3), we first numerically calculate λ∗ of (4.6.3) as the saddle-node point. We give numericalvalues of λ∗ in Table 4.3(a) for exponential profiles and in Table 4.3(b) for power-law profiles,respectively. For the slab domain, in Figure 4.4(a) we plot D = |u(0)| = |u∗(0)| versus α forboth the power-law and the exponential conductivity profile f(x) in the slab domain, whichshow that the pull-in distance D can be increased by increasing the value of α (and hence byincreasing the range of f(x) 1). A similar plot of D = |u(0)| = |u∗(0)| versus α is shownin Figure 4.4(b) for the unit disk. For the power-law profile in the unit disk we observe that|u(0)| ≈ 0.444 for any α > 0. Therefore, rather curiously, the power-law profile does notincrease the pull-in distance for the unit disk. For the exponential profile we observe fromFigure 4.4(b) that the pull-in distance is not a monotonic function of α. The maximum valueoccurs at α ≈ 4.8 where λ∗ ≈ 15.11 (see Figure 2.1(b)) and D = |u(0)| = 0.485. For α = 0,

102


we have λ∗ ≈ 0.789 and |u(0)| = 0.444. Therefore, since λ∗ is proportional to V 2 (cf. §1.1.2)we conclude that the exponential permittivity profile for the unit disk can increase the pull-indistance by roughly 9% if the voltage is increased by roughly a factor of four.

0:50:40:30:20:10:0

0:0 0:2 0:4 0:6 0:8 1:0u

jxj 0:50:40:30:20:10:0

0:0 0:2 0:4 0:6 0:8 1:0u

jxjFigure 4.5: Left figure: plots of u versus |x| at λ = λ∗ for α = 0, α = 1, α = 3, and α = 10, inthe unit disk for the power-law profile. Right figure: plots of u versus |x| at λ = λ∗ for α = 0,α = 2, α = 4, and α = 10, in the unit disk for the exponential profile. In both figures thesolution develops a boundary-layer structure near |x| = 1 as α is increased.

For the unit disk, in Figure 4.5(a) we plot u versus |x| at λ = λ∗ with four values of α forthe power-law profile. Notice that u(0) is the same for each of these values of α. A similarplot is shown in Figure 4.5(b) for the exponential permittivity profile. From these figures, weobserve that u has a boundary-layer structure when α 1. In this limit, f(x) 1 except in anarrow zone near the boundary of the domain. For α 1 the pull-in distance D = |u(0)| alsoreaches some limiting value (see Figure 4.4 & 4.5). For the slab domain with an exponentialpermittivity profile, we remark that the limiting asymptotic behavior of |u(0)| for α 1 isbeyond the range shown in Figure 4.4(a).

0:00:10:20:30:4

0:00 0:05 0:10 0:15 0:20 0:25

0Figure 4.6: Bifurcation diagram of w

′(0) = −γ versus λ0 from the numerical solution of (4.6.6).

For α 1, we now use a boundary-layer analysis to determine a scaling law of λ∗ for bothtypes of permittivity profiles and for either a slab domain or the unit disk. We illustrate the

103

analysis for a power-law permittivity profile in the unit disk. For α 1, there is an outerregion defined by 0 ≤ r 1−O(α−1), and an inner region where r−1 = O(1/α). In the outerregion, where λrα 1, (4.6.3) reduces asymptotically to ∆u = 0. Therefore, the leading-orderouter solution is a constant u = A. In the inner region, we introduce new variables w and ρ by

w(ρ) = u (1− ρ/α) , ρ = α(1− r) . (4.6.5)

Substituting (4.6.5) into (4.6.3) with f(r) = rα, using the limiting behavior (1− ρ/α)α → e−ρ

as α→∞, and defining λ = α2λ0, we obtain the leading-order boundary-layer problem

w′′

=λ0e−ρ

(1 + w)2, 0 ≤ ρ <∞ ; w(0) = 0 , w

′(∞) = 0 , λ = α2λ0 . (4.6.6)

In terms of the solution to (4.6.6), the leading-order outer solution is u = A = w(∞).

050100150200250300

0 5 10 15 20

050100150200250300350

0 10 20 30 40

Figure 4.7: Comparison of numerically computed λ∗ (heavy solid curve) with the asymptoticresult (dotted curve) from (4.6.7) for the unit disk. Left figure: the exponential profile. Rightfigure: the power-law profile.

We define γ by w′(0) = −γ for γ > 0, and we solve (4.6.6) numerically using COLSYS [2] to

determine λ0 = λ0(γ). In Figure 4.6 we plot λ0(γ) and show that this curve has a saddle-nodepoint at λ0 = λ∗0 ≡ 0.1973. At this value, we compute w(∞) ≈ 0.445, which sets the limitingmembrane deflection for α 1. Therefore, (4.6.6) shows that for α 1, the saddle-nodevalue has the scaling law behavior λ∗ ∼ 0.1973α2 for a power-law profile in the unit disk. Asimilar boundary-layer analysis can be done to determine the scaling law of λ∗ when α 1for other cases. In each case we can relate λ∗ to the saddle-node value of the boundary-layerproblem (4.6.6). In this way, for α 1, we obtain

λ∗ ∼ 4(0.1973)α2 , λ2 ∼ 4α2

3, (power-law, slab), (exponential, unit disk) , (4.6.7a)

λ∗ ∼ (0.1973)α2 , λ2 ∼ α2

3, (power-law, unit disk), (exponential, slab) , (4.6.7b)

Notice that λ2 = O(α2), with a factor that is about 5/3 times as large as the multiplier of α2

in the asymptotic formula for λ∗. In Figure 4.7, we compare the computed λ∗ as a saddle-nodepoint with the asymptotic result of λ∗ from (4.6.7).

104


Next we present a few of numerical results for pull-in distance of dynamic problem (4.3.1)by applying the implicit Crank-Nicholson scheme again. Here we always consider the domainand the profile defined by (4.3.2) and (4.6.4), respectively. We choose the meshpoints N = 4000and the applied voltage λ = λ∗ given in Table 4.3:

Figure 4.8: Case of exponential profilesWe consider pull-in distance of (4.3.1) for exponential profiles in the slab or unit disk domain.In Figure 4.8(a) we plot u versus x at the time t = 80 in the slab domain, with α = 0 (solidline), α = 3 (dashed line), α = 6 (dotted line) and α = 10 (dash-dot line), respectively. Thisfigure and Figure 4.4(a) show that pull-in distance is increasing in α. In Figure 4.8(b) weplot u versus |x| at the time t = 80 in the unit disk domain, with α = 0 (dash-dot line),α = 3 (dashed line), α = 4.8 (dotted line) and α = 5.6 (solid line), respectively. In this figurewe observe that the solution develops a boundary-layer structure near the boundary of thedomain as α is increased, and pull-in distance is not a monotonic function of α. Actually fromFigure 4.4(b) we know that pull-in distance is first increasing and then decreasing in α. Themaximum value of pull-in distance occurs at α ≈ 4.8 and λ∗ ≈ 15.114.

−0.5 −0.25 0 0.25 0.5 −0.45

−0.4

−0.3

−0.2

−0.1

0 (a). Slab Domain

x

u(x, 80)

α = 0

α = 3 α = 6

α = 10

0 0.2 0.4 0.6 0.8 1 −0.5

−0.45

−0.4

−0.3

−0.2

−0.1

0(b). Unit Disk Domain

u(|x|, 80)

α = 0

α = 3

α = 4.8 α = 5.6

| x |

Figure 4.8: Left figure: plots of u versus x at λ = λ∗ in the slab domain. Right figure: plotsof u versus |x| at λ = λ∗ in the unit disk domain.

Figure 4.9: Case of power-law profilesWe consider pull-in distance of the membrane for power-law profiles in the slab or unit diskdomain. In Figure 4.9(a) we plot u versus x at the time t = 80 in the slab domain, with α = 0(solid line), α = 1 (dashed line), α = 3 (dash-dot line) and α = 6 (dotted line), respectively.This figure and Figure 4.4(a) show that pull-in distance is increasing in α. In Figure 4.9(b) weplot u versus |x| at the time t = 80 in the unit disk domain, with α = 0 (dotted line), α = 1(dash-dot line), α = 5 (dashed line) and α = 20 (solid line), respectively. For the power-lawprofiles in the unit disk domain, we observe that pull-in distance is a constant for any α ≥ 0.Therefore, with Figure 4.4(b), it is rather curious that power-law profile does not change pull-in distance in the unit disk domain. In both figures, the solution develops a boundary-layerstructure near the boundary of the domain as α in increased.

Since one of the primary goals of MEMS design is to maximize the pull-in distance overa certain allowable voltage range that is set by the power supply, it would be interesting to

105


−0.5 −0.25 0 0.25 0.5 −0.45

−0.4

−0.3

−0.2

−0.1

0 (a). Slab Domain

x

u(x, 80)

α = 0

α = 1

α = 3

α = 6

0 0.2 0.4 0.6 0.8 1 −0.5

−0.45

−0.4

−0.3

−0.2

−0.1

0

| x |

(b). Unit Disk Domain

u(|x|, 80)

α = 0

α = 1

α = 5

α = 20

Figure 4.9: Left figure: plots of u versus x at λ = λ∗ in the slab domain. Right figure: plotsof u versus |x| at λ = λ∗ in the unit disk domain.

formulate an optimization problem that computes a dielectric permittivity f(x) that maximizesthe pull-in distance for a prescribed range of the saddle-node threshold λ∗.

106

Chapter 5

Refined Touchdown Behavior

5.1 Introduction

In this Chapter, we continue the study of dynamic solutions of (1.2.30) in the form

ut −∆u = −λf(x)u2

for x ∈ Ω , (5.1.1a)

u(x, t) = 1 for x ∈ ∂Ω , (5.1.1b)u(x, 0) = 1 for x ∈ Ω , (5.1.1c)

where the permittivity profile f(x) is allowed to vanish somewhere, and will be assumed tosatisfy

f ∈ Cα(Ω) for some α ∈ (0, 1], 0 ≤ f ≤ 1 andf > 0 on a subset of Ω of positive measure.

(5.1.2)

We focus on the case where a unique solution u of (5.1.1) must touchdown at finite timeT = T (λ,Ω, f) in the sense

Definition 5.1.1. A solution u(x, t) of (5.1.1) is said to touchdown at finite time T =T (λ,Ω, f) if the minimum value of u reaches 0 at the time T <∞.

We shall give a refined description of finite-time touchdown behavior for u satisfying (5.1.1),including some touchdown estimates, touchdown rates, as well as some information on theproperties of touchdown set –such as compactness, location and shape.

This Chapter is organized as follows. The purpose of §5.2 is mainly to derive some a prioriestimates of touchdown profiles under the assumption that touchdown set of u is a compactsubset of Ω. In §5.2.1, we establish the following lower bound estimate of touchdown profiles.

