CONTINUOUS APPROACH TO THE LIGHTNING DISCHARGE
By
BEYZA ÇALIŞKAN ASLAN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2007
1
c© 2007 Beyza Çalışkan Aslan
2
To my parents,
Kamile Çalışkan and Yusuf Çalışkan,
and to my husband,
Ömer Deniz Aslan
3
ACKNOWLEDGMENTS
First of all, I would like to express my gratitude to my advisor, Professor William
W. Hager. Without his encouragement, consistent support and guidance, this dissertation
could not have been completed. I am grateful to have had the opportunity to study
under such a caring, intelligent, and energetic advisor. His confidence in me will always
encourage me to move forward on my research.
Second, I would also like to thank Dr. Jayadeep Gopalakrishnan, Dr. Shari Moskow,
Dr. Sergei S. Pilyugin, and Dr. Vladimir A. Rakov for serving on my supervisory
committee. Their valuable suggestions have been very helpful to my research.
Third, thanks go to my officemates (Dr. Hongchao Zhang, Dr. Shu-Jen Huang, and
Sukanya Krishnaswamy), and all colleagues and friends in the Department of Mathematics
at the University of Florida. Their company alleviated the stress and frustration of this
time.
Last, but not least, I wish to express my special thanks to my family: to my husband,
Deniz, for his love and his endless support to pursue and complete my degree; to our
daughter, Erin Başak, for being a glorious joy to us; to my parents for their immeasurable
support and love; to my parents-in-law for their wholehearted understanding and
encouragement; and to my brother for his unstopping support and encouragement.
Without their support and encouragement, this dissertation could not have been
completed successfully.
4
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
CHAPTER
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Lightning Models with Explicit Lightning Channels . . . . . . . . . . . . . 112.1.1 Helsdon’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.2 MacGorman’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.3 Mansell’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Hager’s Model: The Discrete Model . . . . . . . . . . . . . . . . . . . . . . 14
3 THE DISCRETE MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1.2 Derivation of the Governing Equations . . . . . . . . . . . . . . . . 15
3.2 The Model in One-dimension . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Generalization to Three-dimension . . . . . . . . . . . . . . . . . . . . . . 18
4 THE CONTINUOUS MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.1 Formulation of the Equations . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Eigenproblem for AΨ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5 GENERALIZED EIGENPROBLEM FOR THE LAPLACIAN . . . . . . . . . . 23
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.2 Eigenfunctions of Type 1, 2, and 3 . . . . . . . . . . . . . . . . . . . . . . 235.3 Reformulation of Eigenproblem in H Using Double-Layer Potential . . . . 245.4 Eigenvalue Separation and Completeness of Eigenfunctions . . . . . . . . . 32
6 THE LIMIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.2 Reformulation of the Continuous Equation . . . . . . . . . . . . . . . . . . 416.3 Potential Change for the Continuous Equation . . . . . . . . . . . . . . . . 43
7 APPLICATION TO ONE-DIMENSION . . . . . . . . . . . . . . . . . . . . . . 48
7.1 Application of the Generalized Eigenproblem . . . . . . . . . . . . . . . . . 48
5
7.2 Application of the Continuous Model . . . . . . . . . . . . . . . . . . . . . 48
8 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6
LIST OF FIGURES
Figure page
4-1 A sketch of L and Ω for a lightning discharge . . . . . . . . . . . . . . . . . . . 207-1 Eigenfunctions in H in one dimension . . . . . . . . . . . . . . . . . . . . . . . . 49
7
Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy
CONTINUOUS APPROACH TO THE LIGHTNING DISCHARGE
By
Beyza Çalışkan Aslan
August 2007
Chair: William W. HagerMajor: Mathematics
We develop a continuous model for the lightning discharge. We consider Maxwell’s
equations in three dimensions and obtain a formula for the limiting potential as
conductivity tends to infinity in a three-dimensional subdomain (the lightning channel)
of the modeled domain. The limit is expressed in terms of the eigenfunctions for a
generalized eigenvalue problem for the Laplacian operator. The potential in the breakdown
region can be expressed in terms of a harmonic function which is constant in the
breakdown region.
8
CHAPTER 1INTRODUCTION
Lightning is one of the most beautiful displays in nature, however, it is also
frightening. It can destroy buildings and even kill people. It is a costly as well as deadly
natural event that mankind can not avoid.
The fear and respect for lightning attracted many people’s attention over the years.
Today, the physical processes involved in lightning are the focus of intensive research
throughout the world. Lightning is a result of charge separation inside a cloud. As the
graupel and ice particles within a cloud grow in size and increase in number, under the
influence of the wind, collisions between them may occur resulting in charge exchanges
between the particles. In general, smaller particles acquire positive charge, while larger
particles acquire negative charge. The charge separation occurs when these particles
separate under the influence of updrafts and gravity, and as a result, upper portion of the
cloud becomes positively charged and the lower portion of the cloud becomes negatively
charged. This results in huge electrical potential difference within the cloud as well as
between the cloud and the ground causing a flash to occur moving charges between
positive and negative regions of a thunderstorm.
Detailed history of early lightning research can be found in Uman [29]. Benjamin
Franklin was the first person who performed a scientific study of lightning. In the second
half of eighteenth century, he designed an experiment that proved the lightning was
electrical. It was after photographic and spectroscopic tools became available towards the
end of the nineteenth century that more studies about lightning started being conducted.
Lightning current measurements were first proposed by Pockels [26–28]. He estimated the
amount of current by analyzing the magnetic field induced by lightning currents. Later,
Wilson [32, 33] was the first researcher to use the electric field measurements to estimate
the structure of thunderstorm charges involved in lightning discharges. He won the Nobel
Prize for inventing the Cloud Chamber to track high energy particles and made major
9
contributions to our present understanding of lightning. Lightning research has been
particularly active since about 1970. This increased interest was motivated by
• The damage to aircraft or spacecraft due to lightning,
• Vulnerability of solid-state electronics used in computers and other devices,
• Development of new techniques of data taking and improvement of observationalcapabilities.
Most lightning research is done by physicists, chemists, meteorologists, and electrical
engineers. Hager [11, 13, 14] was the first mathematician using Maxwell’s equations to
develop a three-dimensional mathematical model to simulate a lightning discharge. His
discharge model [14] was obtained by discretizing Maxwell’s equations to obtain a relation
between the potential field and current density due to the motion of charged particles
under the influence of the wind. Spatial derivatives in his equation were approximated
by using volume elements in space, while the temporal derivatives were estimated by
a backward Euler scheme in time. Since conductivity is very large in the region where
the electric field reaches the breakdown threshold, he evaluated the solution limit as the
conductivity tends to infinity in the breakdown region. In his model [14], the output was
the electric field as a function of time, and the inputs were currents generated by the flow
of charged particles within the thundercloud under the influence of the wind.
This dissertation is based on Hager’s mathematical model. Some improvements are
made compared to Hager’s earlier work. For example, the solution is computed without
discretizing the equations. Consequently, we do not have huge matrix systems to compute
and therefore it is computationally much more efficient and less expensive.
10
CHAPTER 2LITERATURE REVIEW
Numerous studies in lightning from different aspects have been reported in the past
few decades. This review is focused on different approaches for the change in charge due to
lightning. The neutralization of charge by lightning in the models with explicit lightning
channels is discussed in Section 2.1. The approach used by Hager et al. is discussed briefly
in Section 2.2, and studied further in Chapter 3.
2.1 Lightning Models with Explicit Lightning Channels
2.1.1 Helsdon’s Model
Helsdon et al. [15–18] estimated both the geometry and charge distribution of an
intercloud lightning flash in a two-dimensional Storm Electrification Model (SEM)which
has been extended to a three-dimensional numerical cloud model later. Adapting ideas
from Kasemir [19], the parameterized lightning propagated bidirectionally (initially
parallel and antiparallel to the electric field) from the point of initial breakdown and
developed segments of opposite charge polarity.
Initiation, propagation direction, and termination of the discharge were computed
using the magnitude and direction of the electric field vector as the determining criteria.
The charge redistribution associated with lightning was approximated by assuming that
the channel remained electrically neutral over its total length.Their discharge followed
the electric-field lines until the termination condition was satisfied. Therefore, their
parametrization produced a single, unbranched channel.
As an initial critera, a threshold of electric field of 400 kV/m was chosen. The
channel was extended in both directions along the field line until the ambient electric-field
magnitude fell below a certain threshold (150 kV/m) at the locations of the channel-termination
points. They assumed that the linear charge density at a grid point, P , along the channel
was proportional to the difference between the potential at the point where the discharge
11
originated, and the potential at P . The linear charge density can be given by
QP = −k(ΦP −Φ0),
where QP was the charge density at P , and ΦP and Φ0 were the potentials at P and the
initiation point of the discharge, respectively. The value of this proportionality constant k
controlled the amount of charge transferred by the discharge. They extended the channel
by four grid points at each end and adjusted the charge distribution at each end of the
channel in order to maintain charge neutrality over the channel. In this extended region,
they assumed that the charge density decreased like e−αx2, where x is the distance from
the channel.
2.1.2 MacGorman’s Model
MacGorman et al. [23] suggested a lightning parametrization that was considered
an extension of the parametrization of Helsdon et al. [18] in conjunction with some of
the bulk-lightning parametrization methods presented by Ziegler and MacGorman [34].
