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

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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.

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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.

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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].

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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)

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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 •

• •

.

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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].

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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