Theorem 5.1.1. Assume f satisfies (5.1.2) on a bounded domain Ω, and suppose u is atouchdown solution of (5.1.1) at finite time T . If touchdown set of u is a compact subset of Ω,then

1. any point a ∈ Ω satisfying f(a) = 0 is not a touchdown point for u(x, t);

2. there exists a bounded positive constant M such that

M(T − t) 13 ≤ u(x, t) in Ω× (0, T ) . (5.1.3)

Note that whether the compactness of touchdown set holds for any f(x) satisfying (5.1.2) is aquite challenging problem. We shall prove in Proposition 5.2.1 of §5.2 that the compactness of

107

Chapter 5. Refined Touchdown Behavior

touchdown set holds for the case where the domain Ω is convex and f(x) satisfies the additionalcondition



Here ν is the outward unit norm vector to ∂Ω. On the other hand, when f(x) does not satisfy(5.1.4), the compactness of touchdown set is numerically observed, see Chapter 4 or §5.4.Therefore, it is our conjecture that under the convexity of Ω, the compactness of touchdownset may hold for any f(x) satisfying (5.1.2). In §5.2.2 we estimate the derivatives of touchdownsolution u, see Lemma 5.2.4; and as a byproduct, an integral estimate is also given in §5.2.2,see Theorem 5.2.5.

Motivated by Theorem 5.1.1, the key point of studying touchdown profiles is a similarityvariable transformation of (5.1.1). For the touchdown solution u = u(x, t) of (5.1.1) at finitetime T , we use the associated similarity variables


3wa(y, s) , (5.1.5)

where a is any interior point of Ω. Then wa(y, s) is defined in Wa := (y, s) : a + ye−s/2 ∈Ω, s > s′ = −logT, and it solves

ρ(wa)s −∇ · (ρ∇wa)− 13ρwa +

λρf(a+ ye−s2 )

w2a

= 0 ,

where ρ(y) = e−|y|2/4. Here wa(y, s) is always strictly positive in Wa. The slice of Wa at agiven time s1 is denoted by Ωa(s1) := Wa ∩ s = s1 = es

1/2(Ω − a). Then for any interiorpoint a of Ω, there exists s0 = s0(a) > 0 such that Bs := y : |y| < s ⊂ Ωa(s) for s ≥ s0. Wenow introduce the frozen energy functional

Es[wa](s) =12

∫

Bs

ρ|∇wa|2dy − 16

∫

Bs

ρw2ady −

∫

Bs

λρf(a)wa

dy . (5.1.6)

By estimating the energy Es[wa](s) inBs, one can establish the following upper bound estimate.

Theorem 5.1.2. Assume f satisfies (5.1.2) on a bounded domain Ω in RN , suppose u isa touchdown solution of (5.1.1) at finite time T and wa(y, s) is defined by (5.1.5). Assumetouchdown set of u is a compact subset of Ω. If wa(y, s) → ∞ as s → ∞ uniformly for|y| ≤ C, where C is any positive constant, then a is not a touchdown point for u.

Based on a prior estimates of §5.2, we shall establish refined touchdown profiles in §5.3,where self-similar method and center manifold analysis will be applied. Here is the statementof refined touchdown profiles:

Theorem 5.1.3. Assume f satisfies (5.1.2) on a bounded domain Ω in RN , and suppose uis a touchdown solution of (5.1.1) at finite time T . Assume touchdown set of u is a compactsubset of Ω, then,

1. If N = 1 and x = a is a touchdown point of u, then we have

limt→T−

u(x, t)(T − t)− 13 ≡ (3λf(a)

) 13 (5.1.7)

108


uniformly on |x− a| ≤ C√T − t for any bounded constant C. Moveover, when t→ T−,

u ∼ [3λf(a)(T − t)]1/3(

1− 14| log(T − t)| +

|x− a|28(T − t)| log(T − t)| + · · ·

), N = 1 .

(5.1.8)

2. If Ω = BR(0) ⊂ RN is a bounded ball with N ≥ 2, and f(r) = f(|x|) is radially symmetric.Suppose r = 0 is a touchdown point of u, then we have

limt→T−

u(r, t)(T − t)− 13 ≡ (3λf(0)

) 13 (5.1.9)

uniformly on r ≤ C√T − t for any bounded constant C. Moveover, when t→ T−,

u ∼ [3λf(0)(T − t)]1/3(

1− 12| log(T − t)| +

r2

4(T − t)| log(T − t)| + · · ·), N = 2 .

(5.1.10)

Note that the uniqueness of solutions for (5.1.1) gives the radial symmetry of u in Theorem5.1.3(2). When dimension N ≥ 2, it should remark from Theorem 5.1.3(2) that we are onlyable to discuss the refined touchdown profiles for special touchdown point x = 0 in the radialsituation, and it seems unknown for the general case.

Adapting various analytical and numerical techniques, §5.4 will be focused on the set oftouchdown points. This may provide useful information on the design of MEMS devices. In§5.4.1 we discuss the radially symmetric case of (5.1.1) as follows:

Theorem 5.1.4. Assume f(r) = f(|x|) satisfies (5.1.2) and f ′(r) ≤ 0 in a bounded ballBR(0) ⊂ RN with N ≥ 1, and suppose u is a touchdown solution of (5.1.1) at finite time T .Then, r = 0 is the unique touchdown point of u.

Remark 5.1.1. Assume f(r) = f(|x|) satisfies (5.1.2) and f ′(r) ≤ 0 in a bounded ball BR(0) ⊂RN with N ≥ 1. Together with Proposition 5.2.1 below, Theorems 5.1.1 and 5.1.4 show aninteresting phenomenon: finite-time touchdown point is not the zero point of f(x), but themaximum value point of f(x).

Remark 5.1.2. Numerical simulations in §5.4.1 show that the assumption f ′(r) ≤ 0 in Theorem5.1.4 is sufficient, but not necessary. This gives that Theorem 5.1.3(2) does hold for a largerclass of profiles f(r) = f(|x|).

For one dimensional case, Theorem 5.1.4 already implies that touchdown points must beunique when permittivity profile f(x) is uniform. In §5.4.2 we further discuss one dimensionalcase of (5.1.1) for varying profile f(x), where numerical simulations show that touchdownpoints may be composed of finite points or finite compact subsets of the domain.

5.2 A priori estimates of touchdown behavior

Under the assumption that touchdown set of u is a compact subset of Ω, in this section we studysome a priori estimates of touchdown behavior, and establish the claims in Theorems 5.1.1 and5.1.2. In §5.2.1 we establish a lower bound estimate, from which we complete the proof of

109

Theorem 5.1.1. Using the lower bound estimate, in §5.2.2 we shall prove some estimates forthe derivatives of touchdown solution u, and an integration estimate will be also obtained as abyproduct. In §5.2.3 we shall study the upper bound estimate by energy methods, which givesTheorem 5.1.2.

We first prove the following compactness result for a large class of profiles f(x) satisfying(5.1.2) and



Proposition 5.2.1. Assume f satisfies (5.1.2) and (5.2.1) on a bounded convex domain Ω,and suppose u is a touchdown solution of (5.1.1) at finite time T . Then, the set of touchdownpoints for u is a compact subset of Ω.

Proof: We prove Proposition 5.2.1 by adapting moving plane method from Theorem 3.3 in[27], where it is used to deal with blow-up problems. Take any point y0 ∈ ∂Ω, and assume forsimplicity that y0 = 0 and that the half space x1 > 0 (x = (x1, x

′)) is tangent to Ω at y0. LetΩ+α = Ω∩x1 > α where α < 0 and |α| is small, and also define Ω−α = (x1, x

′) : (2α−x1, x′) ∈

Ω+α , the reflection of Ω+

α with respect to the plane x1 = α, where x′ = (x2, · · · , xN ).Consider the function

w(x, t) = u(2α− x1, x′, t)− u(x1, x

′, t)

for x ∈ Ω−α , then w satisfies

wt −∆w =λ(u(x1, x

′, t) + u(2α− x1, x′, t))f(x)

u2(x1, x′, t)u2(2α− x1, x′, t)w .

It is clear that w = 0 on x1 = α. Since u(x, t) = 1 along ∂Ω and since the maximum principlegives ut < 0 for 0 < t < T , we may choose a small t0 > 0 such that

∂u(x, t0)∂ν

> 0 along ∂Ω , (5.2.2)

where ν is the outward unit norm vector to ∂Ω. Then for sufficiently small |α|, (5.2.2) impliesthat w(x, t0) ≥ 0 in Ω−α and also w = 1−u(x1, x

′, t) > 0 on (∂Ω−α ∩x1 < α)×(t0, T ). Applyingthe maximal principle we now conclude that w > 0 in Ω−α × (t0, T ) and ∂w

∂x1= −2 ∂u

∂x1< 0 on

x1 = α . Since α is arbitrary, it follows by varying α that

∂u

∂x1> 0 , (x, t) ∈ Ω+

α0× (t0, T ) (5.2.3)

provided |α0| = |α0(t0)| > 0 is sufficiently small.Fix 0 < |α0| ≤ δ, where δ is as in (5.2.1), we now consider the function

J = ux1 − ε1(x1 − α0) in Ω+α0× (t0, T ) ,

where ε1 = ε1(α0, t0) > 0 is a constant to be determined later. The direct calculations showthat

Jt −∆J =2λfu3

ux1 −λfx1

u2=

2λfu3

ux1 −λ

u2

∂f

∂ν

∂ν

∂x1≥ 0 in Ω+

α0× (t0, T ) (5.2.4)

110

due to (5.2.1). Therefore, J can not attain negative minimum in Ω+α0× (t0, T ). Next, J > 0 on

x1 = α0 by (5.2.3). Since (5.2.2) gives ∂u(x,t0)∂x1

≥ C > 0 along (∂Ω+α0∩ ∂Ω) for some C > 0,

we have J > 0 on t = t0 provided ε1 = ε1(α0, t0) > 0 is sufficiently small. We now claimthat for small ε1 > 0,

J > 0 on (∂Ω+α0∩ ∂Ω)× (t0, T ) . (5.2.5)

To prove (5.2.5), we compare the solution U := 1− u satisfying

Ut −∆U =λf(x)

(1− U)2(x, t) ∈ Ω× (t0, T ),

U(x, t0) = 1 − u(x, t0) ; U(x, t) = 0 x ∈ ∂Ω

with the solution v of the heat equation

vt = ∆v , (x, t) ∈ Ω× (t0, T ) ,

where 0 ≤ v(x, t0) = U(x, t0) < 1 and v = 0 on ∂Ω. Then we have U ≥ v in Ω × (t0, T ).Consequently,

∂U

∂ν≤ ∂v

∂ν≤ −C0 < 0 on (∂Ω+

α0∩ ∂Ω)× (t0, T ) ,

and hence ∂u∂ν ≥ C0 > 0 on (∂Ω+

α0∩∂Ω)×(t0, T ). It then follows that J ≥ C0

∂ν∂x1−ε1(x1−α0) > 0

provided ε1 = ε1(α0, t0) is small enough, which gives (5.2.5).The maximum principle now yields that there exists ε1 = ε1(α0, t0) > 0 so small that J ≥ 0

in Ω+α0× (t0, T ), i.e.,

ux1 ≥ ε1(x1 − α0) , (5.2.6)

if x′ = 0 and α0 ≤ x1 < 0. Integrating (5.2.6) with respect to x1 on [α0, y1], where α0 < y1 < 0,yields that

u(y1, 0, t)− u(α0, 0, t) ≥ ε1

2|y1 − α0|2.