MacGorman et al. [23] developed a parametrization to enable cloud models to simulate
the location and structure of individual lightning flashes by using the conceptual model of
MacGorman et al. [22] and Williams et al. [31]. Their parametrization proceeded in two
stages. Using the ideas of Helsdon et al. [18], a flash traced the electric-field line from an
initiation point outward in both parallel and antiparallel directions until the magnitude
of the ambient electric field at each end fell below some certain threshold value. When
one end of the channel reached ground, the parametrization terminated at that end, but
allowed the other end to continue developing.
Charge estimation and neutralization were parameterized by applying the technique
proposed by Ziegler and MacGorman [34], except that Ziegler and MacGorman neutralized
charge at all grid points having |ρ(i, j, k)| ≥ ρ1 (where ρ(i, j, k) was the net chargedensity at the grid point (i, j, k) and ρ1 was the minimum |ρ(i, j, k)| for all grid points
12
to be involved in lightning beyond initial propagation) throughout the storm, but their
parametrization neutralized charge only at such grid point within a single localized flash.
2.1.3 Mansell’s Model
Mansell et al. [24] proposed a lightning parametrization derived from the dielectric
breakdown model that was developed by Niemeyer et al. [25] and Wiesmann and Zeller
[30] to simulate electric discharges. They extended the dielectric breakdown model to
a three-dimensional domain to represent more realistic electric field and thunderstorm
dynamics.
In their work, the stochastic lightning model (SLM) was an application of the
Wiesmann-Zeller model to simulate bidirectional discharges in the regions of varying net
charge density (e.g., in an electrified thunderstorm). Procedures for simulating lightning
flashes in the thunderstorm model were as follows. A flash occurred when the magnitude
of the electric field exceeded the initiation threshold Einit anywhere in the model domain.
The lightning initiation point was chosen randomly from among all the points where
the magnitude of the electric field is greater than 0.9Einit. Both decisions for choosing
the initiation threshold and the initiation point were made according to MacGorman et
al. [23]. Positive and negative parts of the flash were propagated independently so that
up to two new channel segments (positive and negative) could be added at each step.
Both ends had default initial propagation thresholds of 0.75Einit. For flash neutrality,
they applied the ideas from Kasemir [19] and assumed that the channel structure would
maintain overall charge neutrality as long as neither end reached the ground. But, for
computational simplicity, their parametrization maintained near-neutrality (within 5%) by
a technique of adjusting the reference potential to the growth of the lightning structure
instead of adjusting the reference potential of the channel.
13
2.2 Hager’s Model: The Discrete Model
Hager et al. [11, 13, 14] proposed a three-dimensional lightning-discharge model that
produced bidirectional IC and -CG flashes. The model generated the discharge region,
charge transfer, and detailed charge rearrangement associated with the flash.
Their approach to lightning was quite different from those in Section 2.1. Their
breakdown model was based on Maxwell’s equations. They assumed that current due to
transport of charge under the influence of wind was known. They obtained an equation
governing the evolution of the electric potential under the assumption that the time
derivative of the magnetic field can be disregarded. After integrating this equation over
boxes and approximating derivatives by finite differences, they obtained an implicit system
of difference equations describing the evolution of the electric field. Their approach to
lightning was to let the conductivity tend to infinity wherever the electric field reached the
breakdown threshold. This approach appeals to our basic conception of nature: When the
electric field reaches breakdown threshold, conductivity becomes very large as a plasma
forms.
When the electric field reaches the breakdown threshold, the electric potential changes
instantaneously everywhere within the thundercloud. The Inverse Matrix Modification
Formula [10] was applied to evaluate this change:
Φafter = Φbefore −A−1U(U>A−1U)−1U>Φbefore, (2–1)
where Φbefore was the electric potential before discharge, Φafter was the electric potential
after discharge, A was the discrete Laplacian, and U was a matrix with a +1 and -1 in
each column corresponding to the arcs associated with the breakdown. There were no
parameters in Equation (2–1) besides the electric potential before discharge. This was
consistent with experimental observations: The charge is controlled predominately by a
single parameter: the local electrostatic field. This was observed in experiments reported
by Williams et al. [31].
14
CHAPTER 3THE DISCRETE MODEL
3.1 Governing Equations
3.1.1 Maxwell’s Equations
Maxwell’s equations are a set of four equations, first written down in complete form
by physicist James Clerk Maxwell, that describe the behavior of both the electric and
magnetic fields. Maxwell’s four equations express how electric charges produce electric
fields (Gauss’s law), the experimental absence of magnetic monopoles, how currents and
changing electric fields produce magnetic fields (Ampre’s law), and how changing magnetic
fields produce electric fields (Faraday’s law of induction).
In the absence of magnetic or polarizable media, the differential form of these
equations are:
1. Gauss’ law for electricity: ∇ · E = ρε
2. Gauss’ law for magnetism: ∇ ·H = 03. Faraday’s law of induction: ∇× E = −∂B
∂t
4. Ampere’s law: ∇×H = J0 + ε∂E∂t
where E is the electric field, H is the magnetic field strength, B is the magnetic flux
density, ρ is the charge density, ε is the permittivity of air, J0 is the current density, ∇· isthe divergence operator, and ∇× is the curl operator.3.1.2 Derivation of the Governing Equations
By Ampere’s law, the curl of the magnetic field strength H is given by
∇×H = J0 + ε∂E∂t
. (3–1)
Since J0 is partly due to the movement of charged ice and water particles in the cloud and
partly due to the electrical conductivity of the cloud, we write
J0 = Jp + σ0E
15
where σ0 is the conductivity of the atmosphere. In this model, we assume the time
derivative of the magnetic flux density is zero, i.e.,
∂B
∂t= 0.
Hence, the curl of E vanishes by Faraday’s law and E is the gradient of a potential φ:
E = −∇φ.
Therefore (3–1) becomes
∇×H = Jp − σ0∇φ− ε∂∇φ∂t
. (3–2)
Taking the divergence of (3–2), it follows that
0 = ∇ · ∇ ×H = ∇ · Jp −∇ · σ0∇φ− ε∇ · ∂∇φ∂t
. (3–3)
Letting σ = σ0/ε and J = Jp/ε, we obtain
∇ · ∂∇φ∂t
+∇ · σ∇φ−∇ · J = 0. (3–4)
In our model we also have the following assumptions:
• Let EB be the breakdown field strength. Then the electric field magnitude is alwaysless than or equal to the breakdown threshold EB. That is, |E| ≤ EB.
• When the electric field reaches the breakdown threshold EB at some point, theconductivity tends to infinity in a small neighborhood of that point. That is,σ(x) →∞ wherever |E(x)| = EB.
3.2 The Model in One-dimension
To illustrate the ideas used to obtain (2–1) in Section 2.2, we now focus on the
equation (3–4) in one dimension. In one-dimension, (3–4) reduces to
φ̇′′ + (σφ′)′ − J ′ = 0, φ(0) = 0, φ(H) = V, (3–5)
16
where H is the distance to ionosphere and the domain is from the surface of the earth
up to the ionosphere. the domain [0, H] is discretized using N discretization points. Let
h = H/N be the mesh spacing, xi be the ith grid point, φi be the approximation to
potential at xi + h/2, σi be the conductivity at xi, and Ji be the current density at xi.
Integrating (3–5) over [xi, xi+1] gives
(φ̇′(xi+1)− φ̇′(xi)
)+ (σi+1φ
′(xi+1)− σiφ′(xi))− (Ji+1 − Ji) = 0
To approximate φ′, [11, 13, 14] used backward differences,
φ′(xi) ≈ φi − φi−1h
,
which yield the discrete equation
(φ̇i+1 − φ̇i
h− φ̇i − φ̇i−1
h
)+ σi+1
φi+1 − φih
− σi φi − φi−1h
− (Ji+1 − Ji) = 0
This can be written in matrix form as
⇒ AΦ̇ + BΦ = h∆J
where
A =
2 −1−1 2 −1
−1 2 −1−1 2 •
• •
and
B =
σ1 + σ2 −σ2−σ2 σ2 + σ3 −σ3
−σ3 σ3 + σ4 −σ4−σ4 σ4 + σ5 •
• •
.
17
Integrating the matrix equation over the interval [0, ∆t] gives
Φ(t + ∆t) = e−A−1B∆tΦ(t) + h
∫ ∆t0
e−A−1B(∆t−s)∆J(s)ds
If
∣∣∣∣φk − φk−1
h
∣∣∣∣ = Eb, then Hager lets σk → σk + τ where τ → +∞. It is shown thatas σk → σk + τ , B → B + τwwT, where wk = 1, wk+1 = −1, wi = 0 otherwise. Next,they calculated the limit as τ →∞ and ∆t → 0 and obtained the formula for the potentialafter the lightning:
Φ(t+) = Φ(t)−A−1w(wTA−1w)−1wTΦ(t).
3.3 Generalization to Three-dimension
Suppose∣∣∣φk−φk−1h
∣∣∣ = Eb at k = k0, k1, . . . , kl. For each such k, σk → +∞,
B → B +l∑
i=0
τiwiwTi .
Here, wi is zero except for +1 and −1 in components ki and ki+1. Taking limits asτi → +∞ and ∆t → 0 yields
Φ(t+) = Φ(t)−A−1Wl(WTl A−1Wl)−1WTl Φ(t)
where Wl = [w0| . . . |wl].
