Nonlocal transmission conditions arising inhomogenization of pε(x)-Laplacian in perforated

domains

V. Prytula1 L. Pankratov2,3

1Universidad de Castilla-La ManchaDepartamento de Matematicas

2B. Verkin Institute for Low Temperature Physics UkraineMathematical Division

3Universite de Pau, Laboratoire de Mathematiques Appliquees

MP2 Workshop, 2009

Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects

Introduction

Applications and physical motivation:

Image restoration based on a variable exponent Laplacian 1

E (u) =

∫Ω

|∇u(x)|p(x) + |u(x)− I (x)|2 dx

Modeling of non–newtonian fluids, in particularelectrorheological fluids (ER).2

−div (K (x)|∇u|p(x)−2∇u) + R(x)|u|σ(x)−2u = g(x),

1Y. Chen, S. Levine, R. Rao, Functionals with p(x)–growth in image processing. Duquesne University, Dep. of

Math. And Comp. Sci. Tech. Rep.2

M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000

Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects

Introduction

Applications and physical motivation:

Image restoration based on a variable exponent Laplacian 1

E (u) =

∫Ω

|∇u(x)|p(x) + |u(x)− I (x)|2 dx

Modeling of non–newtonian fluids, in particularelectrorheological fluids (ER).2

−div (K (x)|∇u|p(x)−2∇u) + R(x)|u|σ(x)−2u = g(x),

1Y. Chen, S. Levine, R. Rao, Functionals with p(x)–growth in image processing. Duquesne University, Dep. of

Math. And Comp. Sci. Tech. Rep.2

M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000

Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects

Introduction

Applications and physical motivation:

Image restoration based on a variable exponent Laplacian 1

E (u) =

∫Ω

|∇u(x)|p(x) + |u(x)− I (x)|2 dx

Modeling of non–newtonian fluids, in particularelectrorheological fluids (ER).2

−div (K (x)|∇u|p(x)−2∇u) + R(x)|u|σ(x)−2u = g(x),

1Y. Chen, S. Levine, R. Rao, Functionals with p(x)–growth in image processing. Duquesne University, Dep. of

Math. And Comp. Sci. Tech. Rep.2

M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000

Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects

Introduction: Mathematical Motivation

We are interested in the study of the following nonlinear elliptical

equations:

− div (F (|∇u|, x),∇u) + R (|u|, x) u = g(x).

Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects

Introduction: Mathematical Motivation

In this work:

pε–Laplacian in perforated domain

−div(|∇uε|pε(x)−2∇uε

)+ |uε|σ(x)−2 uε = g(x), Ωε ⊂ Rn;

uε = Aε on ∂Fε;

uε = 0 on ∂Ω;

∫∂Fε|∇ε|pε(x)−2∂uε

∂~νds = 0.

Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects

Introduction: History of the problem

Existence results:V. V. Zhikov, On some variational problems, J. Math. Phys. 5 (1997),

#1,

E. Acerbi, G Mingione, Regularity results for stationary

electro-rheological fluids, Arch. Ration. Mech. Anal, 164 (2002),

P. Hasto, On the variable exponent Dirichlet energy integral, Comm.

Pure and Appl. Anal, 5 (2006), #3.

Homogenization Dirichlet problem for p-Laplacian (p isconstant)

E. Hruslov, L.Pankratov, Asymptotical behaviour of p-Laplacian

equations in domains with complex boundary, J. Math. Phys., Anal.,

Geom., FTINT, (1987),

I. Skrypnik, Methods for analysis of nonlinear elliptic boundary value

problems, Providence, RI: American Mathematical Society (AMS),

1994,

A. Braides and A. Defranceschi, Homogenization of Multiple Integrals,

Oxford Lecture Ser. Math. Appl. vol. 12, Clarendon Press, Oxford

(1998).

Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects

Introduction: History of the problem

Existence results:V. V. Zhikov, On some variational problems, J. Math. Phys. 5 (1997),

#1,

E. Acerbi, G Mingione, Regularity results for stationary

electro-rheological fluids, Arch. Ration. Mech. Anal, 164 (2002),

P. Hasto, On the variable exponent Dirichlet energy integral, Comm.

Pure and Appl. Anal, 5 (2006), #3.