It follows thatlimt→T−u(0, t) = limt→T− lim

y1→0−u(y1, 0, t) ≥ ε1α

20/2 > 0,

which shows that y0 = 0 can not be a touchdown point of u(x, t).The proof of (5.2.3) can be slightly modified to show that ∂u

∂ν > 0 in Ω+α0× (t0, T ) for any

direction ν close enough to the x1-direction. Together with (5.2.1), this enables us to deducethat any point in x′ = 0, α0 < x1 < 0 can not be a touchdown point. Since above proofshows that α0 can be chosen independently of initial point y0 on ∂Ω, by varying y0 along ∂Ωwe deduce that there is an Ω−Neighborhood Ω′ of ∂Ω such that each point x ∈ Ω′ can not bea touchdown point. This completes the proof of Proposition 5.2.1.

Remark 5.2.1. When f(x) does not satisfy (5.2.1), the compactness of touchdown set is nu-merically observed, see numerical simulations in Chapter 4 or in §5.4 of the present paper.Therefore, it is our conjecture that under the convexity of Ω, the compactness of touchdownset may hold for any f(x) satisfying (5.1.2).

111

5.2.1 Lower bound estimate

Define for η > 0,

Ωη := x ∈ Ω : dist(x, ∂Ω) > η , Ωcη := x ∈ Ω : dist(x, ∂Ω) ≤ η . (5.2.7)

Since touchdown set of u is assumed to be a compact subset of Ω, in the rest of this sectionwe may choose a small η > 0 such that any touchdown point of u must lie in Ωη. Our firstaim of this subsection is to prove that any point x0 ∈ Ωη satisfying f(x0) = 0 can not be atouchdown point of u at finite time T , which then leads to the following proposition.

Proposition 5.2.2. Assume f satisfies (5.1.2) on a bounded domain Ω, and suppose u(x, t)is a touchdown solution of (5.1.1) at finite time T . If touchdown set of u is a compact subsetof Ω, then any point x0 ∈ Ω satisfying f(x0) = 0 cannot be a touchdown point of u(x, t).

Proof: Since touchdown set of u is assumed to be a compact subset of Ω, it now suffices todiscuss the point x0 lying in the interior domain Ωη for some small η > 0, such that there isno touchdown point on Ωc

η.For any t1 < T , we first recall that the maximum principle gives ut < 0 for all (x, t) ∈

Ω × (0, t1). Further, the boundary point lemma shows that the outward normal derivativeof v = ut on ∂Ω is positive for t > 0. This implies that for taking small 0 < t0 < T , thereexists a positive constant C = C(t0, η) such that ut(x, t0) ≤ −C < 0 for all x ∈ Ωη. For any0 < t0 < t1 < T , we next claim that there exists ε = ε(t0, t1, η) > 0 such that

J ε(x, t) = ut +ε

u2≤ 0 for all (x, t) ∈ Ωη × (t0, t1) . (5.2.8)

Indeed, it is now clear that there exists Cη = Cη(t0, t1, η) > 0 such that ut(x, t) ≤ −Cη on theparabolic boundary of Ωη × (t0, t1). And further, we can choose ε = ε(t0, t1, η) > 0 so smallthat J ε ≤ 0 on the parabolic boundary of Ωη × (t0, t1), due to the local boundedness of 1

u2 on∂Ωη × (t0, t1). Also, direct calculations imply that

J εt −∆J ε =2λfu3

J ε − 6ε|∇u|2u4

≤ 2λfu3

J ε .

Now (5.2.8) follows again from the maximum principle.Combining (5.2.8) and (5.1.1) we deduce that for a small neighborhood B of x0 where

λf(x) ≤ ε/2 is in B ⊂ Ωη, we have for v := 1− u,

∆v ≥ ε

21

(1− v)2, (x, t) ∈ B × (t0, t1) .

Now Proposition 5.2.2 is a direct result of Lemma 4.3.2, since t1 < T is arbitrary. Proof of Theorem 5.1.1: In view of Proposition 5.2.2, it now needs only to prove the lowerbound estimate (5.1.3).

Given any small η > 0, applying the same argument used for (5.2.8) yields that for any0 < t0 < t1 < T , there exists ε = ε(t0, t1, η) > 0 such that

ut ≤ − ε

u2in Ωη × (t0, t1).

112

This inequality shows that ut → −∞ as u touchdown, and there exists M > 0 such that

M1(T − t) 13 ≤ u(x, t) in Ωη × (0, T ) (5.2.9)

due to the arbitrary of t0 and t1, where M1 depends only on λ, f and η. Furthermore, one canobtain (5.1.3) because of the boundedness of u on Ωc

η., and the theorem is proved.

5.2.2 Gradient estimates

As a preliminary of next section, it is now important to know a priori estimates for thederivatives of touchdown solution u, which are the contents of this subsection. Following theanalysis in [27], our first lemma is about the derivatives of first order without the compactnessassumption of touchdown set.

Lemma 5.2.3. Assume f satisfies (5.1.2) on a bounded convex domain Ω, and suppose u is atouchdown solution of (5.1.1) at finite time T . Then for any 0 < t0 < T , there exists a boundedconstant C > 0 such that

12|∇u|2 ≤ C

u− C

uin Ω× (0, t0) , (5.2.10)

where u = u(t0) = minx∈Ω u(x, t0), and C depends only on λ, f and Ω.

Proof: Fix any 0 < t0 < T and treat u(t0) as a fixed constant. Let w = u−u, then w satisfies

wt −∆w = − λf(x)(w + u)2

in Ω× (0, t0) ,

w = 1− u in ∂Ω× (0, t0) ,w(x, 0) = 1− u in Ω .

We introduce the functionP =

12|∇w|2 +

C

w + u− C

u, (5.2.11)

where the bounded constant C ≥ 2λ supx∈Ω f will be determined later. Then we have

Pt −∆P =Cλf(x)(w + u)4

− λ∇f(x)∇w(w + u)2

+2(λf(x)− C)|∇w|2

(w + u)3−

N∑

i,j=1

w2ij

≤ λC supx∈Ω f

(w + u)4+−2λ|∇w|2 supx∈Ω f + λ|∇w| supx∈Ω |∇f |

(w + u)3−

N∑

i,j=1

w2ij

≤ λ(C supx∈Ω f + C1)(w + u)4

−N∑

i,j=1

w2ij ,

(5.2.12)

where C1 := (supx∈Ω |∇f |)2

8 supx∈Ω f≥ 0 is bounded. Since (5.2.11) gives

N∑

i=1

(Pi +

C

(w + u)2wi)2 =

N∑

i,j=1

(wjwij)2 ≤ |∇w|2N∑

i,j=1

w2ij , (5.2.13)

113

we now take

C := max

2λ supx∈Ω

f,λ supx∈Ω f + λ

√(supx∈Ω f)2 + 4C1

2

≥ 2λ sup

x∈Ω

f

so that C2 ≥ λ(C supx∈Ω f + C1), where C clearly depends only on λ, f and Ω. From thechoice of C, a combination of (5.2.12) and (5.2.13) gives that

Pt −∆P ≤ −→b · ∇P ,

where−→b = −|∇w|−2(∇P + 2C∇w

(w+u)2 ) is a locally bounded when ∇w 6≡ 0. Therefore, P can onlyattain positive maximum either at the point where ∇w = 0, or on the parabolic boundary ofΩ× (0, t0). But when ∇w = 0, we have P ≤ 0.

On the initial boundary, P = C1+u − C

u < 0 . Let (y, s) be any point on ∂Ω × (0, t0), if wecan prove that

∂P

∂ν≤ 0 at (y, s) , (5.2.14)

it then follows from the maximum principle that P ≤ 0 in Ω × (0, t0). And therefore, theassertion (5.2.10) is reduced from (5.2.11) together with w = u− u.

To prove (5.2.14), we recall the fact that since w = const. on ∂Ω (for t = s), we have

∆w = wνν + (N − 1)κwν at (y, s) ,

where κ is the non-negative mean curvature of ∂Ω at y. It then follows that

∂P

∂ν= wνwνν − Cwν

(w + u)2≤ wν

[∆w − (N − 1)κwν − λf(x)

(w + u)2

]

= wν [wt − (N − 1)κwν ] = −(N − 1)κw2ν ≤ 0

at (y, s), and we are done. The following lemma is dealt with the derivatives of higher order, and the idea of its proof

is similar to Proposition 1 of [35].

Lemma 5.2.4. Assume f satisfies (5.1.2) on a bounded domain Ω, and suppose u is a touch-down solution of (5.1.1) at finite time T . Assume touchdown set of u is a compact subset ofΩ, and x = a is any point of Ωη for some small η > 0. Then there exists a positive constantM ′ such that

|∇mu(x, t)|(T − t)− 13

+m2 ≤M ′ , m = 1, 2 (5.2.15)

holds for |x− a| ≤ R.

Proof: It suffices to consider the case a = 0 by translation, and we may focus on 12R

2 < r2 <R2 and denote Qr = Br × (T [1− ( rR)2], T ).

Our first task is to show that |∇u| and |∇2u| are uniformly bounded on compact subsets ofQR. Indeed, since f(x)/u2 is bounded on any compact subset D of QR, standard Lp estimatesfor heat equations (cf. [47]) gives

∫ ∫

D(|∇2u|p + |ut|p)dxdt < C , 1 < p <∞ .

114

Choosing p to be large enough, we then conclude from Sobolev’s inequality that f(x)/u2 isHolder continuous on D. Therefore, Schauder’s estimates for heat equations (cf. [47]) showthat |∇u| and |∇2u| are uniformly bounded on compact subsets of D. In particular, thereexists M1 such that

|∇u|+ |∇2u| ≤M1 for (x, t) ∈ Br ×(T [1− (

r

R)2], T [1− 1

2(1− r

R)2]), (5.2.16)

where M1 depends only on R, N and M given in (5.1.3).We next prove (5.2.15) for |x| < r and T [1 − 1

2(1 − rR)2] ≤ t < T . Fix such a point (x, t),

let µ = [ 2T (T − t)] 1

2 and consider

v(z, τ) = µ−23u(x+ µz, T − µ2(T − τ)) . (5.2.17)

For above given point (x, t), we now define O := z : (x+µz) ∈ Ω and g(z) := f(x+µz) ≥ 0on O. One can verify that v(z, τ) is a solution of

vτ −∆zv = −λg(z)v2

z ∈ O,v(z, 0) = v0(z) > 0 ; v(z, τ) = µ−

23 z ∈ ∂O ,

(5.2.18)

where ∆z denotes the Laplacian operator with respect to z, and v0(z) = µ−23u(x + µz, T −

µ2T ) > 0 satisfies ∆zv0 − λg(z)v20≤ 0 on O. The formula (5.2.17) implies that T is also the

finite touchdown time of v, and the domain of v includes Qr0 for some r0 = r0(R) > 0. Sincetouchdown set of u is assumed to be a compact subset of Ω, one can observe that touchdownset of v is also a compact subset of O. Therefore, the argument of Theorem 5.1.1(2) can beapplied to (5.2.18), yielding that there exists a constant M2 > 0 such that

v(z, τ) ≥M2(T − τ)13

where M2 depends only on R, λ, f and Ω again. The argument used for (5.2.16) then yieldsthat there exists M ′1 > 0, depending on R, N and M2, such that

|∇zv|+ |∇2zv| ≤M ′1 for (z, τ) ∈ Br ×

(T [1− (

r

r0)2], T [1− 1

2(1− r

r0)2]), (5.2.19)

where we assume 12r

20 < r2 < r2

0. Applying (5.2.17) and taking (z, τ) = (0, T2 ), this estimatereduces to

µ−23

+1|∇u|+ µ−23

+2|∇2u| ≤M ′1 .Therefore, (5.2.15) follows since µ = [ 2

T (T − t)] 12 .