18
CHAPTER 4THE CONTINUOUS MODEL
4.1 Formulation of the Equations
By (3–4), the following equations model the evolution of the electric potential in a
domain Ω, assuming the time derivative of the magnetic flux density can be neglected:
∂∆φ
∂t= −∇ · (σ∇φ) +∇ · J in Ω× [0,∞), (4–1)
φ(x, t) = 0, (x, t) ∈ ∂Ω× [0,∞), (4–2)
φ(x, 0) = φ0(x), x ∈ Ω, (4–3)
where Ω ⊂ Rn is a bounded domain with boundary ∂Ω, σ ≥ 0 lies in the space L∞(Ω)of essentially bounded functions defined on Ω, and the forcing term J lies in L2(Ω), the
usual space of square integrable functions defined on Ω. The divergence ∇·J as well as theequation (4–1) are interpreted in a weak sense, as explained later. The initial condition
φ0 is assumed to lie in H10 (Ω), the Sobolev space consisting of functions which vanish on
∂Ω and with first derivatives in L2(Ω). The evolution problem (4–1)–(4–3) has a solution
φ(x, t) with φ(·, t) ∈ H10 (Ω) and the partial derivative ∂tφ(·, t) ∈ H10 (Ω). AlthoughMaxwell’s equations describe the electromagnetic fields in 3-dimensions, the mathematical
analysis throughout this dissertation is developed in an n-dimensional setting, where n is
an arbitrary positive integer.
In a thunderstorm, σ is the conductivity divided by the permittivity of the atmosphere,
Ω is any large domain extending from the Earth to the ionosphere which contains the
thundercloud in its interior, and J is due to transport by wind of charged ice and water
particles in the cloud. Although J is a function of time, we focus on the potential change
during lightning, which we consider infinitely fast. Hence, during a lightning event, J is
essentially time invariant. The potential of the ionosphere is not zero, however, we can
make a change of variables to subtract off the “fair-field potential” (the potential of the
19
atmosphere when the thundercloud is removed) and transform the problem to the form
(4–1)–(4–3) where the potential vanishes on ∂Ω.
A possible lightning channel is sketched in Figure 4-1. Mathematically, the lightning
L
Ω
Figure 4-1: A sketch of L and Ω for a lightning discharge.
channel L could be any connected, open set contained in Ω with its complement Lc = Ω\Lconnected. More realistically, we should view L as a connected network of thin open tubes.The case where L touches ∂Ω, as would happen during a cloud-to-ground flash, is treatedas a limit in which L approaches arbitrarily closely to the boundary of Ω.
Making the change of variables φ = (−∆)− 12 Φ, we rewrite (4–1) as
∂Φ
∂t= −AσΦ +∇ · J, (4–4)
where
Aσ = −(−∆)− 12 (∇ · (σ∇))(−∆)− 12 .
Here (−∆)− 12 : L2(Ω) → H10 (Ω) denotes the inverse of the square root of the Laplacian([3]).
Let Ψ be the characteristic function for L (Ψ is identically 0 except in L where Ψ is1). The effect of lightning is to ionize the domain L, in essence, replacing σ in (4–4) byσ + τΨ where τ is large. If the lightning occurs at t = 0, then in the moments after the
20
lightning, the electric potential is governed by the equation
∂Φ
∂t= −(Aσ + τAΨ)Φ +∇ · J, (4–5)
where
AΨ = −(−∆)− 12 (∇ · (Ψ∇))(−∆)− 12 ,
in Ω× [0,∞) subject to the boundary conditions (4–2) and (4–3). Here the scalar τ reflectsthe change in conductivity in the lightning channel.
Let Φτ (x, t) denote the solution of (4–5) at time t, and let φτ (x, t) = (−∆)− 12 Φτ (x, t)be the corresponding solution of (4–1). If the lightning occurs at time t = 0, then the
electric potential right after the flash is given by
limt→0+
limτ→∞
φτ (x, t). (4–6)
Using an eigendecomposition for AΨ, the limit (4–6) can be evaluated.
4.2 Eigenproblem for AΨ
In this section, we show ([2]) that the eigenproblem for AΨ reduces to the following
generalized eigenproblem: Find u ∈ H10 (Ω), u 6= 0, and λ ∈ R such that
〈∇u,∇v〉L = λ〈∇u,∇v〉Ω (4–7)
for all v ∈ H10 (Ω), where 〈·, ·〉Ω is the L2(Ω) inner product
〈∇u,∇v〉Ω =∫
Ω
∇u · ∇v dx. (4–8)
We view H10 (Ω) as a Hilbert space for which the inner product between functions u and
v ∈ H10 (Ω) is given by (4–8).The weak form of the eigenproblem for AΨ is to find U ∈ L2(Ω) such that
〈AΨU, V 〉Ω = 〈(Ψ∇)(−∆)− 12 U,∇(−∆)− 12 V 〉Ω = λ〈U, V 〉Ω (4–9)
21
for all V ∈ L2(Ω). Let u = (−∆)−1/2U and v = (−∆)−1/2V denote the correspondingfunctions in H10 (Ω). Consequently, (4–9) reduces to the problem of finding u ∈ H10 (Ω) suchthat
〈∇u,∇v〉L = λ〈(−∆) 12 u, (−∆) 12 v〉Ω (4–10)
for all v ∈ H10 (Ω). If u ∈ C∞0 (Ω), then
〈(−∆) 12 u, (−∆) 12 v〉Ω = −〈∆u, v〉Ω = 〈∇u,∇v〉Ω. (4–11)
Since C∞0 (Ω) is dense in H10 (Ω) and the operators (−∆)
12 and ∇ are both bounded in
H10 (Ω), the identity (4–11) is valid for all u ∈ H10 (Ω). Hence, (4–10) reduces to (4–7).
22
CHAPTER 5GENERALIZED EIGENPROBLEM FOR THE LAPLACIAN
This chapter is based on the paper [2].
5.1 Introduction
Our analysis identifies four classes of eigenfunctions for the generalized eigenproblem
(4–7):
1. The function Π which is 1 on L and harmonic on Ω \ L; the eigenvalue is 0.
2. Functions in H10 (Ω) with support in Ω \ L; the eigenvalue is 0.
3. Functions in H10 (Ω) with support in L; the eigenvalue is 1.
4. Excluding Π, the harmonic extension of the eigenfunctions of a double layer potentialon ∂L. The eigenvalues are contained in the open interval (0, 1). The only possibleaccumulation point is λ = 1/2.
5.2 Eigenfunctions of Type 1, 2, and 3
In this section, we derive the eigenfunctions of types 1, 2, and 3. By (4–7), we have
λ =〈∇u,∇u〉L〈∇u,∇u〉Ω =
〈∇u,∇u〉L〈∇u,∇u〉L + 〈∇u,∇u〉Lc , (5–1)
which implies that 0 ≤ λ ≤ 1. Let H10 (L) ⊂ H10 (Ω) denote the subspace consisting offunctions with support in L. Similarly, let H10 (Lc) ⊂ H10 (Ω) denote the subspace consistingof functions with support in Lc.
Proposition 1. λ = 1 and u ∈ H10 (Ω) is an eigenpair of (4–7) if and only if the supportof u is contained in L. If u ∈ H10 (Lc), then u is an eigenfunction of (4–7) corresponding tothe eigenvalue 0. The only other eigenfunction of (4–7) corresponding to the eigenvalue 0,
which is orthogonal to H10 (Lc), is the solution Π ∈ H10 (Ω) of
〈∇Π,∇v〉Ω = 0 for all v ∈ H10 (Ω), Π = 1 on L. (5–2)
Proof. If λ = 1 and u ∈ H10 (Ω) is an eigenpair of (4–7), then by (5–1), we have
〈∇u,∇u〉Lc = 0.
23
Hence, ∇u = 0 in Lc, which implies that u is constant in Lc since Lc is connected. Sinceu ∈ H10 (Ω), u = 0 in Lc. Conversely, if u = 0 in Lc, then by (4–7), u is an eigenfunctioncorresponding to the eigenvalue 1. If u = 0 in L, then u is an eigenfunction correspondingto the eigenvalue 0. The solution Π of (5–2) is an eigenfunction of (4–7) corresponding to
the eigenvalue 0 since ∇Π = 0 in L.Let w ∈ H10 (Ω) be any eigenfunction of (4–7) corresponding to the eigenvalue 0 which
is orthogonal to H10 (Lc). By (4–7), we have 〈∇w,∇w〉L = 0, which implies that ∇w = 0in L, or w is constant in L since L is connected. Without loss of generality, let us assumethat w = 1 in L. Since w is orthogonal to the functions v ∈ H10 (Lc), we have
〈∇w,∇v〉Ω = 0 for all v ∈ H10 (Lc).
Combining this with (5–2) gives
〈∇(w − Π),∇v〉Ω = 0 for all v ∈ H10 (Lc).
Since ∂Lc = ∂Ω ∪ ∂L and since w − Π vanishes on both ∂Ω and ∂L, it follows thatw = Π.
5.3 Reformulation of Eigenproblem in H Using Double-Layer PotentialProposition 1 describes eigenfunctions of type 1, 2, and 3. In this section, we focus on
type 4 eigenfunctions. Let H be the space which consists of all u ∈ H10 (Ω) satisfying theconditions
〈∇u,∇v〉Ω = 0 for all v ∈ H10 (L) and (5–3)
〈∇u,∇w〉Ω = 0 for all w ∈ H10 (Lc). (5–4)
H is a subspace of H10 (Ω) consisting of functions harmonic in L and Lc (∆u = 0 in L and∆u = 0 in Lc). Note that Π ∈ H. Since H10 (L) and H10 (Lc) are orthogonal with respect tothe H10 (Ω) inner product, and since H is the orthogonal complement of H10 (L)⊕H10 (Lc) in
24
H10 (Ω), we have the orthogonal decomposition
H10 (Ω) = H⊕H10 (L)⊕H10 (Lc).