Homogenization Dirichlet problem for p-Laplacian (p isconstant)

E. Hruslov, L.Pankratov, Asymptotical behaviour of p-Laplacian

equations in domains with complex boundary, J. Math. Phys., Anal.,

Geom., FTINT, (1987),

I. Skrypnik, Methods for analysis of nonlinear elliptic boundary value

problems, Providence, RI: American Mathematical Society (AMS),

1994,

A. Braides and A. Defranceschi, Homogenization of Multiple Integrals,

Oxford Lecture Ser. Math. Appl. vol. 12, Clarendon Press, Oxford

(1998).

Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects

Introduction: History of the problem

Opposite case: logarithmic growth

E. Khruslov, L. Pankratov, Homogenization of the Dirichlet variational

problems in Orlicz–Sobolev spaces, Fields Institute Communications 25

(2000) 345–66,

M. Goncharenko, V. Prytula, Homogenization of the electrostatic

problems in nonlinear medium with thin perfectly conducting grids, J.

Math. Phys. Anal. Geom. 2, 424–448 (2006),

D. Lukkassen, N. Svanstedt, On Γ–convergence in Anisotropic

Orlicz–Sobolev Spaces, Rend. Instit. Mat. Univ. Trieste, XXXIII,

281-287 (2001).

pε(x) case:

B. Amaziane, S. Antontsev, L. Pankratov, A. Piatnitski,

Homogenization of pε(x)–Laplacian in perforated domains, Ann. Institut

H. Poincare (C). Analyse non Lineaire (2009).

Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects

Introduction: History of the problem

Opposite case: logarithmic growth

E. Khruslov, L. Pankratov, Homogenization of the Dirichlet variational

problems in Orlicz–Sobolev spaces, Fields Institute Communications 25

(2000) 345–66,

M. Goncharenko, V. Prytula, Homogenization of the electrostatic

problems in nonlinear medium with thin perfectly conducting grids, J.

Math. Phys. Anal. Geom. 2, 424–448 (2006),

D. Lukkassen, N. Svanstedt, On Γ–convergence in Anisotropic

Orlicz–Sobolev Spaces, Rend. Instit. Mat. Univ. Trieste, XXXIII,

281-287 (2001).

pε(x) case:

B. Amaziane, S. Antontsev, L. Pankratov, A. Piatnitski,

Homogenization of pε(x)–Laplacian in perforated domains, Ann. Institut

H. Poincare (C). Analyse non Lineaire (2009).

Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects

Statement of the problem

−div(|∇uε|pε(x)−2∇uε

)+ |uε|σ(x)−2 uε = g(x), Ωε ⊂ Rn, d ≥ 2;

uε = Aε on ∂Fε;

uε = 0 on ∂Ω;

∫∂Fε|∇ε|pε(x)−2∂uε

∂~νds = 0,

Aε is an unknown constant

Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects

Statement of the problem: Growth Conditions

pε ∈ C (Ω) such that ∀ε > 0:

(i) pε is bounded in Ω, i.e., 1 < p− ≤ p−ε ≡ minx∈Ω pε(x) ≤pε(x) ≤ maxx∈Ω pε(x) ≡ p+

ε ≤ p+ ≤ n;

(ii) pε is log–continuous, i.e., for any x , y ∈ Ω,|pε(x)− pε(y)| ≤ ωε(|x − y |), where limτ→0 ωε(τ) ln

(1τ

)≤ C ;

(iii) pε converges uniformly in Ω to a function p0, where p0 islog–continuous;

(iv) pε satisfies the inequality: pε(x) ≥ p0(x) in Ω.

σ be a log–continuous function in Ω such that ε > 0:

(v) 1 < σ− ≡ minx∈Ω σ(x) ≤ σ(x) ≤ maxx∈Ω σ(x) ≡ σ+ ≤n p0(x)/(n − p0(x)) in Ω.

Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects

Statement of the problem: Growth Conditions

pε ∈ C (Ω) such that ∀ε > 0:

(i) pε is bounded in Ω, i.e., 1 < p− ≤ p−ε ≡ minx∈Ω pε(x) ≤pε(x) ≤ maxx∈Ω pε(x) ≡ p+

ε ≤ p+ ≤ n;

(ii) pε is log–continuous, i.e., for any x , y ∈ Ω,|pε(x)− pε(y)| ≤ ωε(|x − y |), where limτ→0 ωε(τ) ln

(1τ

)≤ C ;

(iii) pε converges uniformly in Ω to a function p0, where p0 islog–continuous;

(iv) pε satisfies the inequality: pε(x) ≥ p0(x) in Ω.