Before concluding this subsection, we now apply gradient estimates to establishing integralestimates.

Theorem 5.2.5. Assume f satisfies (5.1.2) on a bounded domain Ω, and suppose u is atouchdown solution of (5.1.1) at finite time T . Assume touchdown set of u is a compact subsetof Ω, then for γ > 3

2N we have

limt→T−∫

Ωf(x)u−γ(x, t)dx = +∞ .

115

Proof: For any given t0 ∈ (0, T ) close to T , Lemma 5.2.3 implies that

12|∇u|2 ≤ C

u2(u− u) in Ω× (0, t0) (5.2.20)

for some bounded constant C > 0, where u = u(x0, t0) = minx∈Ωu(x, t0). Considering any tsufficiently close to t0, we now introduce polar coordinates (r, θ) about the point x0. Then inany direction θ, there is a smallest value of r0 = r0(θ, t) such that u(r0, t) = 2u. Note that r0

is very small as t < t0 sufficiently approach to T . Furthermore, since x0 approaches to one oftouchdown points of u as t→ T−, Proposition 5.2.2 shows that as t < t0 sufficiently approachto T , we have f(x) ≥ C0 > 0 in r < r0 for some C0 > 0. Since (5.2.20) and the definition

of u imply that ur√u−u ≤

√2Cu , which is 2

√u− u ≤

√2Cu r, we attain

√2Cu

3/2 ≤ r0 by taking

r = r0. Therefore, for γ > 32N we have

∫

Ωu−γdx ≥ C

∫

Ωf(x)u−γdx ≥ CC0

∫

r≤r0u−γdx ≥ C

∫

θdSθ

∫

r≤r0u−γrN−1dr

≥ C∫

θdSθ

∫

r≤r0(2u)−γrN−1dr

≥ C∫

θdSθu

−γrN0 ≥ C∫

θdSθu

−γ+ 32N = +∞

as t→ T−, which completes the proof of Theorem 5.2.5.

5.2.3 Upper bound estimate

In this subsection, we discuss the upper bound estimate of touchdown solution u by applyingenergy methods.

First, we note the following local upper bound estimate.

Proposition 5.2.6. Suppose u is a touchdown solution of (5.1.1) at finite time T . Then, thereexists a bounded constant C = C(λ, f,Ω) > 0 such that

minx∈Ω

u(x, t) ≤ C(T − t) 13 for 0 < t < T . (5.2.21)

Proof: SetU(t) = min

x∈Ωu(x, t), 0 < t < T ,

and let U(ti) = u(xi, ti) (i = 1, 2) with h = t2 − t1 > 0. Then,

U(t2)− U(t1) ≤ u(x1, t2)− u(x1, t1) = hut(x1, t1) + o(h),

U(t2)− U(t1) ≥ u(x2, t2)− u(x2, t1) = hut(x2, t2) + o(h).

It follows that U(t) is lipschitz continuous. Hence, for t2 > t1 we have

U(t2)− U(t1)t2 − t1 ≥ ut(x2, t2) + o(1).

116

On the other hand, since ∆u(x2, t2) ≥ 0 we obtain,

ut(x2, t2) ≥ − λf(x2)u2(x2, t2)

= −λf(x2)U2(t2)

≥ − C

U2(t2)for 0 < t2 < T .

Consequently, at any point of differentiability of U(t), it deduces from above inequalities that

U2Ut ≥ −C a.e. t ∈ (0, T ) . (5.2.22)

Integrating (5.2.22) from t to T we obtain (5.2.21). For the touchdown solution u = u(x, t) of (5.1.1) at finite time T , we now introduce the

associated similarity variables


3wa(y, s) , (5.2.23)

where a is any point of Ωη for some small η > 0. Then wa(y, s) is defined in

Wa := (y, s) : a+ ye−s/2 ∈ Ω, s > s′ = −logT ,

and it solves∂

∂swa −∆wa +

12y · ∇wa − 1

3wa +

λf(a+ ye−s2 )

w2a

= 0 . (5.2.24)

Here wa(y, s) is always strictly positive in Wa. Note that the form of wa defined by (5.2.23) ismotivated by Theorem 5.1.1 and Proposition 5.2.6. The slice of Wa at a given time s1 will bedenoted by Ωa(s1):

Ωa(s1) := Wa ∩ s = s1 = es1/2(Ω− a) .

Then for any a ∈ Ωη, there exists s0 = s0(η, a) > 0 such that

Bs := y : |y| < s ⊂ Ωa(s) for s ≥ s0 . (5.2.25)

From now on, we often suppress the subscript a, writing w for wa, etc.In view of (5.2.23), one can combine Theorem 5.1.1 and Lemma 5.2.4 to reaching the

following estimates on w = wa:

Corollary 5.2.7. Assume f satisfies (5.1.2) on a bounded domain Ω, and suppose u is atouchdown solution of (5.1.1) at finite time T . Assume touchdown set of u is a compact subsetof Ω, then the rescaled solution w = wa satisfies

M ≤ w ≤ e s3 , |∇w|+ |∆w| ≤M ′ in W,

where M is a constant as in Theorem 5.1.1 and while M ′ is a constant as in Lemma 5.2.4.Moreover, it satisfies

M ≤ w(y1, s) ≤ w(y2, s) +M ′|y2 − y1|for any (yi, s) ∈W , i = 1, 2.

117

We now rewrite (5.2.24) in divergence form:

ρws −∇ · (ρ∇w)− 13ρw +

λρf(a+ ye−s2 )

w2= 0 , (5.2.26)

where ρ(y) = e−|y|2/4. We also introduce the frozen energy functional

Es[w](s) =12

∫

Bs

ρ|∇w|2dy − 16

∫

Bs

ρw2dy −∫

Bs

λρf(a)w

dy , (5.2.27)

which is defined in the compact set Bs of Ωa(s) for s ≥ s0.

Lemma 5.2.8. Assume f satisfies (5.1.2) on a bounded domain Ω, and suppose u is a touch-down solution of (5.1.1) at finite time T . Assume touchdown set of u is a compact subset ofΩ, then the rescaled solution w = wa satisfies

12

∫

Bs

ρ|ws|2dy ≤ − d

dsEs[w](s) + gη(s) for s ≥ s0 , (5.2.28)

where gη(s) is positive and satisfies∫∞s0gη(s)ds <∞.

Proof: Multiply (5.2.26) by ws and use integration by parts to get

∫

Bs

ρ|ws|2dy =∫

Bs

ws∇(ρ∇w)dy +13

∫

Bs

ρwwsdy −∫

Bs

λρwsf(a+ ye−s2 )

w2dy

= −12

∫

Bs

d

ds|∇w|2ρdy +

∫

Bs

d

ds

(16w2 +

λf(a)w

)ρdy

+∫

∂Bs

ρws∂w

∂νdS +

∫

Bs

λρws[f(a)− f(a+ ye−s2 )]

w2dy

= − d

dsEs[w](s) +

∫

∂Bs

ρws∂w

∂νdS +

12s

∫

∂Bs

ρ|∇w|2(y · ν)dS

−1s

∫

∂Bs

ρ(1

6w2 +

λf(a)w

)(y · ν)dS +

∫

Bs


w2dy

≤ − d

dsEs[w](s) +

∫

∂Bs

ρws∂w

∂νdS +

12s

∫

∂Bs

ρ|∇w|2(y · ν)dS

+∫

Bs


w2dy

:= − d

dsEs[w](s) + I1 + I2 + I3 ,

(5.2.29)where ν is the exterior unit norm vector to ∂Ω and dS is the surface area element. Thefollowing formula is applied in the third equality of (5.2.29): if g(y, s) : W 7→ R is a smooth

118

function, then

d

ds

∫

Bs

g(y, s)dy =d

ds

∫

B1

g(sz, s)sNdz

= N

∫

B1

g(sz, s)sN−1dz +∫

B1

gs(sz, s)sNdz +∫

B1

(∇yg · z)sNdz

=∫

Bs

gs(y, s)dy +N

∫

Bs

g(y, s)dy

s+∫

Bs

(∇g · ys

)dy

=∫

Bs

gs(y, s)dy +1s

∫

∂Bs

g(y, s)(y · ν)dS .

For s ≥ s0, we next estimate integration terms I1, I2 and I3 as follows:Considering |y| ≤ S in Bs, Corollary 5.2.7 gives

|ws| = |∆w − 12y · ∇w +

13w − λf(a+ ye−

s2 )

w2| ≤ C(1 + |y|) +

13w ≤ C1s+

13es3 ,

which implies

I1 ≤ CsN−1e−s2

4(C1s+

13es3) ≤ C2s

Ne−s2

4+ s

3 . (5.2.30)

It is easy to observe that

I2 ≤ C3sN−1e−

s2

4 . (5.2.31)

As for I3, since w has a lower bound and since f(x) ∈ Cα(Ω) for some α ∈ (0, 1], we applyYoung’s inequality to deduce

I3 ≤ Ce−α2s

∫

Bs

ρ|y|αwsdy ≤ Ce−α2s[ε

∫

Bs

ρw2sdy + C(ε)

∫

Bs

ρ|y|2αdy],

where the constant ε > 0 is arbitrary. Because e−α2s < ∞, one can take sufficiently small ε

such thatI3 ≤ 1

2

∫

Bs

ρw2sdy + C4e

−α2s . (5.2.32)

Combining (5.2.29)− (5.2.32) then yields

12

∫

Bs

ρ|ws|2dy ≤ − d

dsEs[w](s) + C1s

Ne−s2

4+ s

3 + C2e−α

2s

:= − d

dsEs[w](s) + gη(s) ,

where gη(s) is positive and satisfies∫∞s0gη(s)ds <∞, and we are done.

Remark 5.2.2. Supposing the convexity of Ω, one can establish an energy estimate in the wholedomain Ωa(s):

∫

Ωa(s)ρ|ws|2dy ≤ − d

dsEΩa(s)[w](s) +Kη(s) for s ≥ s0 , (5.2.33)

119

where Kη(s) is positive and satisfies∫∞s0Kη(s)ds <∞, and EΩa(s)[w](s) is defined by

EΩa(s)[w](s) =12

∫

Ωa(s)ρ|∇w|2dy − 1

6

∫

Ωa(s)ρw2dy −

∫

Ωa(s)

λρf(a)w

dy . (5.2.34)

However, by estimating the energy functional Es[w](s) in Bs, instead of Ωa(s), it is sufficientto obtain the desirable upper bound estimate of w, see Theorem 5.2.10 below.

The following lemma is also necessary for establishing the desirable upper bound estimate.

Lemma 5.2.9. Assume f satisfies (5.1.2) on a bounded domain Ω, and suppose u is a touch-down solution of (5.1.1) at finite time T . Assume touchdown set of u is a compact subset ofΩ, and a is any point of Ωη for some η > 0. Then there exists a constant ε > 0, dependingonly on λ, f and Ω, such that if

u(x, t)(T − t)− 13 ≥ ε (5.2.35)

for all (x, t) ∈ Qδ := (x, t) : |x− a| < δ, T − δ < t < T, then a is not a touchdown point foru. Here δ > 0 is an arbitrary constant.

Proof: Setting v(x, t) = 1u(x,t) , then v(x, t) blows up at finite time T , and v satisfies

vt −∆v = −2|∇v|2v

+ λf(x)v4 ≤ K(1 + v4) in Qδ , (5.2.36)

where K := λ supx∈Ω f(x) > 0. We now apply Theorem 2.1 of [37] to (5.2.36), which givesthat there exists a constant 1

ε > 0, depending only on λ, f and Ω, such that if

v(x, t) ≤ 1ε

(T − t)− 13 in Qδ ,

then a is not a blow-up point for v, and hence (5.2.35) follows.

Theorem 5.2.10. Assume f satisfies (5.1.2) on a bounded domain Ω, and suppose u is atouchdown solution of (5.1.1) at finite time T . Assume touchdown set of u is a compact subsetof Ω, and a is any point of Ωη for some η > 0. If wa(y, s) → ∞ as s → ∞ uniformly for|y| ≤ C, where C is any positive constant, then a is not a touchdown point for u.

Proof: We first claim that if wa(y, s)→∞ as s→∞ uniformly for |y| ≤ C, then

Es[wa](s)→ −∞ as s→∞ . (5.2.37)

Indeed, it is obvious from Corollary 5.2.7 that the first term and the third term in Es[wa](s)are uniformly bounded. As for the second term, we can write

∫

Bs

ρw2dy =∫

BC

ρw2dy +∫

Bs\BCρw2dy ≥

∫

BC

ρw2dy .

Since wa → ∞ as s → ∞ uniformly on BC , we have∫BC

ρw2dy → ∞ as s → ∞, which gives−1

6

∫BC

ρw2dy → −∞ as s→∞, and hence (5.2.37) follows.

120

Let K be a large positive constant to be determined later. Then (5.2.37) implies that thereexists an s such that Es[wa](s) ≤ −4K. Using the same argument as in [36], it is easy to showthat for any fixed s, Es[wa](s) varies smoothly with a ∈ Ω. Therefore, there exists an r0 > 0such that

Es[wb](s) ≤ −3K for |b− a| < r0 .

Since touchdown set of u is assumed to be a compact subset of Ω, we have dist(a, ∂Ω) > η forsome η > 0. Therefore, it now follows from Lemma 5.2.8 that

Es[wb](s) ≤ −2K for |b− a| < r0 , s ≥ s

provided K ≥ M1 :=∫∞s0gη(s)ds, where gη(s) is as in Lemma 5.2.8. Since the first term and

the third term in Es[wb](s) are uniformly bounded, we have∫

Bs

ρw2bdy ≥ 6K for |b− a| < r0 , s ≥ s . (5.2.38)

Recalling from Corollary 5.2.7,

w2b (y, s) ≤ 2

(w2b (0, s) +M ′2|y|2) ,

we obtain from (5.2.38) that

3K ≤ w2b (0, s)

∫

Bs

ρdy +M ′2∫

Bs

ρ|y|2dy ≤ C1w2b (0, s) + C2 .

We now choose K ≥ maxM1,23C2 so large that

wb(0, s) ≥√

3K2C1

:= ε . (5.2.39)

Setting t := T − e−s, it reduces from (5.2.39) that

u(b, t)(T − t)− 13 ≥ ε for |b− a| < r0 , t < t < T .

Applying Lemma 5.2.9 with a small r0, we finally conclude that a is not a touchdown pointfor u, and the theorem is proved.

5.3 Refined touchdown profiles

In this section we first establish touchdown rates by applying self-similar method [35]. Thenthe refined touchdown profiles for N = 1 and N = 2 will be separately derived by usingcenter manifold analysis of a PDE [25], which will be discussed for N = 1 in §5.3.1 and forN ≥ 2 in §5.3.2, respectively. It should be pointed out that for N = 1 we may establish therefined touchdown profiles for any touchdown point, see Theorem 5.3.3; while for N ≥ 2, weare only able to deal with the refined touchdown profiles in the radial situation for the specialtouchdown point r = 0, see Theorem 5.3.5. Throughout this section and unless mentionedotherwise, touchdown set for u is assumed to be a compact subset of Ω, and a is always

121

assumed to be any touchdown point of u. Therefore, all a priori estimates of last section canbe adapted here.

Our starting point of studying touchdown profiles is a similarity variable transformation of(5.1.1). For the touchdown solution u = u(x, t) of (5.1.1) at finite time T , as before we use theassociated similarity variables


3w(y, s) , (5.3.1)

where a is any touchdown point of u. Then w(y, s) is defined in W = (y, s) : |y| < Res/2, s >s′ = −logT, where R = max|x− a| : x ∈ Ω, and it solves

ws − 1ρ∇(ρ∇w)− 1

3w +

λf(a+ ye−s2 )

w2= 0 (5.3.2)

with ρ(y) = e−|y|2/4, where f(a) > 0 since a is assumed to be a touchdown point. Therefore,studying touchdown behavior of u is equivalent to studying large time behavior of w.

Lemma 5.3.1. Suppose w is a solution of (5.3.2). Then, w(y, s)→ w∞(y) as s→∞ uniformlyon |y| ≤ C, where C > 0 is any bounded constant, and w∞(y) is a bounded positive solution of

∆w − 12y · ∇w +

13w − λf(a)

w2= 0 in RN , (5.3.3)

where f(a) > 0.

Proof: We adapt the arguments from the proofs of Propositions 6 and 7 in [35]: let sj be asequence such that sj →∞ and sj+1 − sj →∞ as j →∞. We define wj(y, s) = w(y, s+ sj).According to Theorem 5.1.1, Corollary 5.2.7 and Arzela-Ascoli theorem, there is a subsequenceof wj, still denoted by wj , such that

wj(y, s)→ w∞(y, s)

uniformly on compact subsets of W , and

∇wj(y,m)→ ∇w∞(y,m)

for almost all y and for each integer m. We obtain from Corollary 5.2.7 that either w∞ ≡ ∞or w∞ < ∞ in RN+1. Since a is a touchdown point for u, the case w∞ ≡ ∞ is ruled out byTheorem 5.2.10, and hence w∞ < ∞ in RN+1. Therefore, we conclude again from Corollary5.2.7 that

w ≤ C1(1 + |y|) (5.3.4)

for some constant C1 > 0.Define the associated energy of w at time s,

ER[w](s) =12

∫

BR

ρ|∇w|2dy − 16

∫

BR

ρw2dy −∫

BR

λρf(a)w

dy . (5.3.5)

122

Taking R(s) = s, the same calculations as in (5.2.29) give

− d

dsEs[w](s) =

∫

Bs

ρ(y)|ws|2dy −K(s) (5.3.6)

with

K(s) =∫

∂Bs

ρws∂w

∂νdS +

12s

∫

∂Bs

ρ|∇w|2(y · ν)dS − 1s

∫

∂Bs

ρ(1

6w2 +

λf(a)w

)(y · ν)dS

+λ∫

Bs

ρws[f(a)− f(a+ ye−s/2)

]

w2dy .

We note that the expression K(s) can be estimated as s 1. Essentially, since f(x) ∈ Cα(Ω)for some α ∈ (0, 1], using (5.3.4) and applying the same estimates as in Lemma 5.2.8 one candeduce that

K(s)− 12

∫

Bs

ρw2sdy ≤ G(s) := C1s

Ne−s2

4 + C2e−α

2s for s 1. (5.3.7)

Together with (5.3.7), integrating (5.3.6) in time yields an energy inequality

12

∫ b

a

∫

Bs

ρ|ws|2dyds ≤ Ea[w](a)− Eb[w](b) +∫ b

aG(s)ds , (5.3.8)

whenever a < b.We now use (5.3.8) to prove that w∞ is independent of s. We set a = sj+m and b = sj+1+m

in (5.3.8) to obtain

12

∫ m+sj+1−sj

m

∫

Bsj+s

ρ|wjs|2dyds ≤ Esj+m[wj ](m)− Esj+1+m[wj+1](m) +∫ sj+1+m

sj+mG(s)ds

(5.3.9)for any integer m, where we use wj(y, s) = w(y, s + sj). Since ∇wj(y,m) is bounded andindependent of j, and since we have assumed that ∇wj(y,m) → ∇w∞(y,m) a.e. as j → ∞,the dominated convergence theorem shows that

∫ρ(y)|∇wj(y,m)|2dy →

∫ρ(y)|∇w∞(y,m)|2dy as j →∞ .

Arguing similarly for the other terms we can deduce that

limj→∞

Esj+m[wj ](m) = limj→∞

Esj+1+m[wj+1](m) := E[w∞] . (5.3.10)

On the other hand, because m+ sj →∞ as j →∞, (5.3.7) assures that the term involving Gin (5.3.9) tends to zero as j →∞. Therefore, the right side of (5.3.9) tends to zero as j →∞.It now follows from sj+1 − sj →∞ that

limj→∞

∫ M

m

∫

Bsj+s

ρ|wjs|2dyds = 0 (5.3.11)

123

for each pair of integers m < M . Further, since (5.3.4) implies |wjs(y, s)| ≤ C(1 + |y|) withC independently of j, one can deduce that wjs converges weakly to w∞s. Because ρ decreasesexponentially as |y| → ∞, the integral of (5.3.11) is lower semi-continuous, and hence

∫ M

m

∫

RNρ|w∞s|2dyds = 0 ,

where m and M are arbitrary, which shows that w∞ is independent of the choice of s.We now notice from (5.3.5) that (5.3.10) defines E[w∞] by

E[v] =12

∫

RNρ|∇yv|2dy − 1

6

∫

RNρ|v|2dy −

∫

RNλρf(a)v

dy .

We claim that E[w∞] is independent of the choice of the sequence sj. If this is not thecase, then there is another sj such that E[w∞] 6= E[w∞], where w∞ = limj→∞ wj withwj(y, s) = w(y, s + sj). Relabeling and passing to a sequence if necessary, we may supposethat E[w∞] < E[w∞] with sj < sj . Now the energy inequality (5.3.8), with a = sj and b = sj ,gives that

12

∫ sj

sj

∫

Bs

ρ|ws|2dyds ≤ Esj [wj ](0)− Esj [wj ](0) +∫ sj

sj

G(s)ds . (5.3.12)

Since Esj [wj ](0) − Esj [wj ](0) → E[w∞] − Ew∞ [w∞] < 0 and∫ sjsjG(s)ds → 0 as j → ∞, the

right side of (5.3.12) is negative for sufficiently large j. This leads to a contradiction, becausethe left side of (5.3.12) is non-negative. Hence E[w∞] = E[w∞], which implies that E[w∞] isindependent of the choice of the sequence sj.

Therefore, we conclude that w(y, s)→ w∞(y) as s→∞ uniformly on |y| ≤ C, where C isany bounded constant, and w∞(y) is a bounded positive solution of (5.3.3).

5.3.1 Refined touchdown profiles for N = 1

In this subsection, we establish refined touchdown profiles for the deflection u = u(x, t) inone dimensional case. We begin with the discussions on the solution w∞(y) of (5.3.3). Forone dimensional case, Fila and Hulshof proved in Theorem 2.1 of [23] that every non-constantsolution w(y) of

wyy − 12ywy +

13w − 1

w2= 0 in (−∞,∞)

must be strictly increasing for all |y| sufficiently large, and w(y) tends to ∞ as |y| → ∞. Soit reduces from Lemma 5.3.1 that it must have w∞(y) ≡ const.. Therefore, by scaling weconclude that

lims→∞w(y, s) ≡ (3λf(a)

) 13

uniformly on |y| ≤ C for any bounded constant C. This gives the following touchdown rate.

Lemma 5.3.2. Assume f satisfies (5.1.2) on a bounded domain Ω ⊂ R1, and suppose u isa unique touchdown solution of (5.1.1) at finite time T . Assume touchdown set for u is acompact subset of Ω. If x = a is a touchdown point of u, then we have

limt→T−

u(x, t)(T − t)− 13 ≡ (3λf(a)

) 13

124

uniformly on |x− a| ≤ C√T − t for any bounded constant C.

We next determine the refined touchdown profiles for one dimensional case. Our methodis based on the center manifold analysis of a PDE that results from a similarity group trans-formation of (5.1.1). Such an approach was used in [32] for the uniform permittivity profilef(x) ≡ 1. A closely related approach was used in [25] to determine the refined blow-up profilefor a semilinear heat equation. We now briefly outline this method and the results that canbe extended to the varying permittivity profile f(x):

Continuing from (5.3.2) with touchdown point x = a, for s 1 and |y| bounded we havew ∼ w∞ + v, where v 1 and w∞ ≡ (3λf(a))1/3 > 0. Keeping the quadratic terms in v, weobtain for N = 1 that

vs − vyy +y

2vy − v =

w∞3[1− f(a+ ye−s/2)

f(a)]

+2[f(a+ ye−s/2)− f(a)

]

3f(a)v

−3λf(a+ ye−s/2)w4∞

v2 +O(v3)

≈ −(3λf(a))− 1

3 v2 +O(v3 + e−

α2s),

(5.3.13)

for s 1 and bounded |y|, due to the assumption (5.1.2) that f(x) ∈ Cα(Ω) for some0 < α ≤ 1. As shown in [25] (see also [32]), the linearized operator in (5.3.13) has a one-dimensional nullspace when N = 1. By projecting the nonlinear term in (5.3.13) against thenullspace of the linearized operator, the following far-field behavior of v for s → +∞ and |y|bounded is obtained (see (1.7) of [25]):

v ∼ −(3λf(a)

) 13

4s

(1− |y|

2

2

), N = 1 . (5.3.14)

The refined touchdown profile is then obtained from w ∼ w∞ + v, (5.3.1) and (5.3.14), whichis for t→ T−,

u ∼ [3λf(a)(T − t)]1/3(

1− 14| log(T − t)| +

|x− a|28(T − t)| log(T − t)| + · · ·

), N = 1 . (5.3.15)

Combining Lemma 5.3.2 and (5.3.15) directly gives the following refined touchdown profile.

Theorem 5.3.3. Assume f satisfies (5.1.2) on a bounded domain Ω in RN , and suppose uis a touchdown solution of (5.1.1) at finite time T . Assume touchdown set of u is a compactsubset of Ω, and suppose N = 1 and x = a is a touchdown point of u. Then we have

limt→T−

u(x, t)(T − t)− 13 ≡ (3λf(a)

) 13 (5.3.16)

uniformly on |x− a| ≤ C√T − t for any bounded constant C. Moveover, when t→ T−,

u ∼ [3λf(a)(T − t)]1/3(

1− 14| log(T − t)| +

|x− a|28(T − t)| log(T − t)| + · · ·

), N = 1 .

(5.3.17)

We finally remark that applying formal asymptotic methods, whenN = 1 the refined touch-down profile of (5.1.1) is also established in (4.5.12). By making a binomial approximation, itis easy to compare that (5.3.15) agrees asymptotically with (4.5.12).

125


5.3.2 Refined touchdown profiles for N ≥ 2

For obtaining refined touchdown profiles in higher dimension, in this subsection we assume thatf(r) = f(|x|) is radially symmetric and Ω = BR(0) is a bounded ball in RN with N ≥ 2. Thenthe uniqueness of solutions for (5.1.1) implies that the solution u of (5.1.1) must be radiallysymmetric. We study the refined touchdown profile for the special touchdown point r = 0 of uat finite time T . In this situation, the fact that the solution u of (5.1.1) is radially symmetricimplies the radial symmetry of w(y, s) in y, and hence the radial symmetry of w∞(y) (cf. [34]).Note that w∞(y) is a radially symmetric solution of

wyy +(N − 1

y− y

2)wy +

13w − λf(0)

w2= 0 for y > 0 , (5.3.18)

where wy(0) = 0 and f(0) > 0. For this case, applying Theorem 1.6 of [39] yields that everynon-constant radial solution w(y) of (5.3.18) must be strictly increasing for all y sufficientlylarge, and w(y) tends to ∞ as y →∞. It now reduces again from Lemma 5.3.1 that

lims→∞w(y, s) ≡ (3λf(0)

) 13

uniformly on |y| ≤ C for any bounded constant C. This gives the following touchdown rate.

Lemma 5.3.4. Assume f(r) = f(|x|) satisfies (5.1.2) on a bounded ball BR(0) ⊂ RN withN ≥ 2, and suppose u is a unique touchdown solution of (5.1.1) at finite time T . Assumetouchdown set for u is a compact subset of Ω. If r = 0 is a touchdown point of u, then we have

limt→T−

u(r, t)(T − t)− 13 ≡ (3λf(0)

) 13

uniformly for r ≤ C√T − t for any bounded constant C.

We next derive a refined touchdown profile (5.3.20). Similar to one dimensional case, indeedwe can establish the refined touchdown profiles for varying permittivity profile f(|x|) defined inhigher dimension N ≥ 2. Specially, applying a result from [25], the refined touchdown profilefor N = 2 is given by

u ∼ [3λf(0)(T − t)]1/3(

1− 12| log(T − t)| +

|x− a|24(T − t)| log(T − t)| + · · ·

), N = 2 .

This leads to the following refined touchdown profile for higher dimensional case.

Theorem 5.3.5. Assume f satisfies (5.1.2) on a bounded domain Ω in RN , and suppose uis a touchdown solution of (5.1.1) at finite time T . Assume touchdown set of u is a compactsubset of Ω, and suppose Ω = BR(0) ⊂ RN is a bounded ball with N ≥ 2 and f(r) = f(|x|) isradially symmetric. If r = 0 is a touchdown point of u, then we have

limt→T−

u(r, t)(T − t)− 13 ≡ (3λf(0)

) 13 (5.3.19)

uniformly on r ≤ C√T − t for any bounded constant C. Moveover, when t→ T−,

u ∼ [3λf(0)(T − t)]1/3(

1− 12| log(T − t)| +

r2

4(T − t)| log(T − t)| + · · ·), N = 2 . (5.3.20)

126


Remark 5.3.1. Applying analytical and numerical techniques, next section we shall show thatTheorem 5.3.5 does hold for a larger class of profiles f(r) = f(|x|).

Before concluding this section, it is interesting to compare the solution of (5.1.1) withthat of the ordinary differential equation obtained by omitting ∆u. For that we focus on onedimensional case, and we compare the solutions of

ut − uxx = −λf(x)u2

in (−a, a) , (5.3.21a)

u(±a, t) = 1 ; u(x, 0) = 1 , (5.3.21b)

and

vt = −λf(x)v2

in (−a, a) , (5.3.22a)

v(±a, t) = 1 ; v(x, 0) = 1 , (5.3.22b)

where f is assumed to satisfy (5.1.2) and (5.2.1). The ordinary differential equation (5.3.22) isexplicitly solvable, and the solution touches down at finite time

v(x, t) =(1− 3λf(x)t

) 13 , (5.3.23)

which shows that touchdown point of v is the maximum value point of f(x). In the partialdifferential equation (5.3.21), there is a contest between the dissipating effect of the Laplacianuxx and the singularizing effect of the nonlinearity f(x)/u2; when u touches down at x = x0 infinite time T , then the nonlinear term dominates (essentially, for some special cases, touchdownpoint x0 of u is also the maximum value point of f(x), see Theorem 5.1.4 for details).

However, we claim that a smoothing effect of the Laplacian can be still observed in thedifferent character of touchdown. Indeed, letting f(y0) = maxf(x) : x ∈ (−a, a), thenf ′(y0) = 0 and f ′′(y0) ≤ 0. And (5.3.23) gives the finite touchdown time T0 for v satisfyingT0 = 1/[3λf(y0)]. Furthermore, we can get from (5.3.23), together with the Taylor series off(x),

limt→T−0

(T0− t)−13 v(y0 + (T0− t)

12 y, t

)=(3λf(y0)

) 13[1− f ′′(y0)

2f2(y0)|y|2]

13 ≥ (3λf(x0)

) 13 . (5.3.24)

And our Theorem 5.3.3 says that for such u we have

limt→T−

(T − t)− 13u(x0 + (T − t) 1

2 y, t)

=(3λf(x0)

) 13 . (5.3.25)

Comparing (5.3.24) with (5.3.25), we see that the touchdown of the partial differential equation(5.3.21) is “flatter” than that of the ordinary differential equation (5.3.22).

5.4 Set of touchdown points

This section is focussed on the set of touchdown points for (5.1.1), which may provide usefulinformation on the design of MEMS devices. In §5.4.1, we consider the radially symmetric casewhere f(r) = f(|x|) with r = |x| is a radial function and Ω is a ball BR = |x| ≤ R ⊂ RNwith N ≥ 1. In §5.4.2, numerically we compute some simulations for one dimensional case,from which we discuss the compose of touchdown points for some explicit permittivity profilesf(x).

127

5.4.1 Radially symmetric case

In this subsection, f(r) = f(|x|) is assumed to be a radial function and Ω is assumed to be aball BR = |x| ≤ R ⊂ RN with any N ≥ 1. For this radially symmetric case, the uniqueness ofsolutions for (5.1.1) implies that the solution u = u(x, t) of (5.1.1) must be radially symmetric.We begin with the following lemma for proving Theorem 5.1.4:

Lemma 5.4.1. Suppose f(r) satisfies (5.1.2) and f ′(r) ≤ 0 in BR, and let u = u(r, t) be atouchdown solution of (5.1.1) at finite time T . Then ur > 0 in 0 < r < R× (t0, T ) for some0 < t0 < T .

Proof: Setting w = rN−1ur, then (5.1.1) gives

ut − 1rN−1

wr = −λf(r)u2

, 0 < t < T . (5.4.1)

Differentiating (5.4.1) with respect to r, we obtain

wt − wrr +N − 1r

wr − 2λfu3

w = −λf′rN−1

u2≥ 0 , 0 < t < T , (5.4.2)

since f ′(r) ≤ 0 in BR. Therefore, w can not attain negative minimum in 0 < r < R× (0, T ).Since w(0, t) = w(r, 0) = 0 and ut < 0 for all t ∈ (0, T ), we have w = rN−1ur > 0 on∂BR× (0, T ). So the maximum principle shows that w ≥ 0 in 0 < r < R× (0, T ). This gives

wt − wrr +N − 1r

wr ≥ 0 in 0 < r < R × (t1, T ) ,

where t1 > 0 is chosen so that w(r, t1) 6≡ 0 in 0 < r < R.Now compare w with the solution z of

zt − zrr +N − 1r

zr = 0 in 0 < r < R × (t1, T )

subject to z(r, t1) = w(r, t1) for 0 ≤ r ≤ R, z(R, t) = w(R, t) > 0 and z(0, t) = 0 for t1 ≤ t < T .The comparison principle yields w ≥ z in 0 < r < R × (t1, T ). On the other hand, for anyt0 > t1 we have z > 0 in 0 < r < R × (t0, T ). Consequently we conclude that w > 0, i.eur > 0 in 0 < r < R × (t0, T ).

Proof of Theorem 5.1.4: For w = rN−1ur, we set J(r, t) = w − ε ∫ rθ0 f(s)ds, where θ ≥ Nand ε = ε(θ) > 0 are constants to be determined. We calculate from (5.4.1) and (5.4.2) that

Jt − Jrr +N − 1r

Jr = b1J +2λεf

∫ rθ0 f(s)dsu3

− λf ′rN−1

u2+ θεrθ−1f ′

≥ b1J − rN−1(λ− θεrθ−N)f ′ ≥ b1J ,

provided ε is sufficiently small, where b1 is a locally bounded function. Here we have appliedthe assumption f ′(r) ≤ 0 and the relations ur = w/rN−1 and w = J + ε

∫ rθ0 f(s)ds. Note that

J(0, t) = 0, and hence it follows that J can not obtain negative minimum in BR × (0, T ).

128

We next observe that J can not obtain negative minimum on r = R provided ε issufficiently small, which comes from the fact

Jr(R, t) = wr − θεRθ−1f(R) =λRN−1f(R)

u2− θεRθ−1f(R) ≥ RN−1f(R)

[λ− θεRθ−N] ≥ 0

for sufficiently small ε > 0, where (5.4.1) is applied. We now choose some 0 < t0 < T suchthat w(r, t0) > 0 for 0 < r ≤ R in view of Lemma 5.4.1. This gives ur(r, t0) > 0 for 0 < r ≤ R.Since ur(0, t0) = 0, there exists some α > 0 such that

urr(0, t0) = limr→0

ur(r, t0)rα

= limr→0

w(r, t0)rN+α−1

> 0 .

We now choose θ = maxN,N + α− 1, from which one can further deduce that J(r, t0) ≥ 0for 0 ≤ r < R provided ε = ε(t0, θ) > 0 is sufficiently small.

−0.5 −0.25 0 0.25 0.50

0.2

0.4

0.6

0.8

1(a). f(x)=1−x2 and λ =8

x

u

0 0.2 0.4 0.6 0.8 1 0

0.2

0.4

0.6

0.8

1 (b). f(r)=1−r2 and λ =8

r=|x|

u

Figure 5.1: Left figure: plots of u versus x at different times with f(x) = 1 − x2 in the slabdomain, where the unique touchdown point is x = 0. Right figure: plots of u versus r = |x| atdifferent times with f(r) = 1− r2 in the unit disk domain, where the unique touchdown pointis r = 0 too.

It now concludes from the maximum principle that J ≥ 0 in BR × (t0, T ) provided ε =ε(t0) > 0 is sufficiently small. This leads to

u(r, t) ≥ u(r, t)− u(0, t) ≥ ε∫ r

0

∫ sθ0 f(µ)dµsN−1

ds. (5.4.3)

Given small C0 > 0, then the assumption of f(r) implies that there exists 0 < r0 = r0(C0) ≤ Rsuch that f(r) ≥ C0 on [0, r0]. Denote rm = minr0, r, and then (5.4.3) gives

u(r, t) ≥ ε∫ rm

0

∫ sθ0 f(µ)dµsN−1

ds ≥ ε∫ rm

0

C0sθ

sN−1ds =

1θ −N + 2

εC0rθ−N+2m , where θ−N+2 ≥ 2 ,

which implies that r = 0 must be the unique touchdown point of u. Before ending this subsection, we now present a few numerical simulations on Theorem

5.1.4. Here we apply the implicit Crank-Nicholson scheme. In the following simulations 1 ∼ 3,

129


−0.5 −0.25 0 0.25 0.50

0.2

0.4

0.6

0.8

1(a). f(x) = e−x

2

and λ = 8

x

u

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1(b). f(r) = e−r

2

and λ = 8

u

r = |x|

Figure 5.2: Left figure: plots of u versus x at different times with f(x) = e−x2in the slab

domain, where the unique touchdown point is x = 0. Right figure: plots of u versus r = |x| atdifferent times with f(r) = e−r2

in the unit disk domain, where the unique touchdown point isr = 0 too.

−0.5 −0.25 0 0.25 0.50

0.2

0.4

0.6

0.8

1(a). f(x) = ex

2−1 and λ = 8

u

x 0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1(b). f(r) = er

2−1 and λ = 8

u

r = |x|

Figure 5.3: Left figure: plots of u versus x at different times with f(x) = ex2−1 in the slab

domain, where the unique touchdown point is still at x = 0. Right figure: plots of u versusr = |x| at different times with f(r) = er

2−1 in the unit disk domain, where the touchdownpoints satisfy r = 0.51952.

130


we always take λ = 8 and the number of meshpoints N = 1000, and consider (5.1.1) in thefollowing symmetric slab or unit disk domains:

Ω : [−1/2, 1/2] (Slab) ; Ω : x2 + y2 ≤ 1 (Unit Disk) . (5.4.4)

Simulation 1: f(|x|) = 1 − |x|2 is chosen as a permittivity profile. In Figure 5.1(a),u versus x is plotted at different times for (5.1.1) in the symmetric slab domain. For thistouchdown behavior, touchdown time is T = 0.044727 and the unique touchdown point isx = 0. In Figure 5.1(b), u versus r = |x| is plotted at different times for (5.1.1) in the unitdisk domain. For this touchdown behavior, touchdown time is T = 0.0455037 and the uniquetouchdown point is r = 0.

Simulation 2: f(|x|) = e−|x|2 is chosen as a permittivity profile. In Figure 5.2(a), u versusx is plotted at different times for (5.1.1) in the symmetric slab domain. For this touchdownbehavior, touchdown time is T = 0.044675 and the unique touchdown point is x = 0. In Figure5.2(b), u versus r = |x| is plotted at different times for (5.1.1) in the unit disk domain. Forthis touchdown behavior, touchdown time is T = 0.0450226 and the unique touchdown pointis r = 0 too.

Simulation 3: f(|x|) = e|x|2−1 is chosen as a permittivity profile. In Figure 5.3(a), u versusx is plotted at different times for (5.1.1) in the symmetric slab domain. For this touchdownbehavior, touchdown time is T = 0.147223 and touchdown point is still uniquely at x = 0. InFigure 5.3(b), u versus r = |x| is plotted at different times for (5.1.1) in the unit disk domain.For this touchdown behavior, touchdown time is T = 0.09065363, but touchdown points areat r0 = 0.51952, which compose into the surface of Br0(0). This simulation shows that theassumption f ′(r) ≤ 0 in Theorem 5.1.4 is just sufficient, not necessary.

−0.5 −0.25 0 0.25 0.50

0.2

0.4

0.6

0.8

1(a). f(x) = 1/2−x/2 and λ = 8

x

u

−0.5 −0.25 0 0.25 0.50

0.2

0.4

0.6

0.8

1(b). f(x) = x+1/2 and λ = 8

u

x

Figure 5.4: Left figure: plots of u versus x at different times with f(x) = 1/2− x/2 in the slabdomain, where the unique touchdown point is x = −0.10761. Right figure: plots of u versusr = |x| at different times with f(x) = x+ 1/2 in the slab domain, where the unique touchdownpoint is x = 0.17467.

5.4.2 One dimensional case

For one dimensional case, Theorem 5.1.4 already gives that touchdown points must be uniqueif the permittivity profile f(x) is uniform. In the following, we choose some explicit varying

131


permittivity profiles f(x) to perform two numerical simulations. Here we apply the implicitCrank-Nicholson scheme again.

Simulation 4: Monotone Function f(x):We take λ = 8 and the number of meshpoints N = 1000, and we consider (5.1.1) in the slabdomain Ω defined in (5.4.4). In Figure 5.4(a), the monotonically decreasing profile f(x) =1/2−x/2 is chosen, and u versus x is plotted for (5.1.1) at different times. For this touchdownbehavior, the touchdown time is T = 0.09491808 and the unique touchdown point is x =−0.10761. In Figure 5.4(b), the monotonically increasing profile f(x) = x+ 1/2 is chosen, andu versus x is plotted for (5.1.1) at different times. For this touchdown behavior, the touchdowntime is T = 0.0838265 and the unique touchdown point is x = 0.17467. For the general casewhere f(x) is monotone in a slab domain, it is interesting to look insights into whether thetouchdown points must be unique.

−0.5 −0.25 0 0.25 0.50

0.2

0.4

0.6

0.8

1(a). α =0.5 and λ = 8

u

x

−0.5 −0.25 0 0.25 0.50

0.2

0.4

0.6

0.8

1(b). α =1 and λ = 8

u

x

−0.5 −0.25 0 0.25 0.50

0.2

0.4

0.6

0.8

1(c). α = 0.785 and λ = 8

x

u

−0.05 −0.025 0 0.025 0.050

0.04

0.08

0.12

0.16

0.2(d). local amplified plots of (c)

x

u

Figure 5.5: Plots of u versus x at different times in the slab domain, for different permittivityprofiles f [α](x) given by (5.4.5). Top left (a): when α = 0.5, two touchdown points are atx = ±0.12631. Top right (b): when α = 1, the unique touchdown point is at x = 0. Bot-tom Left (c): when α = 0.785, touchdown points are observed to consist of a closed interval[−0.0021255, 0.0021255]. Bottom right (d): local amplified plots of (c).

Simulation 5: “M”-Form Function f(x):In this simulation, we consider (5.1.1) in the slab domain Ω defined in (5.4.4). Here we takeλ = 8 and the number of the meshpoints N = 2000, and the varying dielectric permittivity

132

profiles satisfies

f [α](x) =

1− 16(x+ 1/4)2 , if x < −1/4 ;α+ (1− α)|sin(2πx)| , if |x| ≤ 1/4 ;1− 16(x− 1/4)2 , if x > 1/4

(5.4.5)

with α ∈ [0, 1], which has “M”-form. In Figure 5.5, u versus x is plotted at different timesfor (5.1.1) for different α, i.e for different permittivity profiles f [α](x). In Figure 5.5(a): whenα = 0.5, the touchdown time is T = 0.05627054 and two touchdown points are at x = ±0.12631.In Figure 5.5(b): when α = 1, the touchdown time is T = 0.0443323 and the unique touchdownpoint is at x = 0. In Figure 5.5(c): when α = 0.785, the touchdown time is T = 0.04925421and touchdown points are observed to compose into a closed interval [−0.0021255, 0.0021255].In Figure 5.5(d): local amplified plots of (c) at touchdown time t = T . This simulation showsfor dimension N = 1 that the set of touchdown points may be composed of finite points orfinite compact subsets of the domain, if the permittivity profile is ununiform.

133

Chapter 6

Thesis Summary

In this thesis, we have used both analytical and numerical methods to analyze the most basicmathematical model describing the dynamics of an elastic membrane in an electrostatic MEMS.We have answered mathematical questions dealing with existence, uniqueness and regularityof solutions. We have also addressed problems of more applicable nature such as stability, aswell as estimates on pull-in voltages and touchdown times in terms of the shape of the domainand the permittivity profile of the membrane. The thesis consists of two main parts:

6.1 Stationary Case

This reflects the state of the membrane at equilibrium when the voltage is below the criticalthreshold. We have given a detailed and rigorous analysis of the pull-in voltage and the deflec-tion profile, as described by the solutions of the equation that models Electrostatic MEMS.

By applying various numerical and analytic methods, we have given several useful upperand lower bounds for the pull-in voltage λ∗, by using modern mathematical tools such asPohozaev-type estimates and Bandle’s Schwarz symmetrization techniques. We do believehowever that our estimates are not optimal and better ones, depending on the distribution ofthe permittivity profile and the shape of the domain, can still be obtained.

We have studied the branch of stable and semi-stable solutions by using energy estimatesand have shown how the problem is dependent on the dimension, the shape of the membraneand the permittivity profile. We have also used sophisticated blow-up analysis to give a rigorousanalysis of the unstable solutions. We also established partial results about uniqueness andmultiplicity of solutions at various voltage ranges. The case where we have a power lawpermittivity profile on a round ball is however completely understood.

6.2 Dynamic Case

Here we analyze the evolution of the membrane’s deflection with time. The system exhibitsthree types of behavior. We have global convergence to a stable stationary state whenever thevoltage is below the critical threshold. We have touchdown in finite time when we are beyond,and we have possible touchdown in infinite time at the critical pull-in voltage.

We give several useful estimates for the touchdown time in terms of the domain, the per-mittivity profile and the applied voltage. A detailed analysis of the geometry and “size” ofthe touchdown set is given, showing in particular how the permittivity profile and the shapeof the membrane can be used to affect both the duration and the functioning of the MEMS.It is shown for example that the zero points of the profile f cannot be touchdown points.

The pull-in distance is also discussed and several interesting phenomena are observed nu-merically and established mathematically. For example, for the case of a power law profile

134

Chapter 6. Thesis Summary

(f(x) = |x|α) on a 2-dimensional disk, one can show that the pull-in distance is independentof the voltage, while numerical estimates show that the membrane develops a boundary-layerstructure near the boundary of the domain as the power α is increased.

135

Bibliography

[1] A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convexnonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994), 519–543.

[2] U. Ascher, R. Christiansen and R. Russell, Collocation software for boundary valueODE’s, Math. Comp. 33 (1979), 659–679.

[3] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory andapplication, J. Funct. Anal. 14 (1973), 349–381.

[4] S. Alama and G. Tarantello, Elliptic problems with nonlinearities indefinite in sign, J.Funct. Anal. 141 (1996), 159–215.

[5] S. Alama and G. Tarantello, On the solvability of a semilinear elliptic equation via anassociated eigenvalue problem, Math. Z. 221 (1996), 467–493.

[6] C. Bandle, Isoperimetric inequalities and applications, Monographs and Studies inMathematics, Boston, Mass.-London, Pitman, 1980.

[7] R. E. Bank, PLTMG: A software package for solving elliptic partial differential equa-tions, User’s guide 8.0, Software, Environments, and Tools, SIAM, Philadelphia, PA,1998.

[8] H. Bellout, A criterion for blow-up of solutions to semilinear heat equations, SIAM,J. Math. Anal. 18 (1987), 722–727.

[9] D. Bernstein, P. Guidotti and J. A. Pelesko, Analytic and numerical analysis of electro-statically actuated MEMS devices, Proc. of Modeling and Simulation of Microsystems2000 (2000), 489–492.

[10] H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for ut −∆u = g(u)revisited, Adv. Diff. Eqns. 1 (1996), 73–90.

[11] C. M Brauner and B. Nicolaenko, Sur une classe de problemes elliptiques non lineaires,C. R. Acad. Sci. Paris 286 (1978), 1007–1010.

[12] H. Brezis and L. Nirenberg, H1 versus C1 local minimizers, C. R. Math. Acad. Sci.Paris 317 (1993), 465–472.

[13] H. Brezis, L. A. Peletier and D. Terman, A very singular solution of the heat equationwith absorption, Arch. Ration. Mech. Anal. 95 (1986), 185–209.

[14] H. Brezis and J. L. Vazquez, Blow-up solutions of some nonlinear elliptic problems,Rev. Mat. Univ. Compl. Madrid 10 (1997), 443–469.

136

Bibliography

[15] X. Cabre, Extremal solutions and instantaneous complete blow-up for elliptic andparabolic problems, Contemporary Math., Amer. Math. Soc. 2007, in: ”Perspectivesin Nonlinear Partial Differential Equations: In honor of Haim Brezis”, to appear.

[16] X. Cabre and A. Capella, On the stability of radial solutions of semilinear ellipticequations in all of Rn, C. R. Math. Acad. Sci. Paris 338 (2004), 769–774.

[17] E. K. Chan and R. W. Dutton, Effects of capacitors, resistors and residual change onthe static and dynamic performance of electrostatically actuated devices, Proceedingsof SPIE 3680 (1999), 120–130.

[18] X. Cabre and Y. Martel, Weak eigenfunctions for the linearization of extremal ellipticproblems, J. Funct. Anal. 156 (1998), 30–56.

[19] M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvaluesand linearized stability, Arch. Rational Mech. Anal. 52 (1973), 161–180.

[20] M. G. Crandall and P. H. Rabinowitz, Some continuation and variational methods forpositive solutions of nonlinear elliptic eigenvalue problems, Arch. Ration. Mech. Anal.58 (1975), 207–218.

[21] L. Evans, Partial differential equations, Graduate Studies in Mathematics, 19. AMS,Providence, RI, 1998.

[22] P. Esposito, N. Ghoussoub and Y. Guo, Compactness along the branch of semi-stableand unstable solutions for an elliptic problem with a singular nonlinearity, Comm. PureAppl. Math., to appear (2006).

[23] M. Fila and J. Hulshof, A note on the quenching rate, Proc. Amer. Math. Soc. 112(1991), 473–477.

[24] M. Fila, J. Hulshof and P. Quittner, The quenching problem on the N -dimensionalball, Nonlinear Diffusion Equations and their Equilibrium States 3, Birkhauser, Baston,(1991), 183–196.

[25] S. Filippas and R. V. Kohn, Refined asymptotics for the blow up of ut − ∆u = up,Comm. Pure Appl. Math. 45 (1992), 821–869.

[26] G. Flores, G. A. Mercado and J. A. Pelesko, Dynamics and touchdown in electrostaticMEMS, Proceedings of ICMENS 2003 (2003), 182–187.

[27] A. Friedman and B. Mcleod, Blow-up of positive solutions of semilinear heat equations,Indiana Univ. Math. J. 34 (1985), 425–447.

[28] P. Feng and Z. Zhou, Multiplicity and symmetry breaking for positive radial solutions ofsemilinear elliptic equations modelling MEMS on annular domains, Electron. J. Diff.Eqns. 146 (2005), 1–14.

[29] N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMSdevices: stationary case, SIAM, J. Math. Anal. 38 (2007), 1423–1449.

137

Bibliography

[30] N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMSdevices II: dynamic case, NoDEA Nonlinear Diff. Eqns. Appl., to appear (2007).

[31] Y. Guo, On the partial differential equations of electrostatic MEMS devices III: refinedtouchdown behavior, submitted (2006).

[32] Y. Guo, Z. Pan and M. J. Ward, Touchdown and pull-in voltage behavior of a MEMSdevice with varying dielectric properties, SIAM, J. Appl. Math. 66 (2005), 309–338.

[33] F. Gazzola and A. Malchiodi, Some remarks on the equation −∆u = λ(1 + u)p forvarying λ, p and varying domains, Comm. Part. Diff. Eqns. 27 (2002), 809–845.

[34] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maxi-mum principle, Comm. Math. Phys. 68 (1979), 209–243.

[35] Y. Giga and R. V. Kohn, Asymptotically self-similar blow-up of semilnear heat equa-tions, Comm. Pure and Appl. Math. (1985), 297–319.

[36] Y. Giga and R. V. Kohn, Characterizing blow-up using similarity variables, IndianaUniv. Math. J. 36 (1987), 1–40.

[37] Y. Giga and R. V. Kohn, Nondegeneracy of blow-up for semilinear heat equations,Comm. Pure Appl. Math. 42 (1989), 845–884.

[38] J. S. Guo, On the quenching behavior of the solution of a semilinear parabolic equation,J. Math. Anal. Appl. 151 (1990), 58–79.

[39] J. S. Guo, On the semilinear elliptic equation ∆w + 12y · 5w + λw − w−β = 0 in Rn,

Chinese J. Math. 19 (1991), 355–377.

[40] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order,2nd, Springer, Berlin, 1983.

[41] A. Haraux and F. B. Weissler, Non-uniqueness for a semilinear initial value problem,Indiana Univ. Math. J. 31 (1982), 167–189.

[42] J. D. Jackson, Classical electrodynamics, John Wiley, New York, 1999.

[43] D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positivesources, Arch. Ration. Mech. Anal. 49 (1973), 241–268.

[44] J. P. Keener and H. B. Keller, Positive solutions of convex nonlinear eigenvalue prob-lems, J. Diff. Eqns. 16 (1974), 103–125.

[45] J. B. Keller, J. Lowengrub, Asymptotic and numerical results for blowing-up solutionsto semilinear heat equations, Singularities in Fluids, Plasmas, and Optics (1992), 11–38.

[46] H. A. Levine, Quenching, nonquenching, and beyond quenching for solution of someparabolic equations, Annali Matematica Pura Appl. 155 (1989), 243–260.

[47] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uralceva, Linear and quasilinearequations of parabolic type, Translations of Mathematical Monographs AMS 23, 1968.

138

Bibliography

[48] F. Mignot and J. P. Puel, Sur une classe de problemes non lineaires avec non linearitepositive, croissante, convexe. Comm. Partial Differential Equations 5 (1980), 791–836.

[49] H. C. Nathanson, W. E. Newell and R. A. Wickstrom, J. R. Davis, The resonant gatetransistor, IEEE Trans. on Elect. Devices 14 (1967), 117–133.

[50] J. A. Pelesko, Mathematical modeling of electrostatic MEMS with tailored dielectricproperties, SIAM J. Appl. Math. 62 (2002), 888–908.

[51] J. A. Pelesko, D. Bernstein and J. McCuan, Symmetry and symmetry breaking inelectrostatic MEMS, Proceedings of MSM 2003 (2003), 304–307.

[52] J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman Hall andCRC Press, (2002).

[53] J. A. Pelesko and A. A. Triolo, Nonlocal problems in MEMS device control, J. Eng.Math. 41 (2001), 345–366.

[54] I. Stackgold, Green’s functions and boundary value problems, Wiley, New York, (1998).

[55] G. I. Taylor, The coalescence of closely spaced drops when they are at different electricpotentials, Proc. Roy. Soc. A. 306 (1968), 423–434.

[56] G. Zheng, New results on the formation of singularities for parabolic problems, ChineseUniversity of Hongkong, PhD Thesis, 2005.

139

Partial Di erential Equations of Electrostatic MEMS › wp-content › uploads › 2018 › 10 ›...

Documents

Transcript of Partial Di erential Equations of Electrostatic MEMS › wp-content › uploads › 2018 › 10 ›...