The following series of lemmas reformulates the generalized eigenvalue problem (4–7) on Hin terms of an integral operator.
Lemma 1. u ∈ H is a solution of the generalized eigenproblem (4–7) if and only if
∂u
∂n
−= −λ
[∂u
∂n
]on ∂L, (5–5)
where [∂u
∂n
]=
∂u
∂n
+
− ∂u∂n
−∈ H−1/2(∂L).
Here n is the outward unit normal to L and the − and + refer to the limits from theinterior and exterior of L respectively.
Proof. First we show a generalized eigenpair also satisfies (5–5). By (4–7) we have
〈∇u,∇v〉L = λ〈∇u,∇v〉Ω = λ (〈∇u,∇v〉Lc + 〈∇u,∇v〉L) (5–6)
for any v ∈ H10 (Ω). Integrating by parts and utilizing the fact that u is harmonic in bothL and Lc gives ∫
∂Lv∂u
∂n
−dγ = −λ
∫
∂Lv∂u
∂n
+
dγ + λ
∫
∂Lv∂u
∂n
−dγ, (5–7)
where γ denotes the boundary measure on ∂L. Hence, we have∫
∂Lv
[∂u
∂n
−+ λ
(∂u
∂n
+
− ∂u∂n
−)]dγ = 0
for any v ∈ H10 (Ω). Since any v ∈ H1/2(∂L) has an H10 (Ω) extension, (5–5) holds.Conversely, suppose that u satisfies (5–5). As in (5–6)–(5–7), we have
λ〈∇v,∇u〉Ω = −λ∫
∂Lv∂u
∂n
+
dγ + λ
∫
∂Lv∂u
∂n
−dγ
25
Applying (5–5) gives
λ〈∇v,∇u〉Ω =∫
∂Lv∂u
∂n
−= 〈∇v,∇u〉L
since u is harmonic in L. Hence, u satisfies (4–7).
Now let us introduce the Green’s function on Ω:
∆yG(x, y) = δx(y) in Ω, G(x, y) = 0 for y ∈ ∂Ω, (5–8)
where δx is the Dirac delta function located at x. The piecewise harmonic functions u ∈ Hcan be described in terms of the jump on ∂L of the normal derivative.
Lemma 2. Suppose that ∂L and ∂Ω are C2. If u ∈ H, x ∈ Ω, and x /∈ ∂L, then
u(x) =
∫
∂L
[∂u
∂n
](y)G(x, y)dγy. (5–9)
Proof. Since u is harmonic away from ∂L, it is continuous there, and so for x /∈ ∂L,
u(x) =
∫
Ω
u(y)∆yG(x, y)dy
=
∫
Lu(y)∆yG(x, y)dy +
∫
Lcu(y)∆yG(x, y)dy.
Since u is smooth in each subdomain, we can integrate by parts to obtain
u(x) = −∫
L∇u · ∇yGdy +
∫
∂Lu−
∂G
∂nydγy −
∫
Lc∇u · ∇yGdy −
∫
∂Lu+
∂G
∂nydγy.
Since u is smooth on each subdomain and u ∈ H1(Ω), the traces u+ and u− ∈ H1/2(∂L)must satisfy u− = u+ on ∂L. Since ∆u = 0 on each subdomain and G = 0 on ∂Ω, we have
u(x) = −∫
L∇u · ∇yGdy −
∫
Lc∇u · ∇yGdy
= −∫
∂L
∂u
∂n
−(y)G(x, y)dγy +
∫
∂L
∂u
∂n
+
(y)G(x, y)dγy
=
∫
∂L
[∂u
∂n
](y)G(x, y)dγy, (5–10)
which yields (5–9).
26
The following Lemma is well known for free space potentials (see for example
Theorem 3.22 of [8]); we state it here for the case of our Green’s function corresponding to
a bounded outer domain.
Lemma 3. Suppose φ ∈ H1/2(∂L) and both ∂L and ∂Ω are C2. For x ∈ Ω, x /∈ ∂L, letv(x) be defined by
v(x) =
∫
∂Lφ(y)
∂G
∂ny(x, y)dγy.
The trace v+ of v onto ∂L from the exterior of L and the trace v− of v onto ∂L from theinterior L are given by
v+(x) = −12φ(x) +
∫
∂Lφ(y)
∂G
∂ny(x, y)dγy,
and
v−(x) =1
2φ(x) +
∫
∂Lφ(y)
∂G
∂ny(x, y)dγy.
Proof. Let N(x, y) be the free space Green’s function for the Laplacian,
N(x, y) =
|x− y|2−n(2− n)ωn n > 2,
− 12π
log |x− y| n = 2,(5–11)
where ωn is the surface area of the unit sphere in Rn. Recall that
∆yN(x, y) = δx(y).
Define
H(x, y) := G(x, y)−N(x, y).
By the definition of v,
v(x) =
∫
∂Lφ(y)
∂H
∂ny(x, y)dγy +
∫
∂Lφ(y)
∂N
∂ny(x, y)dγy. (5–12)
27
For any x ∈ Ω, H satisfies
∆yH(x, y) = 0 for y ∈ Ω,
H(x, y) = −N(x, y) for y ∈ ∂Ω.
Hence, H(x, y) is harmonic for y ∈ Ω with smooth boundary data. This implies that thefunction w(x) defined by
w(x) =
∫
∂Lφ(y)
∂H
∂ny(x, y)dγy
is continuous in a neighborhood of ∂L since the kernel has no singularity. For the secondterm of (5–12) we can apply the well known result (see [8]) for the limit x → ∂L. ¿Fromthe exterior of L, we have
v+(x) =
∫
∂Lφ(y)
∂H
∂ny(x, y)dγy − 1
2φ(x) +
∫
∂Lφ(y)
∂N
∂ny(x, y)dγy
= −12φ(x) +
∫
∂Lφ(y)
∂G
∂ny(x, y)dγy.
The proof for the interior limit is similar.
Using Lemma 1, 2, and 3, we reformulate the generalized eigenproblem (4–7) on
H in terms of a boundary integral operator. By the trace theorem [1, Thm. 7.53], anyu ∈ H ⊂ H10 (Ω) has a trace on ∂L in H1/2(∂L). Conversely, u ∈ H1/2(∂L) has a uniqueharmonic extension into both L and Lc with u = 0 on ∂Ω. Hence, there is a one-to-onecorrespondence between elements of H and elements of H1/2(∂L).
Define
T : L2(∂L) → L2(∂L)
by
Tφ(x) =
∫
∂Lφ(y)K(x, y)dγy, K(x, y) :=
∂G
∂ny(x, y). (5–13)
By [8, Prop. 3.17], K is a continuous kernel of order n− 2 on ∂L. It follows from [8, Prop.3.12] that T is a compact operator from L2(∂L) to itself.
28
Proposition 2. If both ∂L and ∂Ω are C2, then (u, λ) ∈ H × R is a generalized eigenpairfor (4–7) if and only if the corresponding u ∈ H1/2(∂L) is an eigenfunction of T withassociated eigenvalue 1/2− λ; that is,
Tu = (1/2− λ)u. (5–14)
Proof. First, let us assume that (u, λ) ∈ H × R is a generalized eigenpair for (4−−7). ByLemmas 1 and 2, we have
λu(x) = −∫
∂L
∂u
∂n
−(y)G(x, y)dγy
for x ∈ Ω and x /∈ ∂L. We integrate by parts to obtain
λu(x) = −∫
L∇u(y)∇yG(x, y)dy
= −∫
∂Lu(y)
∂G
∂ny(x, y)dγy +
∫
Lu(y)∆yG(x, y)dy.
If x ∈ Lc, then the second term above disappears, and we have
λu(x) = −∫
∂Lu(y)
∂G
∂ny(x, y)dγy,
an equation for a double layer potential. We let x ∈ Lc approach ∂L. According toLemma 3,
λu(x) =1
2u(x)−
∫
∂Lu(y)
∂G
∂ny(x, y)dγy,
which is equivalent to (5–14).
Conversely, suppose that u ∈ H1/2(∂L) satisfies (5–14). We identify u with itsharmonic extension in H, and we define w(x) by
w(x) =
−∫
∂Lu(y)
∂G
∂ny(x, y)dγy for x ∈ Lc,
−∫
∂Lu(y)
∂G
∂ny(x, y)dγy + u(x) for x ∈ L.
(5–15)
29
In either Lc and L, w is harmonic. By Lemma 3, we have
w+(x) =1
2u(x)−
∫
∂Lu(y)
∂G
∂ny(x, y)dγy
and
w−(x) = u(x)− 12u(x)−
∫
∂Lu(y)
∂G
∂ny(x, y)dγy
=1
2u(x)−
∫
∂Lu(y)
∂G
∂ny(x, y)dγy.
Utilizing (5–14) yields
w+ = w− = (1/2− T )u = λu on ∂L. (5–16)
Observe that w vanishes on ∂Ω due to the symmetry of G(x, y) [8, Lem. 2.33]; that
is, since G(x, y) = 0 when y ∈ ∂Ω, we have by symmetry G(x, y) = 0 when x ∈ ∂Ω.Hence, the normal derivative in (5–15) vanishes when x ∈ ∂Ω. Since w is harmonic in eachsubdomain and it is equal to λu on both ∂L (see (5–16)) and ∂Ω (they both vanish), itfollows that w = λu in Ω. We replace w with λu in (5–15) to obtain
λu(x) =
−∫
∂Lu(y)
∂G
∂ny(x, y)dγy for x ∈ Lc,
−∫
∂Lu(y)
∂G
∂ny(x, y)dγy + u(x) for x ∈ L.
(5–17)
Integrating by parts and using (5–8) gives
∫
∂L
∂u
∂n
−(y)G(x, y)dγy = 〈∇u,∇yG〉L
=
∫
∂Lu(y)
∂G
∂ny(x, y)dγy if x ∈ Lc,
∫
∂Lu(y)
∂G
∂ny(x, y)dγy − u(x) if x ∈ L.
Referring to (5–17), we conclude that
λu(x) = −∫
∂L
∂u
∂n
−(y)G(x, y)dγy. (5–18)
30
By Lemma 2,
λu(x) = λ
∫
∂L
[∂u
∂n
](y)G(x, y)dγy. (5–19)
Subtracting (5–19) from (5–18) gives
s(x) :=
∫
∂Lφ(y)G(x, y)dγy = 0 for any x /∈ ∂L,
where
φ(y) = −∂u∂n
−(y)− λ
[∂u
∂n
](y).
Hence, s = 0 almost everywhere in Ω. If φ = 0, then
∂u
∂n
−= −λ
[∂u
∂n
],
in which case Lemma 1 completes the proof.
To prove that φ = 0, suppose to the contrary that φ does not vanish. Let r be any
smooth function defined on ∂L for which∫
∂Lφ(y)r(y)dγy 6= 0.
Let r also denote any smooth extension in Ω which vanishes on ∂Ω. By the symmetry of
G, we have
r(y) =
∫
Ω
[∆xr(x)]G(x, y)dx.
Forming the L2(Ω) inner product between s (which vanishes almost everywhere) and ∆r
yields
0 = 〈s, ∆r〉Ω =∫
Ω
∫
∂Lφ(y)[∆xr(x)]G(x, y)dγydx = 〈φ, r〉∂L 6= 0.
Hence, we have a contradiction and the proof is complete.
Corollary 1. If both ∂L and ∂Ω are C2, then the eigenvalues of the double layer potentialoperator T in (5–13) are real and contained in the half-open interval (−1/2, 1/2]. The onlypossible accumulation point for the spectrum is 0.
31
Proof. The eigenvalues of the generalized eigenproblem (4–7) are all real due to symmetry
of the inner product. By Proposition 2, the eigenvalues of T are all real. As noted before
Proposition 1, the eigenvalues of (4–7) are contained on the interval [0, 1]. Moreover, by
Proposition 1, the only eigenfunction corresponding to the eigenvalue 1 has support in L.The trace of this eigenfunction on ∂L is 0. The only element in H with vanishing trace on∂L is the zero function. Consequently, there is no eigenfunction in H corresponding to theeigenvalue 1. There is one eigenfunction in H corresponding to the eigenvalue 0, namelythe function Π of Proposition 1. Except for the eigenvalue 0, all the remaining eigenvalues
for the generalized eigenproblem lie in the open interval (0, 1). Since the eigenvalues of
T are 1/2 minus the corresponding eigenvalue of (4–7) in [0, 1), the proof is complete.
Since T is compact on L2(Ω) [8, Prop. 3.12], the only possible accumulation point for the
spectrum is 0.
A lower bound for the separation between the largest and second largest eigenvalues
of T is obtained from Proposition 3.
5.4 Eigenvalue Separation and Completeness of Eigenfunctions
Due to Proposition 1, the generalized eigenproblem (4–7) restricted to H has asimple eigenvalue λ = 0 corresponding to the eigenfunction Π ∈ H while the remainingeigenvalues are positive. By Proposition 2, the only possible accumulation point for the
spectrum is λ = 1/2. Hence, there is an interval (0, ρ), ρ > 0, where the generalized
eigenproblem has no eigenvalues. We now give an explicit positive lower bound for ρ in
terms of three embedding constants:
E1. Let ua denote the constant function on Ω whose value is the average of u ∈ H1(Ω)over L:
ua =1
measure(L)∫
Lu(x)dx.
By [7, Thm. 1, p. 275], there exists a constant θ1 > 0 such that
‖∇u‖2L2(L) ≥ θ1‖u− ua‖2H1(L)for all u ∈ H1(Ω).
32
E2. By [1, Thm. 7.53], there exists a constant θ2 > 0 such that
‖u‖2H1(L) ≥ θ2‖u‖2H1/2(∂L)for all u ∈ H1(Ω).
E3. There exists a constant θ3 > 0 such that
‖u‖2H1/2(∂L) ≥ θ3‖u‖2H1(Lc) (5–20)
for all u ∈ H10 (Ω) which are harmonic in Lc (in other words, (5–4) holds). Thefollowing proof of E3 was suggested by Jayadeep Gopalakrishnan: For u ∈ H10 (Ω),let T (u) ∈ H1/2(∂L) denote the trace of u evaluated on ∂L. By [9, Thm. 1.5.1.3],T has a continuous right inverse which we denote T −1. In other words, for eachg ∈ H1/2(∂L), we have T −1(g) ∈ H10 (Ω), T T −1(g) = g, and
‖T −1(g)‖H1(Ω) ≤ ‖T −1‖‖g‖H1/2(∂L).
Define v0 = u−T −1(g). Since v0 vanishes on both ∂Ω and ∂L, there exists a constantc > 0 such that (see [7, Thm. 3, p. 265])
‖v0‖H1(Lc) ≤ c‖∇v0‖L2(Lc). (5–21)
Moreover, since u is harmonic in Lc and v0 vanishes on both ∂Ω and ∂L, we have
〈∇v0,∇v0〉Lc = 〈∇v0,∇(u− T −1(g))〉Lc = −〈∇v0,∇T −1(g)〉Lc≤ ‖∇v0‖L2(Lc)‖∇T −1(g)‖L2(Lc)≤ ‖∇v0‖L2(Lc)‖T −1(g)‖H1(Lc),
which gives ‖∇v0‖L2(Lc) ≤ ‖∇T −1(g)‖H1(Lc). We combine this with (5–21) to obtain
‖v0‖H1(Lc) ≤ c‖∇T −1(g)‖H1(Lc).
Hence, by the triangle inequality,
‖u‖H1(Lc) ≤ ‖v0‖H1(Lc) + ‖T −1(g)‖H1(Lc)≤ (1 + c)‖T −1(g)‖H1(Lc)≤
((1 + c)‖T −1‖
)‖g‖H1/2(∂L),
which yields (5–20).
33
Proposition 3. If both ∂L and ∂Ω are Lipschitz, then the generalized eigenproblem (4–7)has no eigenvalues in the interval (0, ρ) where
ρ = min{1, θ2θ3}θ1/2.
Proof. Let µ be the smallest positive eigenvalue for the generalized eigenproblem (4–7),
and let u be an associated eigenfunction with normalization 〈∇u,∇u〉Ω = 1. If Π ∈ H isthe eigenfunction described in (5–2), then we have
µ = ‖∇u‖2L2(L)≥ θ1‖u− ua‖2H1(L) (5–22)
= θ1‖u− Πua‖2H1(L) (5–23)
≥ θ1θ2‖u− Πua‖2H
12 (∂L)
(5–24)
≥ θ1θ2θ3‖u− Πua‖2H1(Lc). (5–25)
Above, (5–23) is due to the fact that Π = 1 on L, while (5–22), (5–24), and (5–25) comefrom E1, E2, and E3 respectively.
Suppose that the proposition does not hold, in which case µ < θ1/2 and µ < θ1θ2θ3/2.
By (5–23) and (5–25), we have
‖u− Πua‖2H1(L) < 1/2 and ‖u− Πua‖2H1(Lc) < 1/2.
Combining these gives
‖u− Πua‖2H1(Ω) < 1. (5–26)
On the other hand, u and Π are orthogonal since these eigenfunctions correspond to
distinct eigenvalues. Since uaΠ is a multiple of Π which is orthogonal to u, it follows that
1 ≤ ‖∇(u− Πua)‖2L2(Ω) ≤ ‖∇(u− Πua)‖2H1(Ω). (5–27)
34
Comparing (5–26) and (5–27), we have a contradiction. Hence, either µ ≥ θ1/2 orµ ≥ θ1θ2θ3/2.
We continue to develop properties for the eigenfunctions of the generalized eigenproblem
(4–7) by exploiting the connection, given in Proposition 2, between the eigenfunctions of
the generalized eigenproblem (4–7) and those of the double layer potential T in (5–13).
As noted before Proposition 2, there is a one-to-one correspondence between elements
of H and elements of H1/2(∂L). If u ∈ H1/2(∂L), then the corresponding E(u) ∈ His the harmonic extension of u ∈ H1/2(∂L) into Ω which vanishes on ∂Ω. For anyu, v ∈ H1/2(∂L), we define the inner product
(u, v) = 〈∇E(u),∇E(v)〉Ω. (5–28)
In other words, harmonically extend u and v in Ω and form the H10 (Ω) inner product of
the extended functions. We now show that T is self adjoint and compact relative to this
new inner product.
Lemma 4. The following properties are satisfied:
T1. If ∂Ω and ∂L are Lipschitz, then the norm (·, ·)1/2 is equivalent to the usual norm forH1/2(∂L). That is, there exist positive constants c1 and c2 such that
c1(v, v) ≤ ‖v‖2H1/2(∂L) ≤ c2(v, v)
for all v ∈ H1/2(∂L).
T2. If ∂Ω and ∂L are C2, then the double layer potential operator T in (5–13) is self-adjoint relative to the inner product (5–28).
T3. If ∂Ω is C2 and ∂L is C2,α, then T is a compact operator from H1/2(∂L) intoH1/2(∂L).
Proof. We begin by showing that the norm of H1/2(∂L) and the norm (·, ·)1/2 areequivalent. First, recall [7, p. 265] that there exists a constant θ4 > 0 such that
‖∇v‖2L2(Ω) ≥ θ4‖v‖2H1(Ω)
35
for each v ∈ H which vanishes on ∂Ω. Combining this with E2 gives the lower bound
(v, v) = 〈∇E(v),∇E(v)〉Ω ≥ θ4‖E(v)‖2H1(Ω)≥ θ4‖E(v)‖2H1(L) ≥ θ2θ4‖v‖2H1/2(∂L). (5–29)
An upper bound for (v, v) is obtained from E3:
(v, v) = 〈∇E(v),∇E(v)〉Ω ≤ ‖E(v)‖2H1(Ω)≤ (θ−13 + θ̄−13 )‖v‖2H1/2(∂L). (5–30)
Here θ̄3 > 0 is analogous to θ3 in (5–20) except that it relates L to ∂L:
‖v‖2H1/2(∂L) ≥ θ̄3‖E(v)‖2H1(L)
Relations (5–29) and (5–30) yield T1.
To show that T is self adjoint relative to the inner product (5–28), we must verify the
identity
(Tu, v) = 〈∇E(Tu),∇E(v)〉Ω = 〈∇E(u),∇E(Tv)〉Ω = (u, Tv) (5–31)
for all u and v ∈ H1/2(∂L). We first observe that the extension of Tu has the form
E(Tu) =
1
2E(u(x)) +
∫
∂Lu(y)
∂G(x, y)
∂nydγy for x ∈ Lc,
−12E(u(x)) +
∫
∂Lu(y)
∂G(x, y)
∂nydγy for x ∈ L.
(5–32)
By Lemma 3, the trace of the right side of (5–32) is Tu from either side of ∂L. Moreover,the right side is harmonic and it vanishes on ∂Ω since E(u) vanishes on ∂Ω and G(x, y) =
0, independent of y ∈ Ω, when x ∈ ∂Ω. Since the right side is harmonic and satisfies theboundary conditions associated with E(Tu), it must equal E(Tu).
36
Integrating by parts and utilizing (5–32), we obtain
(Tu, v) = 〈∇E(Tu),∇v〉Ω
= 〈∇E(Tu),∇E(v)〉L + 〈∇E(Tu),∇E(v)〉Lc
= −12
∫
∂L
(∂E(u)−
∂n+
∂E(u)+
∂n
)E(v)dγ. (5–33)
The term in E(Tu) associated with the Green’s function cancels since the normal
derivative of a double layer potential operator is continuous across ∂L (for example,see [6, Thm. 3.1], [5, Thm. 2.21], [21, Thm. 6.13]).
For any p and q ∈ H, we have the identities
〈∇p,∇q〉L =∫
∂Lq∂p
∂n
−dγ =
∫
∂Lp∂q
∂n
−dγ,
and
〈∇p,∇q〉Lc = −∫
∂Lq∂p
∂n
+
dγ = −∫
∂Lp∂q
∂n
+
dγ.
Hence, the normal derivatives in (5–33) can be moved from the u terms to v to obtain
(Tu, v) = −12
∫
∂L
(∂E(v)−
∂n+
∂E(v)+
∂n
)E(u)dγ = (u, Tv),
which establishes T2.
We now show that T is compact on H1/2(∂L). Consider the corresponding free spacedouble layer potential operator TF defined by
TF φ(x) =
∫
∂Lφ(y)
∂N
∂ny(x, y)dγy,
where N is the free space Green’s function defined in (5–11). For n = 2, TF is compact
by [21, Thm. 8.20]. For n ≥ 3, Theorem 4.2 in [20] gives the boundedness of TF as a mapfrom L2(∂L) to H1(∂L). This result extends to our operator T as follows. The difference,
37
T − TF , is an integral operator on ∂L with kernel
∂H
∂ny(x, y) =
∂G
∂ny(x, y)− ∂N
∂ny(x, y).
For x ∈ ∂L, H has no singularity since it is harmonic with smooth boundary data (seethe proof of Lemma 3). Consequently, T − TF is bounded from L2(∂L) to H1(∂L).Since both TF and T − TF are bounded from L2(∂L) to H1(∂L), we conclude that T isbounded from L2(∂L) to H1(∂L). This implies that T is compact on H1/2(∂L) since H1
embeds compactly in H1/2; that is, by [9, Thm. 1.4.3.2] Hs embeds compactly in H t when
s > t ≥ 0. Hence, T is compact on H1/2(∂L).
Theorem 1. If ∂Ω is C2 and ∂L is C2,α, for some α ∈ (0, 1) (the exponent of Höldercontinuity for the second derivative), then any f ∈ H10 (Ω) has an expansion of the form
f =∞∑i=1
φi,
where the φi are eigenfunctions of (4–7) which are orthogonal relative to the inner product
(6–5). Here the convergence is with respect to the norm of H10 (Ω).
Proof. As pointed out earlier, we have the orthogonal decomposition
H10 (Ω) = H⊕H10 (L)⊕H10 (Lc).
By Proposition 1, any complete orthonormal basis for H10 (L) is an eigenfunction basiscorresponding to the eigenvalue 1. Likewise, any complete orthonormal basis for H10 (Lc) isa basis whose elements are eigenfunctions of the generalized eigenproblem corresponding to
the eigenvalue 0. To complete the proof, we need to show that any f ∈ H lies in the spanof the remaining eigenfunctions for (4–7).
By Lemma 4, T is compact and self adjoint relative to the inner product (·, ·) definedin (5–28). Hence, every f ∈ H1/2(∂L) has a unique expansion in terms of orthogonaleigenfunctions of T (for example, see [4, Thm. 1.28]). Given f ∈ H, its restriction to ∂L
38
lies in H1/2(∂L). Therefore, there exist orthogonal eigenfunctions φi, i ≥ 1, of T such that
f =∞∑i=1
φi on ∂L.
By the linearity and boundedness of the extension operator, we have
f =∞∑i=1
E(φi) on Ω.
By Proposition 2, E(φi) is an eigenfunction for the generalized eigenproblem (4–7).
39
CHAPTER 6THE LIMIT
This chapter is based on the paper [12].
6.1 Introduction
We expand the solution to (4–1) in terms of the eigenfunctions of (4–7) and analyze
limits to compute the change in the electric potential due to lightning discharge. Our main
result is the following:
Theorem 2. If ∂Ω is C2 and ∂L is C2,α, for some α ∈ (0, 1) (the exponent of Höldercontinuity for the second derivative), then the electric potential φ+ immediately after the
lightning discharge is given by
φ+(x) =
φL if x ∈ L,
φ0(x) + ξ(x) if x ∈ Lc,(6–1)
where
φL =〈∇φ0,∇Π〉Ω〈∇Π,∇Π〉Ω , (6–2)
and where Π and ξ are harmonic functions in Lc with boundary conditions as specifiedbelow:
∆Π = 0 in Lc, Π = 0 on ∂Ω, Π = 1 in L, (6–3)
∆ξ = 0 in Lc, ξ = 0 on ∂Ω, ξ = φL − φ0 on ∂L. (6–4)
Here 〈·, ·〉Ω is the L2(Ω) inner product
〈∇u,∇v〉Ω =∫
Ω
∇u · ∇v dx. (6–5)
Thus φ+ has the constant value φL along the lightning channel L and the changein the potential due to lightning has been expressed in terms of the potential φ0 before
the lightning and the lightning channel L. When L touches ∂Ω, as it would during acloud-to-ground flash, φL = 0 and Π can be eliminated. That is, as L approaches the
40
boundary of Ω, Π develops a jump singularity since Π = 1 on L and Π = 0 on ∂Ω. Hence,∇Π approaches a delta function as L approaches ∂Ω. Since the delta function is squaredin the denominator of φL while the numerator is finite, φL tends to 0 as L approaches ∂Ω.Thus in a cloud-to-ground flash, the change ξ in electric potential due to the lightning is
the solution to (6–4) with φL = 0.
6.2 Reformulation of the Continuous Equation
Let (φi, λi), i ∈ N, denote a complete orthonormal set of eigenfunctions for thegeneralized eigenproblem (4–7), as given by Theorem 1. We decompose N into the disjoint
union of four sets corresponding to the four classes of eigenfunctions described in the
introduction:
SΠ = {i ∈ N : φi = Π/‖∇Π‖L2(Ω)},
S0 = {i ∈ N : λi = 0, 〈∇φi,∇Π〉Ω = 0},
S1 = {i ∈ N : λi = 1},
S+ = {i ∈ N : 0 < λi < 1}.
The set SΠ contains precisely one element corresponding to the eigenfunction Π given by(6–3). The set S0 corresponds to eigenfunctions supported on Lc, while S1 corresponds toeigenfunctions supported on L. The set S+ corresponds to functions in H10 (Ω) which areharmonic in both L and Lc, and with the eigenvalues uniformly bounded away from 0.
The weak form of (4–5) is to find φ such that
∂
∂t〈∇φ,∇v〉Ω = −〈σ∇φ,∇v〉Ω − τ〈∇φ,∇v〉L + 〈Jp,∇v〉Ω (6–6)
for all v ∈ H10 (Ω). We substitute the eigenexpansion
φ(x, t) =∑
i∈Nαi(t)φi(x)
41
into (6–6). Taking v = φj, j = 1, 2, . . ., and utilizing the orthonormality of the
eigenfunctions yields the linear system
α̇ = −Aα− τDα + f , α(0) = α0, (6–7)
where the dot denotes time derivative and
(α0)i = 〈∇φ0,∇φi〉Ω, i ∈ N, (6–8)
aij = 〈σ∇φi,∇φj〉Ω, (6–9)
dij = 〈∇φi,∇φj〉L,
fi = 〈Jp,∇φi〉Ω. (6–10)
Since the φi are orthonormal eigenfunctions for (4–7), we have
dij = 〈∇φi,∇φj〉L = λi〈φi, φj〉Ω =
λi if i = j,
0 otherwise.
Hence, D is a diagonal matrix with the eigenvalues λi, i ∈ N, on the diagonal. Since theeigenvalues are nonnegative, D is positive semidefinite. We now consider A and f :
Lemma 5. The matrix A is positive semidefinite. The 2-norms of A and f , defined in
(6–9) and (6–10) respectively, are both finite, and we have
‖A‖ ≤ ess supx∈Ω
|σ(x)| := ‖σ‖L∞(Ω) and ‖f‖ = ‖Jp‖L2(Ω). (6–11)
Proof. By the definition of A, we have
xTAx =
〈σ
∞∑i=1
xi∇φi,∞∑i=1
xi∇φi〉
Ω
≥ 0 (6–12)
for all x ∈ `2 since σ ≥ 0. Since A is positive semidefinite, the Cauchy-Schwarz inequalityyields
|yTAx| ≤√
yTAy√
xTAx (6–13)
42
for all x and y ∈ `2. If ‖x‖ = 1, then by (6–12) we have
xTAx =≤ ‖σ‖L∞〈 ∞∑
i=1
xi∇φi,∞∑i=1
xi∇φi〉
Ω
= ‖σ‖L∞ .
Hence, (6–13) yields the first inequality in (6–11).
Let u ∈ H10 (Ω) be the weak solution to ∆u = −∇ · Jp:
〈∇u,∇v〉 = 〈Jp,∇v〉 for all v ∈ H10 (Ω).
We have
‖∇u‖2 = 〈∇u,∇u〉 = 〈Jp,∇u〉 ≤ ‖Jp‖‖∇u‖.
Dividing by ‖∇u‖ gives‖∇u‖ ≤ ‖Jp‖. (6–14)
For f defined in (6–10),
‖f‖2 =∞∑i=1
〈Jp,∇φi〉2Ω =∞∑i=1
〈∇u,∇φi〉2Ω = 〈∇u,∇u〉Ω ≤ ‖Jp‖2.
The last equality is due to the fact that the φi are a complete orthonormal basis relative
to the H10 (Ω) inner product, while the last inequality is (6–14).
6.3 Potential Change for the Continuous Equation
We now prove Theorem 2. Multiply (6–7) by αT and utilize the fact that D and A
are positive semidefinite to obtain
αTα̇ = −αTAα− ταTDα + αTf ≤ αTf .
Hence, we have
1
2
d
dt‖α‖2 = αTα̇ ≤ ‖α‖‖f‖ ≤ 1
2(‖α‖2 + ‖f‖2).
Multiplying by e−t and rearranging yields
d
dt
(e−t‖α‖2) ≤ e−t‖f‖2.
43
Integration over the interval [0, t] gives
‖α(t)‖2 ≤ et‖α(0)‖2 + (et − 1)‖f‖2. (6–15)
This shows that ‖α(·)‖ is uniformly bounded over any finite interval.For the remaining analysis, it is convenient if the eigenvalues are arranged in
decreasing order. Thus without loss of generality, we assume that
D =
Λ 0
0 0
where Λ is a diagonal matrix with strictly positive diagonal and the 0’s represent blocks
whose elements are all zero. The components of α are partitioned in a compatible way
into [p ; z] where p corresponds to the positive eigenvalues λi and z corresponds to the
zero eigenvalues.
Multiply (6–7) by [pT ; 0] to obtain
pTṗ = [p ; 0]Tα̇
= −[p ; 0]TAα− τ [p ; 0]T
Λp
0
+ [p ; 0]Tf .
Taking norms on the right side gives
1
2
d
dt‖p‖2 ≤ ‖α‖2‖A‖ − τpTΛp + ‖p‖‖f‖
≤ ‖α‖2‖A‖ − τλ0‖p‖2 + 12(‖p‖2 + ‖f‖2), (6–16)
where λ0 denotes the smallest positive eigenvalue; a positive lower bound for λ0 is
obtained in Proposition 3. Suppose τ is large enough that τλ0 ≥ 1. Choose t > 0and let c be the constant defined by
c = ‖f‖2 + 2‖A‖ maxs∈[0,t]
‖α(s)‖2,
44
which is finite due to (6–15). By (6–16), we have
d
dt‖p‖2 ≤ c− τλ0‖p‖2
on the interval [0, t] assuming τλ0 ≥ 1. Multiplying both sides by eτλ0t yields
d
dt
(eτλ0t‖p‖2) ≤ eτλ0tc.
Integration over the interval [0, t] gives
‖p(t)‖2 ≤ e−τλ0t‖p(0)‖2 + 1τλ0
(1− e−τλ0t)c
≤ e−τλ0t‖p(0)‖2 + cτλ0
.
Since the right side approaches 0 as τ tends to ∞, we conclude that for any t > 0,
limτ→∞
pτ (t) = 0.
Here we have inserted a τ subscript on p to remind us that the p-component of the
solution α to (6–7) depends on τ .
Now consider the bottom half of the equation (6–7):
żτ = A2ατ − f2, (6–17)
where A2 denotes the bottom half of A and f2 is the bottom half of f . Since the bottom
half of D is zero, the D term of (6–7) is not present in (6–17). Since ατ (·) is bounded overany finite interval, independent of τ by (6–15), it follows from (6–17) that zτ (t) approaches
z(0) as t tends to 0, independent of τ . To summarize, we have
limt→0+
limτ→∞
ατ (t) = [0 ; z(0)],
where z(0) is the vector of coefficients in the eigenfunction expansion of φ0 corresponding
to the eigenvalues λi = 0. These coefficients correspond to the index set SΠ ∪ S0. It follows
45
that
φ+(x) = limt→0+
limτ→∞
φτ (x, t) =∑
i∈SΠ∪S0αi(0)φi(x), (6–18)
where
αi(0) = 〈∇φ0,∇φi〉Ω.
For x ∈ L and i ∈ S0, φi(x) = 0 since φi, i ∈ S0, is supported on Lc. Hence, for x ∈ L,we have
φ+(x) =∑i∈SΠ
αi(0)φi(x).
Since φi for i ∈ SΠ is the normalized Π, αi(0)φi(x) is simply the projection of φ0 along Π:∑i∈SΠ
αi(0)φi(x) =
(〈∇φ0,∇Π〉Ω〈∇Π,∇Π〉Ω
)Π(x) = φLΠ(x). (6–19)
Since Π(x) = 1 for x ∈ L, the top half of (6–1) has been established.Now suppose that x ∈ Lc. By the completeness of the φi, we have
φ0(x) =∑
i∈Nαi(0)φi(x). (6–20)
Consequently, for x ∈ Lc, (6–18) can be rewritten
φ+(x) = φ0(x)−∑
i∈S1∪S+αi(0)φi(x) = φ0(x)−
∑i∈S+
αi(0)φi(x) (6–21)
since φi for i ∈ S1 vanishes on Lc. Let ξ denote the final term in (6–21):
ξ(x) = −∑i∈S+
αi(0)φi(x)
For i ∈ S+, we have ∆φi = 0 on Lc since the eigenfunctions associated with indices in S+are harmonic in either L or Lc. Hence, ∆ξ = 0 in Lc. ξ vanishes on ∂Ω since φi ∈ H10 (Ω).To obtain the boundary values for ξ on ∂L, we examine the eigenexpansion (6–20), whichcan be rearranged in the form
∑i∈S+
αi(0)φi(x) = φ0(x)−∑
i∈SΠ∪S0∪S1αi(0)φi(x). (6–22)
46
For x ∈ ∂L, φi(x) = 0 if i ∈ S0 ∪ S1 since φi for i ∈ S0 is supported on Lc, while φi fori ∈ S1 is supported on L. Consequently, for x ∈ ∂L, it follows from (6–19) and (6–22) that
−ξ(x) =∑i∈S+
αi(0)φi(x) = φ0(x)−∑i∈SΠ
αi(0)φi(x) = φ0(x)− φLΠ(x).
Since Π(x) = 1 for x ∈ ∂L, ξ(x) = φL−φ0(x) on ∂L. This completes the proof of Theorem2.
47
CHAPTER 7APPLICATION TO ONE-DIMENSION
In this chapter, we will present the results in one-dimension for both the generalized
eigenproblem for the Laplacian and the continuous model for the lightning discharge.
7.1 Application of the Generalized Eigenproblem
Let us consider the generalized eigenproblem (4–7) in one dimension where Ω is the
interval [0, 1] and L is a subinterval [a, b] ⊂ (0, 1). In this case, there are precisely 2eigenfunctions in H. The functions which are harmonic on both L and Lc are piecewiselinear. The eigenfunction Π of Proposition 1, corresponding to the eigenvalue 0, is defined
by its boundary values Π(0) = Π(1) = 0 and the values Π(x) = 1 on L. Let s1, s2,and s3 be the slope on the intervals [0, a], [a, b], and [b, 1] respectively of the remaining
eigenfunction u ∈ H. The jump condition of Lemma 1 yields
s2 = −λ(s1 − s2) and s2 = −λ(s3 − s2). (7–1)
Hence, s1 = s3. Let s denote either s1 or s3. The boundary conditions u(0) = u(1) = 0
imply that
0 =
∫ 10
u′(x) dx = s1a + s2(b− a) + s3(1− b) = s(1 + a− b) + s2(b− a).
This gives
s2 = s
(b− a− 1
b− a)
.
With this substitution in (7–1), we have
λ = 1− (b− a).
A sketch of these two eigenfunctions appears in Figure 7-1.
7.2 Application of the Continuous Model
In this section, we focus on Theorem 2 in dimension 1 (n = 1) with Ω the open
interval (0, 1) and L a subinterval (a, b) whose closure is contained in (0, 1). In this case,
48
Π
10
a b
u
Figure 7-1: Eigenfunctions in H in one dimension.
the equations describing Π reduce to
Π′′ = 0 in (0, a) ∪ (b, 1), Π = 1 in [a, b], Π(0) = Π(1) = 0.
The solution is
Π(x) =
s1x if x ∈ (0, a),1 if x ∈ [a, b],
s2(1− x) if x ∈ (b, 1),where
s1 =1
aand s2 =
1
1− b.
Hence, we have
φL =〈φ′0, π′〉Ω〈π′, π′〉Ω =
s1φ0(a) + s2φ0(b)1a
+ 11−b
=(1− b)φ0(a) + aφ0(b)
1− b + a .
Let us define the parameters
θ1 =1− b
1− b + a and θ2 =a
1− b + a.
With these definitions,
φL = θ1φ0(a) + θ2φ0(b), (7–2)
49
where θ1 > 0, θ2 > 0, and θ1 + θ2 = 1. Thus the effect of the lightning is to make the
potential φL on the lightning channel (a, b) a convex combination of the potential φ0(a)
and φ0(b) at the ends of the channel. The coefficients θ1 and θ2 in the convex combination
depend on the distance between the ends of the channel and the boundary of the domain
Ω. It is interesting to note that the potential φL on the lightning channel only depends
on the pre-flash potentials φ0(a) and φ0(b) at the ends of the channel; in other words, the
pre-flash potential at interior points along the channel apparently has no effect on the
potential that is achieved along the lightning channel after the flash. Also, notice that
as one of the channel ends, say a approaches the boundary, φL approaches zero since θ2
and φ0(a) both approach 0 as a approaches 0 (recall that φ0(0) = 0). A more general
discussion of a cloud-to-ground flash is given after Theorem 2.
Now let us focus on the potential change ξ outside the lightning channel L. Accordingto Theorem 2,
ξ′′ = 0 on (0, a), ξ(0) = 0, ξ(a) = φL − φ0(a),
ξ′′ = 0 on (b, 1), ξ(1) = 0, ξ(b) = φL − φ0(b).
The solution is
ξ(x) =
r1x on (0, a),
r2(1− x) on (b, 1),where
r1 =φL − φ0(a)
aand r2 =
φL − φ0(b)1− b .
Substituting for φL using (7–2), we obtain
r1 =(θ1 − 1)φ0(a) + θ2φ0(b)
a=
θ2a
(φ0(b)− φ0(a)) = φ0(b)− φ0(a)1− b + a = −r2.
50
Hence, by Theorem 2, we have
φ+(x) =
φ0(x) + xδφ0
1−|L| if x ∈ (0, a),
φ0(x)− (1− x) δφ01−|L| if x ∈ (b, 1),
where δφ0 = φ0(b) − φ0(a) and |L| = b − a is the length of the lightning channel. Thuslightning causes a linear change in the electric potential, where the size of the linear
perturbation is proportional to the pre-flash potential difference across the ends of the
channel.
51
CHAPTER 8CONCLUSIONS
In this dissertation, Maxwell’s equations are used to establish a continuous lightning
discharge model:
∂∆φ
∂t= −∇ · (σ∇φ) +∇ · J, (x, t) ∈ Ω× [0,∞), (8–1)
φ(x, t) = 0, (x, t) ∈ ∂Ω× [0,∞),
φ(x, 0) = φ0(x), x ∈ Ω,
in a bounded domain Ω ⊂ Rn with a connected subdomain L. When the electric field ina thundercloud reaches the “breakdown threshold,” the atmosphere turns into a plasma,
locally, where conductivity is large. When conditions are right, a lightning discharge can
occur. In the lightning domain L the conductivity σ is essentially infinite. To evaluate thechange in the electric potential due to lightning, we replace σ by σ + τΨ where Ψ is the
characteristic function for L, and we consider the differential equation
∂Φ
∂t= −(Aσ + τAΨ)Φ +∇ · J. (8–2)
where φ = (−∆)− 12 Φ. In Chapter 4 we show that the eigenproblem for AΨ is equivalent toa generalized eigenproblem for the Laplacian (4–7). We analyze the eigenproblem (4–7) in
Chapter 5 and obtain the following results: The elements of H10 (Lc) are eigenfunctionscorresponding to the eigenvalue 0, while the elements of H10 (L) are eigenfunctionscorresponding to the eigenvalue 1. The remaining eigenfunctions are elements of the
piecewise harmonic space H, consisting of functions in H10 (Ω) which are harmonic in bothL and Lc. There is a one-to-one correspondence between eigenfunctions of (4–7) in Hand eigenfunctions of the double layer potential T in (5–13). The eigenfunctions of (4–7)
are the harmonic extensions of the eigenfunctions of T , and if µ is an eigenvalue of T ,
then λ = 1/2 − µ is the corresponding eigenvalue of (4–7). Π ∈ H (see Proposition1) is the only eigenfunction in H corresponding to the eigenvalue 0. All the remaining
52
eigenvalues corresponding to eigenfunctions in H are contained in the open interval(0, 1) and λ = 1/2 is the only possible accumulation point. Since the eigenvalues of the
generalized eigenproblem (4–7) associated with eigenfunctions in H are contained in thehalf-open interval [0, 1), the eigenvalues of the double layer potential T in (5–13) are
contained in [−1/2, 1/2). Proposition 3 gives a lower bound for the positive eigenvaluesof the generalized eigenproblem, or equivalently, a lower bound for the gap between the
largest and the second largest eigenvalue of T . Based on the fact that the double layer
potential T is self adjoint and compact relative to the inner product (5–28), as established
in Lemma 4, we conclude that any f ∈ H10 (Ω) can be expressed as a linear combinationof orthogonal eigenfunctions for (4–7). The potential immediately after the lightning
discharge is computed in Chapter 6 by expanding the potential φ using the orthonormal
eigenfunctions of (4–7) and studying the limits as τ tends to infinity and t tends to zero to
compute the solution to (8–1). We find that the potential immediately after the lightning
discharge is constant throughout the lightning domain and the constant value depends on
the initial potential and the eigenfunction Π of (4–7). Outside the lightning domain, the
change in the potential is the solution to the problem
∆ξ = 0 in Lc, ξ = 0 on ∂Ω, ξ = φL − φ0 on ∂L.
Applications of both the generalized eigenproblem and the continuous model for the
lightning discharge to one dimension are given in Chapter 7.
53
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BIOGRAPHICAL SKETCH
Beyza Çalışkan Aslan was born in Kütahya, Turkey, in 1977. She was awarded a
Bachelor of Science degree in mathematics in 1999 from Middle East Technical University
(METU), Ankara, Turkey. In 2000, she started her graduate study in mathematics at the
University of Florida, from which she received her M.S. in mathematics in 2003 and her
Ph.D. in mathematics in 2007.
57
ACKNOWLEDGMENTSTABLE OF CONTENTSLIST OF FIGURESABSTRACT1 INTRODUCTION2 LITERATURE REVIEW2.1 Lightning Models with Explicit Lightning Channels2.1.1 Helsdon's Model2.1.2 MacGorman's Model2.1.3 Mansell's Model
2.2 Hager's Model: The Discrete Model
3 THE DISCRETE MODEL3.1 Governing Equations3.1.1 Maxwell's Equations3.1.2 Derivation of the Governing Equations
3.2 The Model in One-dimension3.3 Generalization to Three-dimension
4 THE CONTINUOUS MODEL4.1 Formulation of the Equations4.2 Eigenproblem for A
5 GENERALIZED EIGENPROBLEM FOR THE LAPLACIAN5.1 Introduction5.2 Eigenfunctions of Type 1, 2, and 35.3 Reformulation of Eigenproblem in H Using Double-Layer Potential5.4 Eigenvalue Separation and Completeness of Eigenfunctions
6 THE LIMIT6.1 Introduction6.2 Reformulation of the Continuous Equation6.3 Potential Change for the Continuous Equation
7 APPLICATION TO ONE-DIMENSION7.1 Application of the Generalized Eigenproblem7.2 Application of the Continuous Model
8 CONCLUSIONSREFERENCESBIOGRAPHICAL SKETCH
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