σ be a log–continuous function in Ω such that ε > 0:

(v) 1 < σ− ≡ minx∈Ω σ(x) ≤ σ(x) ≤ maxx∈Ω σ(x) ≡ σ+ ≤n p0(x)/(n − p0(x)) in Ω.

Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects

Main steps in the analysis

Consider the associated variational problem

Prove convergence of the minimizers to the minimizer of thehomogenized functional

Obtain homogenized PDE

Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects

Variational Formulation

Jε[u] ≡∫

ΩεFε (x , u,∇u) dx −→ inf, uε ∈W 1,pε(·)(Ωε);

uε = Aε on ∂Fε and uε = 0 on ∂Ω,

where

Fε (x , u,∇u) =1

pε(x)|∇u|pε(x) +

1

σ(x)|u|σ(x) − g(x) u,

Aε unknown; g ∈ C (Ω).

Existence

For each ε > 0, ∃!uε ∈W 1,pε(·)(Ωε).

Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects

Variational Formulation

Jε[u] ≡∫

ΩεFε (x , u,∇u) dx −→ inf, uε ∈W 1,pε(·)(Ωε);

uε = Aε on ∂Fε and uε = 0 on ∂Ω,

where

Fε (x , u,∇u) =1

pε(x)|∇u|pε(x) +

1

σ(x)|u|σ(x) − g(x) u,

Aε unknown; g ∈ C (Ω).

Existence

For each ε > 0, ∃!uε ∈W 1,pε(·)(Ωε).

Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects

Energy Characteristics of the set F ε

Local energy characteristics of the set Fε

For any piece S of Γ, introduce a layer Th(S) generated by thesurfaces Γ−h (S), Γ+

h (S):

C ε,h(S , b) = infvε

∫Th(S)

1

pε(x)|∇v ε|pε(x) + h−p+−γ |v ε − b|pε(x)

dxa

b ∈ R, γ > 0,

vε ∈W 1,pε(·) (Th(S)) , vε = 0, x ∈ Fε.

aV. A. Marchenko, E. Ya. Khruslov, Homogenization of Partial DifferentialEquations, Birkhauser, Berlin, 2006.

Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects

Energy Characteristics of the set F ε

For any piece S of Γ, introduce a layer Th(S) generated by the surfaces Γ−h

(S), Γ+h

(S):

C ε,h(S , b) = infvε

∫Th(S)

1

pε(x)|∇vε|pε(x) + h−p+−γ |vε − b|pε(x)

dx , a

b ∈ R, γ > 0, vε ∈ W 1,pε(·) (Th(S)), vε = 0 on Fε.

aV. A. Marchenko, E. Ya. Khruslov, Homogenization of Partial Differential Equations, Birkhauser, Berlin,

2006.

Condition

(C.1) for any arbitrary piece S ⊂ Γ and any b ∈ R:

limh→0

limε→0

C ε,h(S , b) = limh→0

limε→0

C ε,h(S , b) =

∫S

c(x , b)dΓ,

where c(x , b) is a nonnegative continuous function on Γ such thatc(x , b) ≤ C

(1 + |b|p0(x)−1

)|b|.

Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects

Main Result

Theorem

Let uε be a solution of BVP extended by uε(x) = Aε in Fε. Letgrowth assumptions and (C.1) hold. Then there is a subsequenceuε, ε = εk → 0 that converges weakly in W 1,p0(·)(Ω) to afunction u(x) such that the pair u(x),A is a solution of

Jhom[u,A] ≡∫

ΩF0(x , u,∇u) dx +

∫Γ

c(x , u − A) dx −→ inf,

u ∈W1,p0(·)0 (Ω) with A = lim

ε→0Aε,

where

F0(x , u,∇u) =1

p0(x)|∇u|p0(x) +

1

σ(x)|u|σ(x) − g(x) u,

and constant A is finite.

Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects

Main Result

Remark

Constant A remains unknown. Supposing that the function c(x , b)is differentiable with respect to b we obtain:−div

(|∇u|p0(x)−2∇u

)+ |u|σ(x)−2 u = g(x) in Ω \ Γ;

u = 0 on ∂Ω; [u]±Γ = 0,[|∇u|p0(x)−2 ∂u

∂ν

]±Γ

= c ′u(x , u − A);∫Γ

c ′u(x , u − A) dS = 0,

where ν is a normal vector to Γ, [ · ]±Γ is the jump on Γ, c ′u is thepartial derivative of c with respect to u. This means that problemcontains a non–local transmission condition.

Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects

Main Result

Lemma

Properties of the homogenized functional.Let the conditions of Theorem hold. Then we have the followingproperties:

The functional Jhom[u,A] is convex in A and strictly convex inu;

Jhom[u,A] is continuous in the space W1,p0(·)0 (Ω) with respect

to the variable u.

Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects

Periodic Example

Let Ω be a bounded Lipschitz domain in R3. Fε consists of thinintersecting cylinders of radius

r (ε) = e−1/ε.

The axes of the cylinders belong to a plane Γ b Ω and form anε–periodic lattice in R2. Ωε = Ω \ Fε.Let pε(ε>0) be a class of smooth functions in Ω given by:

pε(x) =

2 + ε `(x) in N (Fε, ε2);2 + `ε(x) elsewhere,

N (Fε, ε2) denotes the cylindrical ε2–neighborhood of the set Fεand where `, `ε are smooth strictly positive functions in Ω,maxx∈Ω `ε(x) = o(1) , ε→ 0. It is clear that pε converges

uniformly in Ω to p0 ≡ 2.

Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects

Periodic Example

Let Ω be a bounded Lipschitz domain in R3. Fε consists of thinintersecting cylinders of radius

r (ε) = e−1/ε.

The axes of the cylinders belong to a plane Γ b Ω and form anε–periodic lattice in R2. Ωε = Ω \ Fε.Let pε(ε>0) be a class of smooth functions in Ω given by:

pε(x) =

2 + ε `(x) in N (Fε, ε2);2 + `ε(x) elsewhere,

N (Fε, ε2) denotes the cylindrical ε2–neighborhood of the set Fεand where `, `ε are smooth strictly positive functions in Ω,maxx∈Ω `ε(x) = o(1) , ε→ 0. It is clear that pε converges

uniformly in Ω to p0 ≡ 2.

Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects

Periodic Example

Theorem

Let uε be the solution extended by the equality uε(x) = Aε in Fε.Then uε converges weakly in H1(Ω) to u the solution of−∆u + |u|σ(x)−2 u = g(x) in Ω \ Γ;

u = 0 on ∂Ω; [u]±Γ = 0 and[∂u∂ν

]±Γ

= 4π (u(z)− Al )µ(z) on Γ,

where

Al =

(∫Γµ(s) ds

)−1 ∫Γµ(s)u(s) ds and µ(z) =

e l(z) − 1

l(z).

Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects

Periodic Example: Remark

Remark

In the case of a surface distribution of Fε, with a constant growthpε(x) = 2 + α, α > 0 is a parameter independent of ε, there is no3D lattice for the corresponding problem which leads tohomogenization because the capacity of the lattice goes to infinityas ε→ 0. However our Theorem gives an example of the growthpε ∼ 2 + ε (in a small neighborhood of the lattice) which leads toa non trivial homogenization result.

Polynomial growth faster then 2 : no 3D lattice,

Logarithmic growth : nontrivial homogenization results,

Our case, pε ∼ 2 + ε : nontrivial homogenization results.

Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects

Periodic Example: Remark

Remark

In the case of a surface distribution of Fε, with a constant growthpε(x) = 2 + α, α > 0 is a parameter independent of ε, there is no3D lattice for the corresponding problem which leads tohomogenization because the capacity of the lattice goes to infinityas ε→ 0. However our Theorem gives an example of the growthpε ∼ 2 + ε (in a small neighborhood of the lattice) which leads toa non trivial homogenization result.

Polynomial growth faster then 2 : no 3D lattice,

Logarithmic growth : nontrivial homogenization results,

Our case, pε ∼ 2 + ε : nontrivial homogenization results.

Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects