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Continuous Approach to the Lightning Discharge

Permanent Link: http://ufdc.ufl.edu/UFE0019659/00001

Material Information

Title: Continuous Approach to the Lightning Discharge
Physical Description: 1 online resource (57 p.)
Language: english
Creator: Aslan, Beyza Caliskan
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: complete, eigenbasis, eigenproblem, electric, generalized, laplacian, lightning, maxwell, potential
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: 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.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Beyza Caliskan Aslan.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Hager, William W.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0019659:00001

Permanent Link: http://ufdc.ufl.edu/UFE0019659/00001

Material Information

Title: Continuous Approach to the Lightning Discharge
Physical Description: 1 online resource (57 p.)
Language: english
Creator: Aslan, Beyza Caliskan
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: complete, eigenbasis, eigenproblem, electric, generalized, laplacian, lightning, maxwell, potential
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: 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.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Beyza Caliskan Aslan.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Hager, William W.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0019659:00001


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CONTINUOUS APPROACH TO THE LIGHTNING DISCHARGE


By

BEYZA CALISKAN ASLAN



















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007

































2007 B.--I Calihkan Asian


























To my parents,

Kamile Qallkan and Yusuf Qahskan,

and to my husband,

Omer Deniz Asian









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 ahv---

encourage me to move forward on my research.

Second, I would also like to thank Dr. .1i1-,d.-.p Gopalakrishnan, Dr. Shari Moskow,

Dr. Sergei S. Pilyugin, and Dr. Vladimir A. Rakov for serving on my supervisory

committee. Their valuable -Ir.-. -Ii..i, have been very helpful to my research.

Third, thanks go to my officemates (Dr. Hongchao Zhang, Dr. Shu-Jen Huang, and

Sukanya K i-li:i ,i'.--v i), and all colleagues and friends in the Department of Mathematics

at the University of Florida. Their c~ rl'' -: 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 Basak, 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.









TABLE OF CONTENTS


page


ACKNOWLEDGMENTS ...............

LIST OF FIGURES ...................

ABSTRACT .......................

CHAPTER

1 INTRODUCTION .................

2 LITERATURE REVIEW ............

2.1 Lightning Models with Explicit Lightning (C!
2.1.1 Helsdon's Model ............
2.1.2 MacGorman's Model .........
2.1.3 Mansell's Model ............
2.2 Hager's Model: The Discrete Model .....

3 THE DISCRETE MODEL .............

3.1 Governing Equations .............
3.1.1 Maxwell's Equations .........
3.1.2 Derivation of the Governing Equations
3.2 The Model in One-dimension .. ......
3.3 Generalization to Three-dimension .....

4 THE CONTINUOUS MODEL .. ........

4.1 Formulation of the Equations .. ......
4.2 Eigenproblem for A .............

5 GENERALIZED EIGENPROBLEM FOR THE L.

5.1 Introduction .. .. .. ... .. .. .. ..
5.2 Eigenfunctions of Type 1, 2, and 3 .....
5.3 Reformulation of Eigenproblem in 7- Using D
5.4 Eigenvalue Separation and Completeness of E

6 THE LIMIT .....................

6.1 Introduction .. .. .. ... .. .. .. ..
6.2 Reformulation of the Continuous Equation .
6.3 Potential C!i ,. for the Continuous Equatio

7 APPLICATION TO ONE-DIMENSION .....

7.1 Application of the Generalized Eigenproblem


ill., . . .




















PLACIAN . .



'ouble-Layer Potential
.igenfunctions ..





n . . .
. . . .









7.2 Application of the Continuous Model ................ .... .. 48

8 CONCLUSIONS ............... ................ ..52

REFERENCES .. . ....... .. ........ ....... 54

BIOGRAPHICAL SKETCH ............... . . ..57









LIST OF FIGURES
Figure page

4-1 A sketch of L and Q for a lightning discharge .................. 20

7-1 Eigenfunctions in -F in one dimension .................. .... 49









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

CONTINUOUS APPROACH TO THE LIGHTNING DISCHARGE

By

B.-i- Cahlkan Asian

August 2007

C'!h wi: William W. Hager
Major: 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.









CHAPTER 1
INTRODUCTION

Lightning is one of the most beautiful di-p-v'1 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.

Tod iw, 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]. Be-,i i,,ii

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 C'i i,: her to track high energy particles and made 1i i ir









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 observational
capabilities.


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.









CHAPTER 2
LITERATURE 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 ('! Ilpter 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 criteria, 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









originated, and the potential at P. The linear charge density can be given by


Qp -k(p Do),


where Qp was the charge density at P, and 4)p and 4o 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-,"2, where x is the distance from

the channel.

2.1.2 MacGorman's Model

MacGorman et al. [23] slr.-.- -1 .I 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.

('C! Irge 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 Ip(i,j, k) > pi (where p(i,j, k) was the net charge

density at the grid point (i,j, k) and pi was the minimum |p(i, j, k) for all grid points









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 Eiu 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.









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:

After Ibefore A- U(UTA-1U) -1UTbefore, (2-1)


where before was the electric potential before discharge, after 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].









CHAPTER 3
THE 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 (FEaYd'i's law of induction).

In the absence of magnetic or polarizable media, the differential form of these

equations are:

1. Gauss' law for electricity: V E =

2. Gauss' law for magnetism: V H = 0
9B
3. Faraday's law of induction: V x E =
at
9~E
4. Ampere's law: V x H = Jo + E-

where E is the electric field, H is the magnetic field strength, B is the magnetic flux

density, p is the charge density, E is the permittivity of air, Jo is the current density, V. is

the divergence operator, and Vx 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

dE
Vx H Jo + a (3-1)
at *

Since Jo 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


Jo Jp + aoE









where or is the conductivity of the atmosphere. In this model, we assume the time

derivative of the magnetic flux density is zero, i.e.,

9B
0.
at

Hence, the curl of E vanishes by Faraday's law and E is the gradient of a potential Q:


E -Vl.


Therefore (3-1) becomes


V x H = Jp aoV0


avo
at


(3-2)


Taking the divergence of (3-2), it follows that


0 -VVxH V-Jp -V-a oV


avt
at


(3-3)


Letting a = ao/ and J = Jp/E, we obtain


V +aV V- V J 0.
at't


(3-4)


In our model we also have the following assumptions:

Let EB be the breakdown field strength. Then the electric field magnitude is alvb--,
less than or equal to the breakdown threshold EB. That is, |E| < EB.

When the electric field reaches the breakdown threshold EB at some point, the
conductivity tends to infinity in a small neighborhood of that point. That is,
a(x) oo 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


" + (,7')' = o, 0(0) 0, (H) = V,


(3-5)









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, 4i be the approximation to

potential at xi + h/2, ai be the conductivity at xi, and Ji be the current density at xi.

Integrating (3-5) over [x', xi+,] gives


((x+i) )'(Xi)) + (i+1'(X+) i4'(X)) (J+ Ji) 0

To approximate 0', [11, 13, 14] used backward differences,

wci te-1
0((Xi) i ,
h

which yield the discrete equation


i hi-1 \ +1 i
h h


This can be written in matrix form as


> A + B4 = hAJ


where


-1

2
So


and


-1 + -2 -72

-72 92 + a -3 3

-a3 U3 + 4 -94

-O4 a4 + 5
S S


(i Oi-1 -
o"- I -- (J4+1 J) 0
&I


Si+1 i
h h









Integrating the matrix equation over the interval [0, At] gives

4(t + At) e-A Bt (t) + h C-A-1B(zt-s)AJ(s)ds


If = Eb, then Hager lets Jk -- Tk + 7 where 7 -+ +oo. It is shown that
h
as 7k k-- k + T, B B + TwwT, where ,' = 1, ',,+1i -1, = 0 otherwise. Next,

they calculated the limit as 7 -- o0 and At -+ 0 and obtained the formula for the potential

after the lightning:

4(t+) (t) A- w(wTA- w)-wT(t).

3.3 Generalization to Three-dimension

Suppose =-l b Eb at k = ko, k,... ki. For each such k, k -- +00,


B -+ B + Y wT WT.
i=0
Here, wi is zero except for +1 and -1 in components ki and ki+l. Taking limits as

-Ti +oo and At 0 yields

4(t+) = )(t) A-1WI(WTA-'1W)- W 4t)


where WI = [wol... wi].









CHAPTER 4
THE CONTINUOUS MODEL

4.1 Formulation of the Equations

By (3-4), the following equations model the evolution of the electric potential in a

domain Q, assuming the time derivative of the magnetic flux density can be neglected:

aA-V (aV) + V J in 2x [0, o), (4-1)
at
0(x, t) 0, (x, t) e 09 x [0, oo), (4-2)

(x, 0) -= o(x), x e (4-3)

where 2 C PR is a bounded domain with boundary 02, a > 0 lies in the space L(Q2)

of essentially bounded functions defined on 2, and the forcing term J lies in L2(2), the

usual space of square integrable functions defined on 2. The divergence V J as well as the

equation (4-1) are interpreted in a weak sense, as explained later. The initial condition

Qo is assumed to lie in H o'(), the Sobolev space consisting of functions which vanish on

0L and with first derivatives in L2(2). The evolution problem (4-1)-(4-3) has a solution

0(x,t) with (., t) E Hol(0) and the partial derivative At(-, t) E H0o'(). Although
Maxwell'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, a is the conductivity divided by the permittivity of the atmosphere,

2 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









atmosphere when the thundercloud is removed) and transform the problem to the form

(4-1)-(4-3) where the potential vanishes on 9Q.

A possible lightning channel is sketched in Figure 4-1. Mathematically, the lightning














Figure 4-1: A sketch of L and 2 for a lightning discharge.


channel L could be any connected, open set contained in 2 with its complement C = 2\

connected. More realistically, we should view L as a connected network of thin open tubes.

The case where L touches 02, as would happen during a cloud-to-ground flash, is treated

as a limit in which L approaches arbitrarily closely to the boundary of 2.

Making the change of variables 0 = (-A)- ~, we rewrite (4-1) as

A, + V J, (4-4)
at

where

A, -(-A)-7(V.(aV))(-A) 2

Here (-A)-2 : L2() H- Ho(2) denotes the inverse of the square root of the Laplacian

([3]).

Let T be the characteristic function for L (T is identically 0 except in L where T is

1). The effect of lightning is to ionize the domain in essence, replacing a in (4-4) by

a + 7r where r is large. If the lightning occurs at t = 0, then in the moments after the









lightning, the electric potential is governed by the equation


S-(A, + TAv) + V. J, (4-5)

where

A, -(- A)-(V.7 (TV))(-A)-2,

in 2 x [0, oc) subject to the boundary conditions (4-2) and (4-3). Here the scalar r reflects

the change in conductivity in the lightning channel.

Let 4,(x, t) denote the solution of (4-5) at time t, and let ,-(x, t) = (-A)-'-,(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

lim lim r (x,t). (4-6)
t- + -Too

Using an eigendecomposition for Ap, the limit (4-6) can be evaluated.

4.2 Eigenproblem for A1

In this section, we show ([2]) that the eigenproblem for Ap reduces to the following

generalized eigenproblem: Find u E Hi(0), u / 0, and A E IR such that


(Vu, Vv), A(Vu, Vv)n (4-7)

for all v e Hs(Q), where (, .)n is the L2(Q) inner product

(Vu, Vv)n Vu Vv dx. (4-8)

We view Hi(Q) as a Hilbert space for which the inner product between functions u and

v C Hi(Q2) is given by (4-8).

The weak form of the eigenproblem for A, is to find U e L2(2) such that


(A, U, V) ((V)(-A)- U, V(-A)- V), A(U, V)Q (4-9)








for all V E L2(Q). Let u = (-A)-1/2U and v = (-A)-1/2V denote the corresponding
functions in H0(/). Consequently, (4-9) reduces to the problem of finding u H0(Q/ ) such
that
(Vu, Vv) = A((-A) ~, (-A)'v) (4-10)

for all v c H(Q). If u c Co-(Q), then

((-A)> (-A)v)n -(Au, v)A = (Vu, Vv)n. (4-11)

Since Co"(Q) is dense in H (Q) and the operators (-A)i and V are both bounded in
HO(Q), the identity (4-11) is valid for all u c H1(Q). Hence, (4-10) reduces to (4-7).









CHAPTER 5
GENERALIZED 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 H which is 1 on L and harmonic on 2 \ L; the eigenvalue is 0.

2. Functions in H (Q) with support in 2 \ L; the eigenvalue is 0.

3. Functions in H/(0) with support in L; the eigenvalue is 1.

4. Excluding H, the harmonic extension of the eigenfunctions of a double li. r potential
on 9. The eigenvalues are contained in the open interval (0, 1). The only possible
accumulation point is A = 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

A (Vu, Vu) (Vu, Vu) )
A =w(5+1)
(Vu, Vu)n (Vu, Vu)L + (Vu, Vu) '

which implies that 0 < A < 1. Let Ho([) C Ho(Q) denote the subspace consisting of

functions with support in L. Similarly, let Ho([c) C Hol() denote the subspace consisting

of functions with support in Lc.

Proposition 1. A = 1 and u E H (Q) is an eigenpair of (4-7) if and only if the support

of u is contained in L. If u E H (c)0, then u is an .:, ,. function of (4-7) corresponding to

the .: ,': ;in .l, 0. The only other .:, ,function of (4-7) corresponding to the ':, 'o.: ;. : 0,

which is or'l,. ',j.,,.'1 to H([ c), is the solution HI c Ho(') of


(VI, Vv)n 0 for all v E Hol(), H = 1 on L. (5-2)


Proof. If A = 1 and u E Ho() is an eigenpair of (4-7), then by (5-1), we have


(Vu, Vu)c = 0.









Hence, Vu = 0 in c, which implies that u is constant in L since L' is connected. Since

u E H1J(), u 0 in c. Conversely, if u 0 in Lc, then by (4-7), u is an eigenfunction

corresponding to the eigenvalue 1. If u 0 in L, then u is an eigenfunction corresponding

to the eigenvalue 0. The solution H of (5-2) is an eigenfunction of (4-7) corresponding to

the eigenvalue 0 since VH = 0 in .

Let w E H1(Q) be any eigenfunction of (4-7) corresponding to the eigenvalue 0 which

is orthogonal to H1(Lc). By (4-7), we have (Vw, Vw) = 0, which implies that Vw 0

in L, or w is constant in L since L is connected. Without loss of generality, let us assume

that w 1 in L. Since w is orthogonal to the functions v c H[(c), we have


(Vw, Vv)Q 0 for all v H(C).

Combining this with (5-2) gives

(V(w n),Vv)n 0 for all ve HC (.


Since 0I = 9fQ U 9i and since w H vanishes on both fQ and 9, it follows that

w = H. O

5.3 Reformulation of Eigenproblem in -H Using Double-Layer Potential

Proposition 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 E H (Q) satisfying the

conditions


(Vu, Vv)n 0 for all v H0() and (5-3)

(Vu, Vw)n 0 for all w E H(Lc). (5-4)

H is a subspace of H (Q) consisting of functions harmonic in L and Lc (Au = 0 in L and

Au = 0 in c). Note that II e U. Since H0() and H0(c) are orthogonal with respect to

the Ho(Q) inner product, and since 1H is the orthogonal complement of Ho() E Hl(c) in








H01(), we have the orthogonal decomposition


H'(Q) = E H'() H0'(c).

The following series of lemmas reformulates the generalized eigenvalue problem (4-7) on R
in terms of an integral operator.

Lemma 1. u cE is a solution of the generalized .:i. ',,.'l. 11 ,, (4-7) if and only if

O- A -O on (L, (55)
On [9n\

where
S E H-1/2 ().
on] an On
Here n is the outward unit normal to L and the and + refer to the limits from the
interior and exterior of L -1./,' ;

Proof. First we show a generalized eigenpair also satisfies (5-5). By (4-7) we have

(Vu, Vv) =- A(Vu, Vv)n Q A ((Vu, Vv)Lc + (Vu, Vv)L) (5-6)

for any v c Ho'(Q). Integrating by parts and utilizing the fact that u is harmonic in both
L and Lc gives
f u- ou+ f u-
SV d -jA v d7 + A v- d', (57)
JL On JQ On 11L On
where 7 denotes the boundary measure on ai. Hence, we have

[F u +A dOu+ u
JQC Ln On n
for any v c Hd(Q). Since any v E H1/2(aO) has an H{(Q) extension, (5-5) holds.
Conversely, suppose that u satisfies (5-5). As in (5-6)-(5-7), we have

A(Vv, Vu)Q --A v d7+AJ vJU d7
far On J" On








Applying (5-5) gives
A(Vv,Vu)Q v O- (Vv, Vu)

since u is harmonic in L. Hence, u satisfies (4-7). D

Now let us introduce the Green's function on Q:

AyG(x, y)= 6,(y) in Q, G(x, y) = 0 for ye 9, (5-8)

where 6x is the Dirac delta function located at x. The piecewise harmonic functions u CE 7
can be described in terms of the jump on L of the normal derivative.

Lemma 2. Suppose that 9 and 89 are C2. If u cE -, x cE and x 9i, then

u(x) "a I (y)cG(x, y)d^y. (5-9)


Proof. Since u is harmonic away from 9, it is continuous there, and so for x 9,

u(x) = I u(y)AyG(x, y)dy

= u(y)AG(x,y)dy+ j u(y)AyG(x,y)dy.
SLc

Since u is smooth in each subdomain, we can integrate by parts to obtain

u(x) = Vu VGdy + U dyd Vu -VGdy- u dy.

Since u is smooth on each subdomain and u E H1(Q), the traces u+ and u- e H1/2(aL)
must satisfy u- = u on 9. Since Au = 0 on each subdomain and G = 0 on 89, we have

u(x) = Vu VGdy Vu VGdy
JI Jc
u (y)G(x, y)dy,+ / u (y)G(x, y)dy,
Jwh On l 5 O9n
= [ (y) G (x, y) d- (5-10)

which yields (5-9). D









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 E H1/2([) and both a and OQ are C2. For x E Q, x f 9 let

v(x) be /. F, .1 by

S anC
v~x) = (y) 4){y {(x,y)dy,.

The trace v+ of v onto 9 from the exterior of L and the trace v- of v onto 9 from the
interior L are given by

1 OG
v+(x) =- (x) + (y) (x, y)d
2 J i I Yn,

and

v-(x) = -(x)W + 0(y)() (x,y)dy,.
2 J+ J a ay
Proof. Let N(x, y) be the free space Green's function for the Laplacian,

|X y>2-n
Sn > 2,
N(x, y) (2=) (5-11)
-- log Ix yl n = 2,
27r
where w is the surface area of the unit sphere in R". Recall that


avN(x, y) = 6,(y).

Define

H(x,y):= G(x, y)-N(x,y).

By the definition of v,

v(x)= y) (x, y)dy + 0(y) (x,y)dy,. (512)
JQC any JQC any









For any x E Q, H satisfies


AH(x, y) = 0 for y E ,

H(x, y) = -N(x, y) for y e 9.

Hence, H(x, y) is harmonic for y E Q with smooth boundary data. This implies that the

function w(x) defined by

w(x) = 0(y) OH(x, y) dy

is continuous in a neighborhood of 0i since the kernel has no singularity. For the second
term of (5-12) we can apply the well known result (see [8]) for the limit x -- a0. iFrom

the exterior of we have

O "H ) ON
v+(x) = (y) (x, y)dy,- -b(x) + (y) (x, y)dy,
at ha 2 OG Bu
SOW + (y) n (x, y)dy,.
z'\ Jd any

The proof for the interior limit is similar. D

Using Lemma 1, 2, and 3, we reformulate the generalized eigenproblem (4-7) on

R- in terms of a boundary integral operator. By the trace theorem [1, Thm. 7.53], any
u c R c Ho'() has a trace on 9A in H1/2 (A). Conversely, u c H1/2(/[) has a unique

harmonic extension into both L and [c with u = 0 on a9. Hence, there is a one-to-one

correspondence between elements of and elements of H1/2(/[).

Define

T: L2(a!) L- L2(a2)

by
T(x)= (y)K(x, y)d K(x, y) : (x, Y). (5-13)

By [8, Prop. 3.17], K is a continuous kernel of order n 2 on 9[. It follows from [8, Prop.
3.12] that T is a compact operator from L2( /) to itself.








Proposition 2. If both a and OQ are C2, then (u, A) cE H x R is a generalized .,. ,l"'.:'
for (4-7) if and only if the corresponding u E H1/2(a) is an .u ,. function ofT with
associated .:I ,.'', ; l;,. 1/2 A; that is,

Tu = (1/2 A)u. (5-14)


Proof. First, let us assume that (u, A) E -H x R is a generalized eigenpair for (4 -7). By
Lemmas 1 and 2, we have

Au(x) U (y)G(x,y) dy

for x E Q and x 5. We integrate by parts to obtain

Au(x) = Vu(y) VG(x, y)dy
Jc
u(y) (x, y)d y + u(y)AG(x, y)dy.

If x e [C, then the second term above disappears, and we have

Au(x)= () (x,y)d-

an equation for a double liv-r potential. We let x E [C approach L. According to
Lemma 3,
1 OG
Au(x) = u(x) (y) x, y)d-
2 uYn,
which is equivalent to (5-14).
Conversely, suppose that u E H1/2 ([) satisfies (5-14). We identify u with its
harmonic extension in H-, and we define w(x) by

u (x, y)dy for x e C,
W(X) = (5-15)
u(y) (x, y)dy, + u(x) for x e L.
/1 aGny








In either I' and w is harmonic. By Lemma 3, we have


w(x)= u(x) u(y) (ary)dcy
W 21 JaL (x, y) dy

and

w- () = ux)- () (x, y)dy,
1 OG
-u2() u (y) an (x,y)dy,.

Utilizing (5-14) yields

w+ w- (1/2- T)u = Au on 9. (5-16)

Observe that w vanishes on 9Q due to the symmetry of G(x, y) [8, Lem. 2.33]; that
is, since G(x, y) = 0 when y E 0Q, we have by symmetry G(x, y) = 0 when x E 0Q.
Hence, the normal derivative in (5-15) vanishes when x cE O. Since w is harmonic in each
subdomain and it is equal to Au on both 9 (see (5-16)) and 0Q (they both vanish), it
follows that w = Au in Q. We replace w with Au in (5-15) to obtain

j u(y) (.x,y)dy, for x e C,







a u )(y)G(x,y)d( (VudfG),
\u( (xy if (5x 17)





) f 9 u
Au(x) (y) G(x, y)d (5 8)
JQC 8n~









By Lemma 2,

Au(x) =A "~n (y)G(x,y)d7,. (5-19)

Subtracting (5-19) from (5-18) gives

s(x): j (y)G(x,y)dy= 0 for any x a[,
J8

where

O( -u (y) (?)- [A ] (y).

Hence, s = 0 almost everywhere in Q. If =- 0, then

(9u(- 9u
an Ln

in which case Lemma 1 completes the proof.

To prove that = 0, suppose to the contrary that 0 does not vanish. Let r be any
smooth function defined on aL for which

j (y)r(y)d7y/ 0.

Let r also denote any smooth extension in Q which vanishes on a0. By the symmetry of

G, we have

r(y) I' [Axr (x)]G(x, y)dx.

Forming the L2(Q) inner product between s (which vanishes almost everywhere) and Ar

yields

0 (s, Ar)n- (y) [Axr(x)]G(x, y)d- Jr (, r)aL / 0.

Hence, we have a contradiction and the proof is complete. E

Corollary 1. If both aO and OQ are C2, then the ,. :'ii ,,.i, I of the double '7.;,. potential

operator T in (5-13) are real and contained in the half-open interval (-1/2, 1/2]. The only
possible accumulation point for the spectrum is 0.









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 9iC is 0. The only element in R- with vanishing trace on

at is the zero function. Consequently, there is no eigenfunction in R- corresponding to the

eigenvalue 1. There is one eigenfunction in R- corresponding to the eigenvalue 0, namely

the function H 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(Q) [8, Prop. 3.12], the only possible accumulation point for the

spectrum is 0. O

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 R- has a

simple eigenvalue A = 0 corresponding to the eigenfunction HIE while the remaining

eigenvalues are positive. By Proposition 2, the only possible accumulation point for the

spectrum is A = 1/2. Hence, there is an interval (0, p), p > 0, where the generalized

eigenproblem has no eigenvalues. We now give an explicit positive lower bound for p in

terms of three embedding constants:

El. Let Ua denote the constant function on 2 whose value is the average of u E H1(2)
over L:
Ua measure-t) f u(x)dx.
measure(L) I
By [7, Thm. 1, p. 275], there exists a constant 01 > 0 such that

Vfr al ) > 01 HI(U

for all u c H1(Q).








E2. By [1, Thm. 7.53], there exists a constant 02 > 0 such that


|H ||H (>) 2 2 H1/2(ac)

for all u c H1(Q).

E3. There exists a constant 03 > 0 such that

I 1/|2I/ >( 0 H llHi(c (5-20)

for all u E HiJ(Q) which are harmonic in c (in other words, (5-4) holds). The
following proof of E3 was sil-'-, -1'. by .1 v-'- p Gopalakrishnan: For u E H (Q),
let T(u) e H1/2(L) denote the trace of u evaluated on ai. By [9, Thm. 1.5.1.3],
T has a continuous right inverse which we denote T-1. In other words, for each
g e H1/2(a[), we have T-l(g) e H'(Q), TT- (g) and

IT -(g)IHI(Q) < i-1 H1/2(A)*
Define vo u-T-l(g). Since vo vanishes on both 0Q and 9, there exists a constant
c > 0 such that (see [7, Thm. 3, p. 265])

'I,, H1(c ) c< c IV ,,, LL2(L). (5-21)
Moreover, since u is harmonic in [c and i,, vanishes on both 0Q and [, we have

(VT,, V,,)>, (V,, V(u T- (g)))c -(V,, VT- (g))Lc
< lVvo oILc2(C) Ir-(g) IL2(c)
< |V ,,|L2(C.L )iT- (- )||HI(c),
which gives ||V,, lL2(L) < I 1VT-'(g)11H1(Lc). We combine this with (5-21) to obtain

',,I HI(oc) < c 1 T-l (g)llH (o)-
Hence, by the triangle inequality,

I|u|HI( c) < ,,11 HI(c) + 7-1'(g\)\H1Lc)
< (1 +c)llr-l(g)llH- (c)

< ((1+ c)7 -1|) gH/2(),

which yields (5-20).









Proposition 3. If both a and OQ are Lipschitz, then the generalized .:j, ig,'l. 'i,, (4-7)

has no .:j, n' in the interval (0,p) where

p =min{l, 0203}01/2.


Proof. Let p be the smallest positive eigenvalue for the generalized eigenproblem (4-7),

and let u be an associated eigenfunction with normalization (Vu, Vu)Q = 1. If IIE 7- is

the eigenfunction described in (5-2), then we have




> Oiu a H1() (5-22)

0111u- U HI() (5-23)

> 0102 1U FUa 1u2 1 (5-24)

> 0028 i -nUa 2HI(. (5-25)

Above, (5-23) is due to the fact that II 1 on L, while (5-22), (5-24), and (5-25) come

from El, E2, and E3 respectively.

Suppose that the proposition does not hold, in which case p < 01/2 and p < 010203/2.

By (5-23) and (5-25), we have


H(Lu- H ) < 1/2 and Iu Hwll1i|c) < 1/2.

Combining these gives

1U- nu ll||1() < 1. (5-26)

On the other hand, u and II are orthogonal since these eigenfunctions correspond to

distinct eigenvalues. Since Hll is a multiple of H which is orthogonal to u, it follows that

1 < I|V(u InU)|| 2() < |IV( nUf ) ||H(). (5-27)









Comparing (5-26) and (5-27), we have a contradiction. Hence, either p > 01/2 or

P > 010203/2. EO

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 1v,--r potential T in (5-13).
As noted before Proposition 2, there is a one-to-one correspondence between elements

of R and elements of H1/2(O). If u e H1/2([), then the corresponding E(u) E R
is the harmonic extension of u e H1/2( O) into Q which vanishes on 90. For any
u,v E H1/2(O[), we define the inner product


(u, v) (VE(u), VE(v))n. (5-28)

In other words, harmonically extend u and v in 2 and form the H1J() 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 -,;.:/7. ,1

T1. If 9Q and a are Lipschitz, then the norm (., )1/2 is equivalent to the usual norm for
H1/2(aL). That is, there exist positive constants cl and c2 such that

C (v, v) < I, II1 /2(aC) < C2( v, v)

for all v E H1/2(a).

T2. If 9Q and 9 are C2, then the double '7..;. potential operator T in (5-13) is self-
adjoint relative to the inner product (5-28).

T3. If O9 is C2 and a Z is C2'0, then T is a compact operator from H1/2(a[) into
H1/2(aO).


Proof. We begin by showing that the norm of H1/2([) and the norm (., .)1/2 are
equivalent. First, recall [7, p. 265] that there exists a constant 04 > 0 such that

IIV 11L 2(Q) > 04 I :|i1(Q)









for each v e H- which vanishes on 9Q. Combining this with E2 gives the lower bound

(v, v) (VE(v), VE(v)) > 04 1E(v) H\\(Q)

> e04E(v)i HI(f) > ~H4 /2(). (5-29)

An upper bound for (v, v) is obtained from E3:


(v, v) (VE(v), VE(v))n < ||E(v) ||2H(Q)

< (031 +01) ,l/yC). (5 30)

Here 03 > 0 is analogous to 03 in (5-20) except that it relates L to [L:


1 H111 2(aL) >03|E(v) HW(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) (VE(Tu),VE(v))n = (VE(u),VE(Tv))Q = (u,Tv) (5-31)

for all u and v E H1/2(O). We first observe that the extension of Tu has the form

-E(u(x)) + u) (Y )dy for x c C',
E(Tu) 2 (5-32)
--E(u(x)) + U(y) G( dy for x c L.
2 JQ any

By Lemma 3, the trace of the right side of (5-32) is Tu from either side of 9L. Moreover,
the right side is harmonic and it vanishes on 9Q since E(u) vanishes on 9Q and G(x, y) =
0, independent of y cE 2, when x e 92. Since the right side is harmonic and satisfies the
boundary conditions associated with E(Tu), it must equal E(Tu).









Integrating by parts and utilizing (5-32), we obtain


(Tu, v) = (VE(Tu), Vv)n

= (VE(Tu), VE(v)), + (VE(Tu), VE(v)),c

t f E(u)- OE(u)+
a- a +E(v)dy. (5-33)
2 O n On j

The term in E(Tu) associated with the Green's function cancels since the normal

derivative of a double l1-cv-r potential operator is continuous across a0 (for example,

see [6, Thm. 3.1], [5, Thm. 2.21], [21, Thm. 6.13]).

For any p and q e R-, we have the identities


(Vp, Vq)c = q- 7 d7,
JaL On JaL On

and
f ap' + Oq+
(Vp, Vq)LI = q d7 = p- d.
JaL On OQ on
Hence, the normal derivatives in (5 33) can be moved from the u terms to v to obtain

(Tu, (OE(v)- OE, E(u)d7 (u, Tv),

which establishes T2.

We now show that T is compact on H1/2 (a). Consider the corresponding free space

double l-1.,-r potential operator TF defined by


JQC any
TF(x) aN) (x, y)d

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 map

from L2(a) to H1(a). This result extends to our operator T as follows. The difference,









T TF, is an integral operator on aL with kernel


anH anG any


For x cE 0, H has no singularity since it is harmonic with smooth boundary data (see

the proof of Lemma 3). Consequently, T TF is bounded from L2(t) to H1(9).

Since both TF and T TF are bounded from L2(a) to H'1(), we conclude that T is

bounded from L2(t) to H'1(). This implies that T is compact on H1/2( ) since H1

embeds compactly in H1/2; that is, by [9, Thm. 1.4.3.2] H8 embeds compactly in Ht when

s > t > 0. Hence, T is compact on H1/2(a). E

Theorem 1. If 98 is C2 and at is C2',, for some a E (0, 1) (the exponent of Holder

conl/.,',iI, for the second derivative), then I w.; f c Ho(Q) has an expansion of the form
00
f Z Yi,
i=1

where the Qi are :,. ,functions of (4-7) which are orl,. .,. ,,,al relative to the inner product

(6-5). Here the convergence is with respect to the norm of Hdo(Q).

Proof. As pointed out earlier, we have the orthogonal decomposition

H/(Q) = E H'() E H'(c).

By Proposition 1, any complete orthonormal basis for H'() is an eigenfunction basis

corresponding to the eigenvalue 1. Likewise, any complete orthonormal basis for Ho1(gC) is

a 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 E R- lies in the span

of the remaining eigenfunctions for (4-7).

By Lemma 4, T is compact and self adjoint relative to the inner product (., .) defined

in (5-28). Hence, every f e H1/2(t9) has a unique expansion in terms of orthogonal

eigenfunctions of T (for example, see [4, Thm. 1.28]). Given f E R-, its restriction to aL










lies in H1/2([C). Therefore, there exist orthogonal eigenfunctions 0i, i > 1, of T such that

00
f = i on L.
i= 1

By the linearity and boundedness of the extension operator, we have

OO
f Z= E(0) on Q.
i= 1

By Proposition 2, E( i) is an eigenfunction for the generalized eigenproblem (4-7). D









CHAPTER 6
THE 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 02 is C2 and Z is C2',, for some ca E (0, 1) (the exponent of Holder

conlii,',il', for the second derivative), then the electric potential + :ii,,/,, .:al. /;/l after the

lightning 1.:-. lI.,r, ,- is given by

(x ) L if x E L,
+(x) (6 -1)
o (x) + (x) if x e c,

where
(Vo, vn),
L V (6 2)
(vn, vn), '
and where II and are harmonic functions in c with boundary conditions as ... i'', ,

below:

AII 0 in c, II =0 on 0n II 1 in L, (6-3)

AO = 0 in = oon Q, L 4o on (6-4)

Here (-, .)Q is the L2(Q) inner product

(Vu, Vv)n Vu Vv dx. (6-5)


Thus 0+ has the constant value OL along the lightning channel L and the change

in the potential due to lightning has been expressed in terms of the potential 4o before

the lightning and the lightning channel L. When L touches 0Q, as it would during a

cloud-to-ground flash, L = 0 and II can be eliminated. That is, as [ approaches the









boundary of Q, H develops a jump singularity since I = 1 on C and H = 0 on a9. Hence,

VII approaches a delta function as approaches a0. Since the delta function is squared

in the denominator of QL while the numerator is finite, QL tends to 0 as approaches aQ.

Thus in a cloud-to-ground flash, the change ( in electric potential due to the lightning is

the solution to (6-4) with L 0.

6.2 Reformulation of the Continuous Equation

Let (0i, Ai), i E N, denote a complete orthonormal set of eigenfunctions for the

generalized 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:


Sn = {i i N: O 11/VIII||L2(Q)},

So = {i N: A, 0, (VQ, Vn) = 0},

S1 {i N: Ai },

S+ = N: 0 < A{ < A 1}.

The set Sn contains precisely one element corresponding to the eigenfunction I given by

(6-3). The set So corresponds to eigenfunctions supported on [c, while S1 corresponds to

eigenfunctions supported on L. The set S+ corresponds to functions in Ho'1() which are

harmonic in both L and [C, and with the eigenvalues uniformly bounded away from 0.

The weak form of (4-5) is to find Q such that


a (VQ, V)n = -(aVQ, V)n T ,(VQ, Vv) + (Jp, VV)> (6-6)

for all v E Ho(Q). We substitute the eigenexpansion


(x, t) aM(t)(x)
iEN









into (6-6). Taking v = j, j = 1, 2,..., and utilizing the orthonormality of the
eigenfunctions yields the linear system

&a -Aa Da + f, a(0) = ao, (6-7)

where the dot denotes time derivative and

(ao)i (V- o, Vo)Q, i N, (6-8)

ail = <(7V4, V )n, (6-9)

dij = (V V, j)V ,

fi = (Jp, Vd)n. (6-10)

Since the Oi are orthonormal eigenfunctions for (4-7), we have

J A\ if i j,
dij = (VA Vj) Ai(0i, 0j) if i
0 otherwise.

Hence, D is a diagonal matrix with the eigenvalues Ai, i E N, on the diagonal. Since the
eigenvalues are nonnegative, D is positive semidefinite. We now consider A and f:

Lemma 5. The matrix A is positive -, i,,../. f,i../ The 2-norms of A and f, /. I,.. in
(6-9) and (6-10) ,'. /.. -/'.; /1,; are both finite, and we have

IIA | < ess sup la(x)| := ||Ia|- and fl = JJp 112( ). (6-11)
xen

Proof. By the definition of A, we have

xTAx a7 xiV ,i, xiV)i >0 (6-12)
i= 1 i= 1 9

for all x e (2 since a > 0. Since A is positive semidefinite, the C li. !:-Schwarz inequality
yields

yTAxx < yTAyvxTAx (6 13)









for all x and y e 2. If |x|| = 1, then by (6-12) we have

XTAx =I< |1| 7||1 xvo, xioV I Io-I \1,o.
i=1 i= 1 9
Hence, (6-13) yields the first inequality in (6-11).

Let u E Ho(Q) be the weak solution to Au = -V Jp:

(Vu, Vv) (Jp, Vv) for all v e HI(Q).

We have

IIVi, l-= (Vu ) = (Jp,Vu) < II||J ||||V ||.

Dividing by |IVull gives

I|VUl < I< llJ. (6-14)

For f defined in (6-10),


llfll2 Z(Jp, V7)2 (Vu7,V )2 (Vu:,VU) < IIJ P 2.
i= 1 i 1
The last equality is due to the fact that the Oi are a complete orthonormal basis relative
to the Ho(Q) inner product, while the last inequality is (6-14). E

6.3 Potential Change for the Continuous Equation

We now prove Theorem 2. Multiply (6-7) by aT and utilize the fact that D and A

are positive semidefinite to obtain

aTd -c aTAa T-TDDa + aTf < aTf.

Hence, we have
1d 1
|dlla 2 aT < jllall l l< (11a 112 + lfll2).
2 dt 2
Multiplying by e-t and rearranging yields

S(e-lal) < e-lfll2.









Integration over the interval [0, t] gives


Ija(t)|2 _< et a(0)||2 + (t- 1) lf2 (6 15)

This shows that I|a(.-) 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

A 0
D
0 0

where A is a diagonal matrix with strictly positive diagonal and the 0's represent blocks

whose elements are all zero. The components of a are partitioned in a compatible way

into [p ; z] where p corresponds to the positive eigenvalues Ai and z corresponds to the

zero eigenvalues.

Multiply (6-7) by [pT ; 0] to obtain


pTp = [p; O]T

[p; O]TAa- r[p; 0T Ap + [p; 0]Tf.
0

Taking norms on the right side gives

ld
dtllP 2 _< alla2IAII Tp p+llpll Ifl
2 dt
< la1211AII TAollP+12 + (lll2 +lf12), (6-16)
2

where Ao denotes the smallest positive eigenvalue; a positive lower bound for Ao is

obtained in Proposition 3. Suppose 7 is large enough that TAo > 1. ('!....-- t > 0

and let c be the constant defined by

c -lf|2 + 211All max lla(s)l2,
sE[o,t]









which is finite due to (6-15). By (6-16), we have


dt
llpll2 C- T- 0A0pll2

on the interval [0, t] assuming TAo > 1. Multiplying both sides by e'xot yields

dt (TAOtpll 2) < CAOtC.


Integration over the interval [0, t] gives

1
IlP(t) 2 < e-TA7t (0) 12 + ( 1 e- -TOt)

< e-TAOt||p( (0)||2 +
TAo

Since the right side approaches 0 as 7 tends to oo, we conclude that for any t > 0,


lim p,(t) = 0.
T---00

Here we have inserted a 7 subscript on p to remind us that the p-component of the

solution a to (6-7) depends on 7.

Now consider the bottom half of the equation (6-7):


iz = A2a, 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 ,(.) is bounded over

any finite interval, independent of 7 by (6-15), it follows from (6-17) that z.(t) approaches

z(0) as t tends to 0, independent of 7. To summarize, we have


lim lim a,() = [0; z(0)],
t-0+ T--oo

where z(0) is the vector of coefficients in the eigenfunction expansion of 0o corresponding

to the eigenvalues A = 0. These coefficients correspond to the index set Sn U So. It follows









that


(x) = lim lim r (x, t) a (0) 4O(x), (6-18)
t-0+ T-Oo
iESnUSo
where

ai(0)= (Vo, Voi)n.

For x E L and i E So, Oi(x) = 0 since Oi, i E So, is supported on Lc. Hence, for x E L,

we have

0+(x) = Y (O)a,( x).
iESn
Since O for i E Sn is the normalized II, ai(0))O(x) is simply the projection of Oo along I:

Y a(0)(x) ( oVn)o, n)Q )(x) On(x). (6-19)
.( (VII, VfII) )
Since H(x) = 1 for x E the top hal' of (6 ) has been established.
Since II(x) = 1 for x E L, the top half of (6-1) has been established.

Now suppose that x e Lc. By the completeness of the ji, we have


0o(X) a(0) ) (x). (6-20)
iEN

Consequently, for x E [c, (6-18) can be rewritten


+(x) o(x) a(0) (x) 0o(x) a(0) (x) (6-21)
iES uS+ iES+

since Oi for i e S1 vanishes on '. Let ( denote the final term in (6-21):


(x) a(0)O(x)
iES+

For i E S+, we have Ai = 0 on LC since the eigenfunctions associated with indices in S+

are harmonic in either L or '. Hence, A =- 0 in c. ( vanishes on fQ since i e Hl(fQ).

To obtain the boundary values for ( on 9, we examine the eigenexpansion (6-20), which

can be rearranged in the form


Y a(0o (x) o(x) 5 a,(0) O(x). (6-22)
iES+ iESnUSoUSi









For x E 9I, Oi(x) = 0 if i E So U Si since Oi for i E So is supported on Lc, while Oi for

i SE S is supported on L. Consequently, for x E a0, it follows from (6-19) and (6-22) that


-(x) =- ai(0)O(x)= o(x) ai(O)dO(x) o(x) OLn(x).
iES+ iESn

Since n (x) = 1 for x e aC, (x) = O 0o(x) on 9C. This completes the proof of Theorem
2.









CHAPTER 7
APPLICATION 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 2 is the

interval [0, 1] and L is a subinterval [a, b] C (0, 1). In this case, there are precisely 2

eigenfunctions in -H. The functions which are harmonic on both L and [c are piecewise

linear. The eigenfunction H of Proposition 1, corresponding to the eigenvalue 0, is defined

by its boundary values H(0) = H(1) = 0 and the values H(x) = 1 on L. Let si, s2,

and S3 be the slope on the intervals [0, a], [a, b], and [b, 1] respectively of the remaining

eigenfunction u e H-. The jump condition of Lemma 1 yields


s2 = -A(s S2) and 2 = -A(s3 S2). (71)

Hence, si = 83. Let s denote either s, or 83. The boundary conditions u(0) = u() = 0

imply that

0 j '(x) dx s1a + s2(b- a) + S3(1 b) = (1 + a b) + 2(b- a).


This gives
(b-a-l1
S2 S b a )

With this substitution in (7-1), we have


A= 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 2 the open

interval (0, 1) and L a subinterval (a, b) whose closure is contained in (0, 1). In this case,



























Figure 7-1: Eigenfunctions in Fi in one dimension.


the equations describing II reduce to


I" 0 in(0, a) U(b, ), II in


[a, b], P1(o) P 1(1) 0 .


The solution is


n H(x)


Six

1

s2(1 -


if xE (0, a),

if xe [a, b],

x) if x E (b, 1),


where


1 1
s8 and S2
a l-b


Hence, we have


(/ 1"')Q sio8(a) + 8200(b)
LL / t
(Tr', 7')Q a + 1-b

Let us define the parameters


1-b
01 -
S-b+a


a
and 02 =
1 -b+a


With these definitions,


OL O 0o(a) + 0200(b),


(1 b)Oo(a) + aOo(b)
1 b+a


\ b
0\ 1


(7-2)









where 01 > 0, 02 > 0, and 01 + 02 = 1. Thus the effect of the lightning is to make the

potential QL on the lightning channel (a, b) a convex combination of the potential Qo(a)

and 0o(b) at the ends of the channel. The coefficients 01 and 02 in the convex combination

depend on the distance between the ends of the channel and the boundary of the domain

Q. It is interesting to note that the potential QL on the lightning channel only depends

on the pre-flash potentials Qo(a) and Qo(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, a approaches the boundary, QL approaches zero since 02

and Qo(a) both approach 0 as a approaches 0 (recall that 0o(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 ( outside the lightning channel L. According

to Theorem 2,


(" =0 on (0,a), ((0) = 0, ((a) =L Qo(a),

1/ 0 on (b, ), (1) 0, ((b) =fL Qo(b).


The solution is
S rx on (0, a),
r2(1 x) on (b, 1),

where
L o(a) a L 0(b)
r.= and r2
a 1-b
Substituting for QL using (7-2), we obtain

(01 1)yo(a) + 02o(b) 02 ) 0(b) yo(a)
r = ( o(b) 1o(a)) -+2.
a a 1-b+a









Hence, by Theorem 2, we have


(X) 0 () + x 1- if x (0, a),

0o(x) ( x) if x (b, 1),

where 6o0 = Oo(b) Oo(a) and I|| b a is the length of the lightning channel. Thus

lightning 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.









CHAPTER 8
CONCLUSIONS

In this dissertation, Maxwell's equations are used to establish a continuous lightning

discharge model:

aA -V (aV) + V J, (x,t) eQx [0,oo), (8-1)

0(x, t) 0, (x, t) e 9 x [0, oo),

(x, 0) -= o(x), x Q,

in a bounded domain f C R" with a connected subdomain When the electric field in

a thundercloud reaches the Ii o, i.1lwn 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 a is essentially infinite. To evaluate the

change in the electric potential due to lightning, we replace a by a + Tr where T is the

characteristic function for L, and we consider the differential equation

=4 -(A, + TA- ) + V.J. (8-2)


where = (-A)-2'. In C'! lpter 4 we show that the eigenproblem for AP is equivalent to

a generalized eigenproblem for the Laplacian (4-7). We analyze the eigenproblem (4-7) in

C'! lpter 5 and obtain the following results: The elements of Ho (,c) are eigenfunctions

corresponding to the eigenvalue 0, while the elements of Hi(C) are eigenfunctions

corresponding to the eigenvalue 1. The remaining eigenfunctions are elements of the

piecewise harmonic space 9-, consisting of functions in Hlo'() which are harmonic in both

L and Lc. There is a one-to-one correspondence between eigenfunctions of (4-7) in 1H

and eigenfunctions of the double 1 vrir potential T in (5-13). The eigenfunctions of (4-7)

are the harmonic extensions of the eigenfunctions of T, and if p is an eigenvalue of T,

then A = 1/2 p is the corresponding eigenvalue of (4-7). H E G (see Proposition

1) is the only eigenfunction in 1H corresponding to the eigenvalue 0. All the remaining









eigenvalues corresponding to eigenfunctions in H are contained in the open interval

(0, 1) and A = 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 the

half-open interval [0, 1), the eigenvalues of the double 1 V-,-r potential T in (5-13) are

contained in [-1/2, 1/2). Proposition 3 gives a lower bound for the positive eigenvalues

of 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 l1- -r

potential T is self adjoint and compact relative to the inner product (5-28), as established

in Lemma 4, we conclude that any f c H0(Q) can be expressed as a linear combination

of orthogonal eigenfunctions for (4-7). The potential immediately after the lightning

discharge is computed in C'!i plter 6 by expanding the potential Q using the orthonormal

eigenfunctions of (4-7) and studying the limits as T 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 II of (4-7). Outside the lightning domain, the

change in the potential is the solution to the problem


A =- 0 in L', 0 on 0Q, = IL 0o on aL.


Applications of both the generalized eigenproblem and the continuous model for the

lightning discharge to one dimension are given in ('!, plter 7.









REFERENCES


[1] R. A. ADAMS, Sobolev Spaces, Academic Press, New York, 1975.

[2] B. C. ASLAN, W. W. HAGER, AND S. MOSKOW, A generalized eigenproblem for
the Laplacian which arises in lightning, J. Math. Pures Appl., (2007, submitted).

[3] P. AUSCHER AND P. TCHAMITCHIAN, Square roots of elliptic second order diver-
gence operators on strongly Lipchitz domains: L2 I,,,,. J. Anal. Math., 90 (2003),
pp. 1-12.

[4] F. CAKONI AND D. COLTON, Qualitative Methods in Inverse Scattering Th(..,;, An
Introduction, Springer, New York, 2006.

[5] D. COLTON AND R. KRESS, Il,/, g,, Equation Methods in Scattering Th(..,;I John
Wiley-Interscience, New York, 1983.

[6] Inverse Acoustic and Electromagnetic Scattering Ti, .., Springer-V, i1 -
Berlin, 1998.

[7] L. C. EVANS, Partial Differential Equations, American Mathematical Society,
Providence, RI, 1998.

[8] G. B. FOLLAND, An Introduction to Partial Differential Equations, Princeton
University Press, New Jersey, 1976.

[9] P. GRISVARD, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.

[10] W. W. HAGER, Updating the inverse of a matrix, SIAM Review, 31 (1989),
pp. 221-239.

[11] A discrete model for the lightning 1.:. li,,hg J. Comput. Phys., 144 (1998),
pp. 137-150.

[12] W. W. HAGER AND B. C. ASLAN, The, l.,i,,g, in electric potential due to lightning
1.- ,ig. SIAM J. Appl. Math., (2007, submitted).

[13] W. W. HAGER, J. S. NISBET, AND J. R. KASHA, The evolution and 1.:- I,,,g of
electric fields within a thunderstorm, J. Comput. Phys., 82 (1989), pp. 193-217.

[14] W. W. HAGER, J. S. NISBET, J. R. KASHA, AND W.-C. SHANN, Simulations of
electric fields within a thunderstorm, J. Atmos. Sci., 46 (1989), pp. 3542-3558.

[15] J. H. HELSDON AND R. D. FARLEY, A numerical modeling study of a montana
thunderstorm: 1. model results versus observations involving nonelectrical aspects,
J. Geophys. Res., 92 (1987), pp. 5645-5659.

[16] A numerical modeling study of a montana thunderstorm: 2. model results versus
observations involving electrical aspects, J. Geophys. Res., 92 (1987), pp. 5661-5675.









[17] J. H. HELSDON, R. D. FARLEY, AND G. FU, Lightning parametrization in a storm
electric i..r.I, model, in Proceedings on the Conference on Atmospheric Electricity,
1988, pp. 849-854.

[18] J. H. HELSDON, G. FU, AND R. D. FARLEY, An intercloud lightning parametriza-
tion scheme for a storm electr':f;. li..., model, J. Geophys. Res., 97 (1992),
pp. 5865-5884.

[19] H. W. KASEMIR, A contribution to the electrostatic theory of a lightning 1.:. h.llg
J. Geophys. Res., 65 (1960), pp. 1873-1878.

[20] A. KIRSCH, S, f,., : gradients and coni ,,,i., properties for some '*,.I, gi,,, operators in
classical scattering i',, .,;; Math. Methods Appl. Sci., 11 (1989), pp. 789-804.

[21] R. KRESS, Linear I,.,I gpd Equations, Springer-Verlag, Berlin, 1989.

[22] D. R. MACGORMAN, A. A. FEW, AND T. L. TEER, I.;, ,, 1.l lightning i.:/;.:'; J.
Geophys. Res., 81 (1986), pp. 9900-9910.

[23] D. R. MACGORMAN, J. M. STRAKA, AND C. L. ZIEGLER, A lightning
parametrization for numerical cloud models, J. Appl. Meteorol., 40 (2001),
pp. 459-478.

[24] E. R. MANSELL, D. R. MACGORMAN, C. L. ZIEGLER, AND J. M. STRAKA,
Simulated three-dimensional branched lightning in a numerical thunderstorm model,
J. Geophys. Res., 107 (2002), pp. 4075-4086.

[25] L. NIEMEYER, L. PIETRONERO, AND H. J. WIESMAN, Fractal dimension of
dielectric breakdown, Physical Review Letter, 52 (1984), pp. 1033-1036.

[26] F. POCKELS, Uber das magnetische verhalten einger basaltischer gesteien, Ann. Phys.
C'!. n 63 (1897), pp. 195-201.

[27] Bestimmung maximaler /1,'l;J,,ig,-stromstdrken aus ihrer magnetisirenden
.:,l.,, Ann. Phys. C'!. ii 65 (1898), pp. 458-475.

[28] Uber die '1l..: ,,,'iI,, l/,,.. ,. erreicht stronstdrke, Phys. Z., 2 (1900), pp. 307-307.

[29] M. A. UMAN, The Lightning D.>. lrg, Academic Press, San Diego, CA, 1987.

[30] H. J. WIESMANN AND H. R. ZELLER, A fractal model of dielectric breakdown
and prebreakdown in and around space lIrg, clouds, J. Geophys. Res., 90 (1985),
pp. 6054-6070.

[31] E. R. WILLIAMS, Electrical .:-. hnrl-g propagation in and around space I,,.g, clouds,
J. Geophys. Res., 90 (1985), pp. 6059-6070.

[32] C. T. R. WILSON, On some determinations of the sign and magnitude of electric
1.:. I,,,,. in lightning flashes, Proc. R. Soc., Ser. A, 92 (1916), pp. 555-574.









[33] Investigations on lightning I.:,. Li.,r and on the electric field of thunderstorms,
Phil. Trans. R. Soc., Ser. A, 221 (1920), pp. 73-115.

[34] C. L. ZIEGLER AND D. R. MACGORMAN, Observed lightning me, //,. I ,I;/ relative
to modeled space .,-/,l/.- and electric field distributions in a tornadic storm, J. Atmos.
Sci., 51 (1994), pp. 833-851.









BIOGRAPHICAL SKETCH

B.vi, Cahllkan Asian was born in Kiitahya, Turkey, in 1977. She was awarded a

Bachelor of Science degree in mathematics in 1999 from Middle East Technical University

(\ IlTU), 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.





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KamileCalskanandYusufCalskan, andtomyhusband, OmerDenizAslan 3

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Firstofall,Iwouldliketoexpressmygratitudetomyadvisor,ProfessorWilliamW.Hager.Withouthisencouragement,consistentsupportandguidance,thisdissertationcouldnothavebeencompleted.Iamgratefultohavehadtheopportunitytostudyundersuchacaring,intelligent,andenergeticadvisor.Hiscondenceinmewillalwaysencouragemetomoveforwardonmyresearch.Second,IwouldalsoliketothankDr.JayadeepGopalakrishnan,Dr.ShariMoskow,Dr.SergeiS.Pilyugin,andDr.VladimirA.Rakovforservingonmysupervisorycommittee.Theirvaluablesuggestionshavebeenveryhelpfultomyresearch.Third,thanksgotomyocemates(Dr.HongchaoZhang,Dr.Shu-JenHuang,andSukanyaKrishnaswamy),andallcolleaguesandfriendsintheDepartmentofMathematicsattheUniversityofFlorida.Theircompanyalleviatedthestressandfrustrationofthistime.Last,butnotleast,Iwishtoexpressmyspecialthankstomyfamily:tomyhusband,Deniz,forhisloveandhisendlesssupporttopursueandcompletemydegree;toourdaughter,ErinBasak,forbeingagloriousjoytous;tomyparentsfortheirimmeasurablesupportandlove;tomyparents-in-lawfortheirwholeheartedunderstandingandencouragement;andtomybrotherforhisunstoppingsupportandencouragement.Withouttheirsupportandencouragement,thisdissertationcouldnothavebeencompletedsuccessfully. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFFIGURES .................................... 7 ABSTRACT ........................................ 8 CHAPTER 1INTRODUCTION .................................. 9 2LITERATUREREVIEW .............................. 11 2.1LightningModelswithExplicitLightningChannels ............. 11 2.1.1Helsdon'sModel ............................. 11 2.1.2MacGorman'sModel .......................... 12 2.1.3Mansell'sModel ............................. 13 2.2Hager'sModel:TheDiscreteModel ...................... 14 3THEDISCRETEMODEL .............................. 15 3.1GoverningEquations .............................. 15 3.1.1Maxwell'sEquations .......................... 15 3.1.2DerivationoftheGoverningEquations ................ 15 3.2TheModelinOne-dimension ......................... 16 3.3GeneralizationtoThree-dimension ...................... 18 4THECONTINUOUSMODEL ........................... 19 4.1FormulationoftheEquations ......................... 19 4.2EigenproblemforA 21 5GENERALIZEDEIGENPROBLEMFORTHELAPLACIAN .......... 23 5.1Introduction ................................... 23 5.2EigenfunctionsofType1,2,and3 ...................... 23 5.3ReformulationofEigenprobleminHUsingDouble-LayerPotential .... 24 5.4EigenvalueSeparationandCompletenessofEigenfunctions ......... 32 6THELIMIT ...................................... 40 6.1Introduction ................................... 40 6.2ReformulationoftheContinuousEquation .................. 41 6.3PotentialChangefortheContinuousEquation ................ 43 7APPLICATIONTOONE-DIMENSION ...................... 48 7.1ApplicationoftheGeneralizedEigenproblem ................. 48 5

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..................... 48 8CONCLUSIONS ................................... 52 REFERENCES ....................................... 54 BIOGRAPHICALSKETCH ................................ 57 6

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Figure page 4-1AsketchofLandforalightningdischarge ................... 20 7-1EigenfunctionsinHinonedimension ........................ 49 7

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Wedevelopacontinuousmodelforthelightningdischarge.WeconsiderMaxwell'sequationsinthreedimensionsandobtainaformulaforthelimitingpotentialasconductivitytendstoinnityinathree-dimensionalsubdomain(thelightningchannel)ofthemodeleddomain.ThelimitisexpressedintermsoftheeigenfunctionsforageneralizedeigenvalueproblemfortheLaplacianoperator.Thepotentialinthebreakdownregioncanbeexpressedintermsofaharmonicfunctionwhichisconstantinthebreakdownregion. 8

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Lightningisoneofthemostbeautifuldisplaysinnature,however,itisalsofrightening.Itcandestroybuildingsandevenkillpeople.Itisacostlyaswellasdeadlynaturaleventthatmankindcannotavoid. Thefearandrespectforlightningattractedmanypeople'sattentionovertheyears.Today,thephysicalprocessesinvolvedinlightningarethefocusofintensiveresearchthroughouttheworld.Lightningisaresultofchargeseparationinsideacloud.Asthegraupelandiceparticleswithinacloudgrowinsizeandincreaseinnumber,undertheinuenceofthewind,collisionsbetweenthemmayoccurresultinginchargeexchangesbetweentheparticles.Ingeneral,smallerparticlesacquirepositivecharge,whilelargerparticlesacquirenegativecharge.Thechargeseparationoccurswhentheseparticlesseparateundertheinuenceofupdraftsandgravity,andasaresult,upperportionofthecloudbecomespositivelychargedandthelowerportionofthecloudbecomesnegativelycharged.Thisresultsinhugeelectricalpotentialdierencewithinthecloudaswellasbetweenthecloudandthegroundcausingaashtooccurmovingchargesbetweenpositiveandnegativeregionsofathunderstorm. DetailedhistoryofearlylightningresearchcanbefoundinUman[ 29 ].BenjaminFranklinwastherstpersonwhoperformedascienticstudyoflightning.Inthesecondhalfofeighteenthcentury,hedesignedanexperimentthatprovedthelightningwaselectrical.Itwasafterphotographicandspectroscopictoolsbecameavailabletowardstheendofthenineteenthcenturythatmorestudiesaboutlightningstartedbeingconducted.LightningcurrentmeasurementswererstproposedbyPockels[ 26 { 28 ].Heestimatedtheamountofcurrentbyanalyzingthemagneticeldinducedbylightningcurrents.Later,Wilson[ 32 33 ]wastherstresearchertousetheelectriceldmeasurementstoestimatethestructureofthunderstormchargesinvolvedinlightningdischarges.HewontheNobelPrizeforinventingtheCloudChambertotrackhighenergyparticlesandmademajor 9

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Mostlightningresearchisdonebyphysicists,chemists,meteorologists,andelectricalengineers.Hager[ 11 13 14 ]wastherstmathematicianusingMaxwell'sequationstodevelopathree-dimensionalmathematicalmodeltosimulatealightningdischarge.Hisdischargemodel[ 14 ]wasobtainedbydiscretizingMaxwell'sequationstoobtainarelationbetweenthepotentialeldandcurrentdensityduetothemotionofchargedparticlesundertheinuenceofthewind.Spatialderivativesinhisequationwereapproximatedbyusingvolumeelementsinspace,whilethetemporalderivativeswereestimatedbyabackwardEulerschemeintime.Sinceconductivityisverylargeintheregionwheretheelectriceldreachesthebreakdownthreshold,heevaluatedthesolutionlimitastheconductivitytendstoinnityinthebreakdownregion.Inhismodel[ 14 ],theoutputwastheelectriceldasafunctionoftime,andtheinputswerecurrentsgeneratedbytheowofchargedparticleswithinthethundercloudundertheinuenceofthewind. ThisdissertationisbasedonHager'smathematicalmodel.SomeimprovementsaremadecomparedtoHager'searlierwork.Forexample,thesolutioniscomputedwithoutdiscretizingtheequations.Consequently,wedonothavehugematrixsystemstocomputeandthereforeitiscomputationallymuchmoreecientandlessexpensive. 10

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Numerousstudiesinlightningfromdierentaspectshavebeenreportedinthepastfewdecades.Thisreviewisfocusedondierentapproachesforthechangeinchargeduetolightning.TheneutralizationofchargebylightninginthemodelswithexplicitlightningchannelsisdiscussedinSection2.1.TheapproachusedbyHageretal.isdiscussedbrieyinSection2.2,andstudiedfurtherinChapter3. 2.1.1Helsdon'sModel 15 { 18 ]estimatedboththegeometryandchargedistributionofanintercloudlightningashinatwo-dimensionalStormElectricationModel(SEM)whichhasbeenextendedtoathree-dimensionalnumericalcloudmodellater.AdaptingideasfromKasemir[ 19 ],theparameterizedlightningpropagatedbidirectionally(initiallyparallelandantiparalleltotheelectriceld)fromthepointofinitialbreakdownanddevelopedsegmentsofoppositechargepolarity. Initiation,propagationdirection,andterminationofthedischargewerecomputedusingthemagnitudeanddirectionoftheelectriceldvectorasthedeterminingcriteria.Thechargeredistributionassociatedwithlightningwasapproximatedbyassumingthatthechannelremainedelectricallyneutraloveritstotallength.Theirdischargefollowedtheelectric-eldlinesuntiltheterminationconditionwassatised.Therefore,theirparametrizationproducedasingle,unbranchedchannel. Asaninitialcritera,athresholdofelectriceldof400kV/mwaschosen.Thechannelwasextendedinbothdirectionsalongtheeldlineuntiltheambientelectric-eldmagnitudefellbelowacertainthreshold(150kV/m)atthelocationsofthechannel-terminationpoints.Theyassumedthatthelinearchargedensityatagridpoint,P,alongthechannelwasproportionaltothedierencebetweenthepotentialatthepointwherethedischarge 11

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23 ]suggestedalightningparametrizationthatwasconsideredanextensionoftheparametrizationofHelsdonetal.[ 18 ]inconjunctionwithsomeofthebulk-lightningparametrizationmethodspresentedbyZieglerandMacGorman[ 34 ].MacGormanetal.[ 23 ]developedaparametrizationtoenablecloudmodelstosimulatethelocationandstructureofindividuallightningashesbyusingtheconceptualmodelofMacGormanetal.[ 22 ]andWilliamsetal.[ 31 ].Theirparametrizationproceededintwostages.UsingtheideasofHelsdonetal.[ 18 ],aashtracedtheelectric-eldlinefromaninitiationpointoutwardinbothparallelandantiparalleldirectionsuntilthemagnitudeoftheambientelectriceldateachendfellbelowsomecertainthresholdvalue.Whenoneendofthechannelreachedground,theparametrizationterminatedatthatend,butallowedtheotherendtocontinuedeveloping. ChargeestimationandneutralizationwereparameterizedbyapplyingthetechniqueproposedbyZieglerandMacGorman[ 34 ],exceptthatZieglerandMacGormanneutralizedchargeatallgridpointshavingj(i;j;k)j1(where(i;j;k)wasthenetchargedensityatthegridpoint(i;j;k)and1wastheminimumj(i;j;k)jforallgridpoints 12

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24 ]proposedalightningparametrizationderivedfromthedielectricbreakdownmodelthatwasdevelopedbyNiemeyeretal.[ 25 ]andWiesmannandZeller[ 30 ]tosimulateelectricdischarges.Theyextendedthedielectricbreakdownmodeltoathree-dimensionaldomaintorepresentmorerealisticelectriceldandthunderstormdynamics. Intheirwork,thestochasticlightningmodel(SLM)wasanapplicationoftheWiesmann-Zellermodeltosimulatebidirectionaldischargesintheregionsofvaryingnetchargedensity(e.g.,inanelectriedthunderstorm).Proceduresforsimulatinglightningashesinthethunderstormmodelwereasfollows.AashoccurredwhenthemagnitudeoftheelectriceldexceededtheinitiationthresholdEinitanywhereinthemodeldomain.Thelightninginitiationpointwaschosenrandomlyfromamongallthepointswherethemagnitudeoftheelectriceldisgreaterthan0.9Einit.BothdecisionsforchoosingtheinitiationthresholdandtheinitiationpointweremadeaccordingtoMacGormanetal.[ 23 ].Positiveandnegativepartsoftheashwerepropagatedindependentlysothatuptotwonewchannelsegments(positiveandnegative)couldbeaddedateachstep.Bothendshaddefaultinitialpropagationthresholdsof0.75Einit.Forashneutrality,theyappliedtheideasfromKasemir[ 19 ]andassumedthatthechannelstructurewouldmaintainoverallchargeneutralityaslongasneitherendreachedtheground.But,forcomputationalsimplicity,theirparametrizationmaintainednear-neutrality(within5%)byatechniqueofadjustingthereferencepotentialtothegrowthofthelightningstructureinsteadofadjustingthereferencepotentialofthechannel. 13

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11 13 14 ]proposedathree-dimensionallightning-dischargemodelthatproducedbidirectionalICand-CGashes.Themodelgeneratedthedischargeregion,chargetransfer,anddetailedchargerearrangementassociatedwiththeash. TheirapproachtolightningwasquitedierentfromthoseinSection2.1.TheirbreakdownmodelwasbasedonMaxwell'sequations.Theyassumedthatcurrentduetotransportofchargeundertheinuenceofwindwasknown.Theyobtainedanequationgoverningtheevolutionoftheelectricpotentialundertheassumptionthatthetimederivativeofthemagneticeldcanbedisregarded.Afterintegratingthisequationoverboxesandapproximatingderivativesbynitedierences,theyobtainedanimplicitsystemofdierenceequationsdescribingtheevolutionoftheelectriceld.Theirapproachtolightningwastolettheconductivitytendtoinnitywherevertheelectriceldreachedthebreakdownthreshold.Thisapproachappealstoourbasicconceptionofnature:Whentheelectriceldreachesbreakdownthreshold,conductivitybecomesverylargeasaplasmaforms. Whentheelectriceldreachesthebreakdownthreshold,theelectricpotentialchangesinstantaneouslyeverywherewithinthethundercloud.TheInverseMatrixModicationFormula[ 10 ]wasappliedtoevaluatethischange: wherebeforewastheelectricpotentialbeforedischarge,afterwastheelectricpotentialafterdischarge,AwasthediscreteLaplacian,andUwasamatrixwitha+1and-1ineachcolumncorrespondingtothearcsassociatedwiththebreakdown.TherewerenoparametersinEquation( 2{1 )besidestheelectricpotentialbeforedischarge.Thiswasconsistentwithexperimentalobservations:Thechargeiscontrolledpredominatelybyasingleparameter:thelocalelectrostaticeld.ThiswasobservedinexperimentsreportedbyWilliamsetal.[ 31 ]. 14

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3.1.1Maxwell'sEquations Intheabsenceofmagneticorpolarizablemedia,thedierentialformoftheseequationsare: 1. Gauss'lawforelectricity:rE= Gauss'lawformagnetism:rH=0 3. Faraday'slawofinduction:rE=@B Ampere'slaw:rH=J0+"@E SinceJ0ispartlyduetothemovementofchargediceandwaterparticlesinthecloudandpartlyduetotheelectricalconductivityofthecloud,wewriteJ0=Jp+0E

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3{1 )becomes @t:(3{2) Takingthedivergenceof( 3{2 ),itfollowsthat 0=rrH=rJpr0r"r@r @t:(3{3) Letting=0="andJ=Jp=",weobtain @t+rrrJ=0:(3{4) Inourmodelwealsohavethefollowingassumptions: 2{1 )inSection2:2,wenowfocusontheequation( 3{4 )inonedimension.Inone-dimension,( 3{4 )reducesto _00+(0)0J0=0;(0)=0;(H)=V;(3{5) 16

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Integrating( 3{5 )over[xi;xi+1]gives_0(xi+1)_0(xi)+(i+10(xi+1)i0(xi))(Ji+1Ji)=0 Toapproximate0,[ 11 13 14 ]usedbackwarddierences,0(xi)ii1 Thiscanbewritteninmatrixformas)A_+B=hJ

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whereWl=[w0j:::jwl]. 18

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3{4 ),thefollowingequationsmodeltheevolutionoftheelectricpotentialinadomain,assumingthetimederivativeofthemagneticuxdensitycanbeneglected: @t=r(r)+rJin[0;1); whereRnisaboundeddomainwithboundary@,0liesinthespaceL1()ofessentiallyboundedfunctionsdenedon,andtheforcingtermJliesinL2(),theusualspaceofsquareintegrablefunctionsdenedon.ThedivergencerJaswellastheequation( 4{1 )areinterpretedinaweaksense,asexplainedlater.Theinitialcondition0isassumedtolieinH10(),theSobolevspaceconsistingoffunctionswhichvanishon@andwithrstderivativesinL2().Theevolutionproblem( 4{1 ){( 4{3 )hasasolution(x;t)with(;t)2H10()andthepartialderivative@t(;t)2H10().AlthoughMaxwell'sequationsdescribetheelectromagneticeldsin3-dimensions,themathematicalanalysisthroughoutthisdissertationisdevelopedinann-dimensionalsetting,wherenisanarbitrarypositiveinteger. Inathunderstorm,istheconductivitydividedbythepermittivityoftheatmosphere,isanylargedomainextendingfromtheEarthtotheionospherewhichcontainsthethundercloudinitsinterior,andJisduetotransportbywindofchargediceandwaterparticlesinthecloud.AlthoughJisafunctionoftime,wefocusonthepotentialchangeduringlightning,whichweconsiderinnitelyfast.Hence,duringalightningevent,Jisessentiallytimeinvariant.Thepotentialoftheionosphereisnotzero,however,wecanmakeachangeofvariablestosubtractothe\fair-eldpotential"(thepotentialofthe 19

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4{1 ){( 4{3 )wherethepotentialvanisheson@. ApossiblelightningchannelissketchedinFigure 4-1 .Mathematically,thelightning Figure4-1: AsketchofLandforalightningdischarge. channelLcouldbeanyconnected,opensetcontainedinwithitscomplementLc=nLconnected.Morerealistically,weshouldviewLasaconnectednetworkofthinopentubes.ThecasewhereLtouches@,aswouldhappenduringacloud-to-groundash,istreatedasalimitinwhichLapproachesarbitrarilycloselytotheboundaryof. Makingthechangeofvariables=()1 2,werewrite( 4{1 )as whereA=()1 2(r(r))()1 2: 2:L2()!H10()denotestheinverseofthesquarerootoftheLaplacian([ 3 ]). LetbethecharacteristicfunctionforL(isidentically0exceptinLwhereis1).TheeectoflightningistoionizethedomainL,inessence,replacingin( 4{4 )by+whereislarge.Ifthelightningoccursatt=0,theninthemomentsafterthe 20

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whereA=()1 2(r(r))()1 2; 4{2 )and( 4{3 ).Herethescalarreectsthechangeinconductivityinthelightningchannel. Let(x;t)denotethesolutionof( 4{5 )attimet,andlet(x;t)=()1 2(x;t)bethecorrespondingsolutionof( 4{1 ).Ifthelightningoccursattimet=0,thentheelectricpotentialrightaftertheashisgivenby limt!0+lim!1(x;t):(4{6) UsinganeigendecompositionforA,thelimit( 4{6 )canbeevaluated. 2 ])thattheeigenproblemforAreducestothefollowinggeneralizedeigenproblem:Findu2H10(),u6=0,and2Rsuchthat forallv2H10(),whereh;iistheL2()innerproduct WeviewH10()asaHilbertspaceforwhichtheinnerproductbetweenfunctionsuandv2H10()isgivenby( 4{8 ). TheweakformoftheeigenproblemforAistondU2L2()suchthat 2U;r()1 2Vi=hU;Vi(4{9) 21

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4{9 )reducestotheproblemofndingu2H10()suchthat 2u;()1 2vi(4{10) forallv2H10().Ifu2C10(),then 2u;()1 2vi=hu;vi=hru;rvi:(4{11) SinceC10()isdenseinH10()andtheoperators()1 2andrarebothboundedinH10(),theidentity( 4{11 )isvalidforallu2H10().Hence,( 4{10 )reducesto( 4{7 ). 22

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Thischapterisbasedonthepaper[ 2 ]. 4{7 ): 1.Thefunctionwhichis1onLandharmoniconnL;theeigenvalueis0. 2.FunctionsinH10()withsupportinnL;theeigenvalueis0. 3.FunctionsinH10()withsupportinL;theeigenvalueis1. 4.Excluding,theharmonicextensionoftheeigenfunctionsofadoublelayerpotentialon@L.Theeigenvaluesarecontainedintheopeninterval(0;1).Theonlypossibleaccumulationpointis=1=2. 4{7 ),wehave whichimpliesthat01.LetH10(L)H10()denotethesubspaceconsistingoffunctionswithsupportinL.Similarly,letH10(Lc)H10()denotethesubspaceconsistingoffunctionswithsupportinLc. 4{7 )ifandonlyifthesupportofuiscontainedinL.Ifu2H10(Lc),thenuisaneigenfunctionof( 4{7 )correspondingtotheeigenvalue0.Theonlyothereigenfunctionof( 4{7 )correspondingtotheeigenvalue0,whichisorthogonaltoH10(Lc),isthesolution2H10()of 4{7 ),thenby( 5{1 ),wehavehru;ruiLc=0:

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4{7 ),uisaneigenfunctioncorrespondingtotheeigenvalue1.Ifu=0inL,thenuisaneigenfunctioncorrespondingtotheeigenvalue0.Thesolutionof( 5{2 )isaneigenfunctionof( 4{7 )correspondingtotheeigenvalue0sincer=0inL. Letw2H10()beanyeigenfunctionof( 4{7 )correspondingtotheeigenvalue0whichisorthogonaltoH10(Lc).By( 4{7 ),wehavehrw;rwiL=0,whichimpliesthatrw=0inL,orwisconstantinLsinceLisconnected.Withoutlossofgenerality,letusassumethatw=1inL.Sincewisorthogonaltothefunctionsv2H10(Lc),wehavehrw;rvi=0forallv2H10(Lc): 5{2 )giveshr(w);rvi=0forallv2H10(Lc): 1 describeseigenfunctionsoftype1,2,and3.Inthissection,wefocusontype4eigenfunctions.LetHbethespacewhichconsistsofallu2H10()satisfyingtheconditions (5{3) 24

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4{7 )onHintermsofanintegraloperator. 4{7 )ifandonlyif @n=@u @non@L;(5{5) @n=@u @n+@u @n2H1=2(@L): Proof. 5{5 ).By( 4{7 )wehave foranyv2H10().IntegratingbypartsandutilizingthefactthatuisharmonicinbothLandLcgives @nd=Z@Lv@u @n+d+Z@Lv@u @nd;(5{7) wheredenotestheboundarymeasureon@L.Hence,wehaveZ@Lv@u @n+@u @n+@u @nd=0 foranyv2H10().Sinceanyv2H1=2(@L)hasanH10()extension,( 5{5 )holds. Conversely,supposethatusatises( 5{5 ).Asin( 5{6 ){( 5{7 ),wehavehrv;rui=Z@Lv@u @n+d+Z@Lv@u @nd

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5{5 )giveshrv;rui=Z@Lv@u @n=hrv;ruiL 4{7 ). NowletusintroducetheGreen'sfunctionon: yG(x;y)=x(y)in;G(x;y)=0fory2@;(5{8) wherexistheDiracdeltafunctionlocatedatx.Thepiecewiseharmonicfunctionsu2Hcanbedescribedintermsofthejumpon@Lofthenormalderivative. @n(y)G(x;y)dy:(5{9) @nydyZLcruryGdyZ@Lu+@G @nydy: @n(y)G(x;y)dy+Z@L@u @n+(y)G(x;y)dy=Z@L@u @n(y)G(x;y)dy; whichyields( 5{9 ). 26

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8 ]);westateithereforthecaseofourGreen'sfunctioncorrespondingtoaboundedouterdomain. @ny(x;y)dy: 2(x)+Z@L(y)@G @ny(x;y)dy; 2(x)+Z@L(y)@G @ny(x;y)dy: 2logjxyjn=2;(5{11) where!nisthesurfaceareaoftheunitsphereinRn.RecallthatyN(x;y)=x(y): @ny(x;y)dy+Z@L(y)@N @ny(x;y)dy:(5{12) 27

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@ny(x;y)dy 5{12 )wecanapplythewellknownresult(see[ 8 ])forthelimitx!@L.>FromtheexteriorofL,wehave @ny(x;y)dy1 2(x)+Z@L(y)@N @ny(x;y)dy=1 2(x)+Z@L(y)@G @ny(x;y)dy: UsingLemma 1 2 ,and 3 ,wereformulatethegeneralizedeigenproblem( 4{7 )onHintermsofaboundaryintegraloperator.Bythetracetheorem[ 1 ,Thm.7.53],anyu2HH10()hasatraceon@LinH1=2(@L).Conversely,u2H1=2(@L)hasauniqueharmonicextensionintobothLandLcwithu=0on@.Hence,thereisaone-to-onecorrespondencebetweenelementsofHandelementsofH1=2(@L). DeneT:L2(@L)!L2(@L) by @ny(x;y):(5{13) By[ 8 ,Prop.3.17],Kisacontinuouskernelofordern2on@L.Itfollowsfrom[ 8 ,Prop.3.12]thatTisacompactoperatorfromL2(@L)toitself. 28

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4{7 )ifandonlyifthecorrespondingu2H1=2(@L)isaneigenfunctionofTwithassociatedeigenvalue1=2;thatis, 47 ).ByLemmas 1 and 2 ,wehaveu(x)=Z@L@u @n(y)G(x;y)dy @ny(x;y)dy+ZLu(y)yG(x;y)dy: @ny(x;y)dy; 3 ,u(x)=1 2u(x)Z@Lu(y)@G @ny(x;y)dy; 5{14 ). Conversely,supposethatu2H1=2(@L)satises( 5{14 ).WeidentifyuwithitsharmonicextensioninH,andwedenew(x)by @ny(x;y)dyforx2Lc;Z@Lu(y)@G @ny(x;y)dy+u(x)forx2L:(5{15) 29

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3 ,wehavew+(x)=1 2u(x)Z@Lu(y)@G @ny(x;y)dy 2u(x)Z@Lu(y)@G @ny(x;y)dy=1 2u(x)Z@Lu(y)@G @ny(x;y)dy: 5{14 )yields Observethatwvanisheson@duetothesymmetryofG(x;y)[ 8 ,Lem.2.33];thatis,sinceG(x;y)=0wheny2@,wehavebysymmetryG(x;y)=0whenx2@.Hence,thenormalderivativein( 5{15 )vanisheswhenx2@.Sincewisharmonicineachsubdomainanditisequaltouonboth@L(see( 5{16 ))and@(theybothvanish),itfollowsthatw=uin.Wereplacewwithuin( 5{15 )toobtain @ny(x;y)dyforx2Lc;Z@Lu(y)@G @ny(x;y)dy+u(x)forx2L:(5{17) Integratingbypartsandusing( 5{8 )givesZ@L@u @n(y)G(x;y)dy=hru;ryGiL=8>>><>>>:Z@Lu(y)@G @ny(x;y)dyifx2Lc;Z@Lu(y)@G @ny(x;y)dyu(x)ifx2L: 5{17 ),weconcludethat @n(y)G(x;y)dy:(5{18) 30

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2 @n(y)G(x;y)dy:(5{19) Subtracting( 5{19 )from( 5{18 )givess(x):=Z@L(y)G(x;y)dy=0foranyx=2@L; @n(y)@u @n(y): @n=@u @n; 1 completestheproof. Toprovethat=0,supposetothecontrarythatdoesnotvanish.Letrbeanysmoothfunctiondenedon@LforwhichZ@L(y)r(y)dy6=0: 5{13 )arerealandcontainedinthehalf-openinterval(1=2;1=2].Theonlypossibleaccumulationpointforthespectrumis0.

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4{7 )areallrealduetosymmetryoftheinnerproduct.ByProposition 2 ,theeigenvaluesofTareallreal.AsnotedbeforeProposition 1 ,theeigenvaluesof( 4{7 )arecontainedontheinterval[0;1].Moreover,byProposition 1 ,theonlyeigenfunctioncorrespondingtotheeigenvalue1hassupportinL.Thetraceofthiseigenfunctionon@Lis0.TheonlyelementinHwithvanishingtraceon@Listhezerofunction.Consequently,thereisnoeigenfunctioninHcorrespondingtotheeigenvalue1.ThereisoneeigenfunctioninHcorrespondingtotheeigenvalue0,namelythefunctionofProposition 1 .Exceptfortheeigenvalue0,alltheremainingeigenvaluesforthegeneralizedeigenproblemlieintheopeninterval(0;1).SincetheeigenvaluesofTare1/2minusthecorrespondingeigenvalueof( 4{7 )in[0;1),theproofiscomplete.SinceTiscompactonL2()[ 8 ,Prop.3.12],theonlypossibleaccumulationpointforthespectrumis0. AlowerboundfortheseparationbetweenthelargestandsecondlargesteigenvaluesofTisobtainedfromProposition 3 1 ,thegeneralizedeigenproblem( 4{7 )restrictedtoHhasasimpleeigenvalue=0correspondingtotheeigenfunction2Hwhiletheremainingeigenvaluesarepositive.ByProposition 2 ,theonlypossibleaccumulationpointforthespectrumis=1=2.Hence,thereisaninterval(0;),>0,wherethegeneralizedeigenproblemhasnoeigenvalues.Wenowgiveanexplicitpositivelowerboundforintermsofthreeembeddingconstants: E1. Letuadenotetheconstantfunctiononwhosevalueistheaverageofu2H1()overL:ua=1 measure(L)ZLu(x)dx: 7 ,Thm.1,p.275],thereexistsaconstant1>0suchthatkruk2L2(L)1kuuak2H1(L) 32

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By[ 1 ,Thm.7.53],thereexistsaconstant2>0suchthatkuk2H1(L)2kuk2H1=2(@L) E3. Thereexistsaconstant3>0suchthat forallu2H10()whichareharmonicinLc(inotherwords,( 5{4 )holds).ThefollowingproofofE3wassuggestedbyJayadeepGopalakrishnan:Foru2H10(),letT(u)2H1=2(@L)denotethetraceofuevaluatedon@L.By[ 9 ,Thm.1.5.1.3],ThasacontinuousrightinversewhichwedenoteT1.Inotherwords,foreachg2H1=2(@L),wehaveT1(g)2H10(),TT1(g)=g,andkT1(g)kH1()kT1kkgkH1=2(@L): 7 ,Thm.3,p.265]) Moreover,sinceuisharmonicinLcandv0vanishesonboth@and@L,wehavehrv0;rv0iLc=hrv0;r(uT1(g))iLc=hrv0;rT1(g)iLckrv0kL2(Lc)krT1(g)kL2(Lc)krv0kL2(Lc)kT1(g)kH1(Lc); 5{21 )toobtainkv0kH1(Lc)ckrT1(g)kH1(Lc): 5{20 ). 33

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4{7 )hasnoeigenvaluesintheinterval(0;)where=minf1;23g1=2: 4{7 ),andletubeanassociatedeigenfunctionwithnormalizationhru;rui=1.If2Histheeigenfunctiondescribedin( 5{2 ),thenwehave =1kuuak2H1(L) 2(@L) Above,( 5{23 )isduetothefactthat=1onL,while( 5{22 ),( 5{24 ),and( 5{25 )comefromE1,E2,andE3respectively. Supposethatthepropositiondoesnothold,inwhichcase<1=2and<123=2.By( 5{23 )and( 5{25 ),wehavekuuak2H1(L)<1=2andkuuak2H1(Lc)<1=2: Ontheotherhand,uandareorthogonalsincetheseeigenfunctionscorrespondtodistincteigenvalues.Sinceuaisamultipleofwhichisorthogonaltou,itfollowsthat 1kr(uua)k2L2()kr(uua)k2H1():(5{27) 34

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5{26 )and( 5{27 ),wehaveacontradiction.Hence,either1=2or123=2. Wecontinuetodeveloppropertiesfortheeigenfunctionsofthegeneralizedeigenproblem( 4{7 )byexploitingtheconnection,giveninProposition 2 ,betweentheeigenfunctionsofthegeneralizedeigenproblem( 4{7 )andthoseofthedoublelayerpotentialTin( 5{13 ).AsnotedbeforeProposition 2 ,thereisaone-to-onecorrespondencebetweenelementsofHandelementsofH1=2(@L).Ifu2H1=2(@L),thenthecorrespondingE(u)2Histheharmonicextensionofu2H1=2(@L)intowhichvanisheson@.Foranyu;v2H1=2(@L),wedenetheinnerproduct (u;v)=hrE(u);rE(v)i:(5{28) Inotherwords,harmonicallyextenduandvinandformtheH10()innerproductoftheextendedfunctions.WenowshowthatTisselfadjointandcompactrelativetothisnewinnerproduct. 5{13 )isself-adjointrelativetotheinnerproduct( 5{28 ). Proof. 7 ,p.265]thatthereexistsaconstant4>0suchthatkrvk2L2()4kvk2H1()

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(v;v)=hrE(v);rE(v)i4kE(v)k2H1()4kE(v)k2H1(L)24kvk2H1=2(@L): Anupperboundfor(v;v)isobtainedfromE3: (v;v)=hrE(v);rE(v)ikE(v)k2H1()(13+13)kvk2H1=2(@L): Here3>0isanalogousto3in( 5{20 )exceptthatitrelatesLto@L:kvk2H1=2(@L)3kE(v)k2H1(L) 5{29 )and( 5{30 )yieldT1. ToshowthatTisselfadjointrelativetotheinnerproduct( 5{28 ),wemustverifytheidentity (Tu;v)=hrE(Tu);rE(v)i=hrE(u);rE(Tv)i=(u;Tv)(5{31) foralluandv2H1=2(@L).WerstobservethattheextensionofTuhastheform 2E(u(x))+Z@Lu(y)@G(x;y) 2E(u(x))+Z@Lu(y)@G(x;y) ByLemma 3 ,thetraceoftherightsideof( 5{32 )isTufromeithersideof@L.Moreover,therightsideisharmonicanditvanisheson@sinceE(u)vanisheson@andG(x;y)=0,independentofy2,whenx2@.SincetherightsideisharmonicandsatisestheboundaryconditionsassociatedwithE(Tu),itmustequalE(Tu). 36

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5{32 ),weobtain (Tu;v)=hrE(Tu);rvi=hrE(Tu);rE(v)iL+hrE(Tu);rE(v)iLc=1 2Z@L@E(u) TheterminE(Tu)associatedwiththeGreen'sfunctioncancelssincethenormalderivativeofadoublelayerpotentialoperatoriscontinuousacross@L(forexample,see[ 6 ,Thm.3.1],[ 5 ,Thm.2.21],[ 21 ,Thm.6.13]). Foranypandq2H,wehavetheidentitieshrp;rqiL=Z@Lq@p @nd=Z@Lp@q @nd; @n+d=Z@Lp@q @n+d: 5{33 )canbemovedfromtheutermstovtoobtain(Tu;v)=1 2Z@L@E(v) WenowshowthatTiscompactonH1=2(@L).ConsiderthecorrespondingfreespacedoublelayerpotentialoperatorTFdenedbyTF(x)=Z@L(y)@N @ny(x;y)dy; 5{11 ).Forn=2,TFiscompactby[ 21 ,Thm.8.20].Forn3,Theorem4.2in[ 20 ]givestheboundednessofTFasamapfromL2(@L)toH1(@L).ThisresultextendstoouroperatorTasfollows.Thedierence, 37

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@ny(x;y)=@G @ny(x;y)@N @ny(x;y): 3 ).Consequently,TTFisboundedfromL2(@L)toH1(@L).SincebothTFandTTFareboundedfromL2(@L)toH1(@L),weconcludethatTisboundedfromL2(@L)toH1(@L).ThisimpliesthatTiscompactonH1=2(@L)sinceH1embedscompactlyinH1=2;thatis,by[ 9 ,Thm.1.4.3.2]HsembedscompactlyinHtwhens>t0.Hence,TiscompactonH1=2(@L). 4{7 )whichareorthogonalrelativetotheinnerproduct( 6{5 ).HeretheconvergenceiswithrespecttothenormofH10(). Proof. 1 ,anycompleteorthonormalbasisforH10(L)isaneigenfunctionbasiscorrespondingtotheeigenvalue1.Likewise,anycompleteorthonormalbasisforH10(Lc)isabasiswhoseelementsareeigenfunctionsofthegeneralizedeigenproblemcorrespondingtotheeigenvalue0.Tocompletetheproof,weneedtoshowthatanyf2Hliesinthespanoftheremainingeigenfunctionsfor( 4{7 ). ByLemma 4 ,Tiscompactandselfadjointrelativetotheinnerproduct(;)denedin( 5{28 ).Hence,everyf2H1=2(@L)hasauniqueexpansionintermsoforthogonaleigenfunctionsofT(forexample,see[ 4 ,Thm.1.28]).Givenf2H,itsrestrictionto@L

PAGE 39

2 ,E(i)isaneigenfunctionforthegeneralizedeigenproblem( 4{7 ). 39

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Thischapterisbasedonthepaper[ 12 ]. 4{1 )intermsoftheeigenfunctionsof( 4{7 )andanalyzelimitstocomputethechangeintheelectricpotentialduetolightningdischarge.Ourmainresultisthefollowing: =0inLc;=0on@;=L0on@L: Thus+hastheconstantvalueLalongthelightningchannelLandthechangeinthepotentialduetolightninghasbeenexpressedintermsofthepotential0beforethelightningandthelightningchannelL.WhenLtouches@,asitwouldduringacloud-to-groundash,L=0andcanbeeliminated.Thatis,asLapproachesthe 40

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6{4 )withL=0. 4{7 ),asgivenbyTheorem 1 .WedecomposeNintothedisjointunionoffoursetscorrespondingtothefourclassesofeigenfunctionsdescribedintheintroduction:S=fi2N:i==krkL2()g;S0=fi2N:i=0;hri;ri=0g;S1=fi2N:i=1g;S+=fi2N:0
PAGE 42

6{6 ).Takingv=j,j=1;2;:::,andutilizingtheorthonormalityoftheeigenfunctionsyieldsthelinearsystem wherethedotdenotestimederivativeand (0)i=hr0;rii;i2N; Sincetheiareorthonormaleigenfunctionsfor( 4{7 ),wehavedij=hri;rjiL=ihi;ji=8><>:iifi=j;0otherwise. Hence,Disadiagonalmatrixwiththeeigenvaluesi,i2N,onthediagonal.Sincetheeigenvaluesarenonnegative,Dispositivesemidenite.WenowconsiderAandf: 6{9 )and( 6{10 )respectively,arebothnite,andwehave forallx2`2since0.SinceAispositivesemidenite,theCauchy-Schwarzinequalityyields 42

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6{12 )wehavexTAx=kkL1*1Xi=1xiri;1Xi=1xiri+=kkL1: 6{13 )yieldstherstinequalityin( 6{11 ). Letu2H10()betheweaksolutiontou=rJp:hru;rvi=hJp;rviforallv2H10(): Forfdenedin( 6{10 ),kfk2=1Xi=1hJp;rii2=1Xi=1hru;rii2=hru;ruikJpk2: 6{14 ). 2 .Multiply( 6{7 )byTandutilizethefactthatDandAarepositivesemidenitetoobtainT_=TATD+TfTf: 2d dtkk2=T_kkkfk1 2(kk2+kfk2): dtetkk2etkfk2:

PAGE 44

Thisshowsthatk()kisuniformlyboundedoveranyniteinterval. Fortheremaininganalysis,itisconvenientiftheeigenvaluesarearrangedindecreasingorder.Thuswithoutlossofgenerality,weassumethatD=264000375 Multiply( 6{7 )by[pT;0]toobtainpT_p=[p;0]T_=[p;0]TA[p;0]T264p0375+[p;0]Tf: 1 2d dtkpk2kk2kAkpTp+kpkkfkkk2kAk0kpk2+1 2(kpk2+kfk2); where0denotesthesmallestpositiveeigenvalue;apositivelowerboundfor0isobtainedinProposition 3 .Supposeislargeenoughthat01.Chooset>0andletcbetheconstantdenedbyc=kfk2+2kAkmaxs2[0;t]k(s)k2;

PAGE 45

6{15 ).By( 6{16 ),wehaved dtkpk2c0kpk2 dte0tkpk2e0tc: 0: 6{7 )dependson. Nowconsiderthebottomhalfoftheequation( 6{7 ): whereA2denotesthebottomhalfofAandf2isthebottomhalfoff.SincethebottomhalfofDiszero,theDtermof( 6{7 )isnotpresentin( 6{17 ).Since()isboundedoveranyniteinterval,independentofby( 6{15 ),itfollowsfrom( 6{17 )thatz(t)approachesz(0)asttendsto0,independentof.Tosummarize,wehavelimt!0+lim!1(t)=[0;z(0)]; 45

PAGE 46

wherei(0)=hr0;rii: Since(x)=1forx2L,thetophalfof( 6{1 )hasbeenestablished. Nowsupposethatx2Lc.Bythecompletenessofthei,wehave Consequently,forx2Lc,( 6{18 )canberewritten sinceifori2S1vanishesonLc.Letdenotethenaltermin( 6{21 ):(x)=Xi2S+i(0)i(x) Fori2S+,wehavei=0onLcsincetheeigenfunctionsassociatedwithindicesinS+areharmonicineitherLorLc.Hence,=0inLc.vanisheson@sincei2H10().Toobtaintheboundaryvaluesforon@L,weexaminetheeigenexpansion( 6{20 ),whichcanberearrangedintheform 46

PAGE 47

6{19 )and( 6{22 )that(x)=Xi2S+i(0)i(x)=0(x)Xi2Si(0)i(x)=0(x)L(x): 2 47

PAGE 48

Inthischapter,wewillpresenttheresultsinone-dimensionforboththegeneralizedeigenproblemfortheLaplacianandthecontinuousmodelforthelightningdischarge. 4{7 )inonedimensionwhereistheinterval[0;1]andLisasubinterval[a;b](0;1).Inthiscase,thereareprecisely2eigenfunctionsinH.ThefunctionswhichareharmoniconbothLandLcarepiecewiselinear.TheeigenfunctionofProposition 1 ,correspondingtotheeigenvalue0,isdenedbyitsboundaryvalues(0)=(1)=0andthevalues(x)=1onL.Lets1,s2,ands3betheslopeontheintervals[0;a],[a;b],and[b;1]respectivelyoftheremainingeigenfunctionu2H.ThejumpconditionofLemma 1 yields Hence,s1=s3.Letsdenoteeithers1ors3.Theboundaryconditionsu(0)=u(1)=0implythat0=Z10u0(x)dx=s1a+s2(ba)+s3(1b)=s(1+ab)+s2(ba): 7{1 ),wehave=1(ba): 7-1 2 indimension1(n=1)withtheopeninterval(0;1)andLasubinterval(a;b)whoseclosureiscontainedin(0;1).Inthiscase, 48

PAGE 49

EigenfunctionsinHinonedimension. theequationsdescribingreduceto00=0in(0;a)[(b;1);=1in[a;b];(0)=(1)=0: 1b: 1b=(1b)0(a)+a0(b) 1b+a: 49

PAGE 50

2 NowletusfocusonthepotentialchangeoutsidethelightningchannelL.AccordingtoTheorem 2 ,00=0on(0;a);(0)=0;(a)=L0(a);00=0on(b;1);(1)=0;(b)=L0(b): 1b: 7{2 ),weobtainr1=(11)0(a)+20(b) 1b+a=r2:

PAGE 51

2 ,wehave+(x)=8><>:0(x)+x0 51

PAGE 52

Inthisdissertation,Maxwell'sequationsareusedtoestablishacontinuouslightningdischargemodel: @t=r(r)+rJ;(x;t)2[0;1); where=()1 2.InChapter4weshowthattheeigenproblemforAisequivalenttoageneralizedeigenproblemfortheLaplacian( 4{7 ).Weanalyzetheeigenproblem( 4{7 )inChapter5andobtainthefollowingresults:TheelementsofH10(Lc)areeigenfunctionscorrespondingtotheeigenvalue0,whiletheelementsofH10(L)areeigenfunctionscorrespondingtotheeigenvalue1.TheremainingeigenfunctionsareelementsofthepiecewiseharmonicspaceH,consistingoffunctionsinH10()whichareharmonicinbothLandLc.Thereisaone-to-onecorrespondencebetweeneigenfunctionsof( 4{7 )inHandeigenfunctionsofthedoublelayerpotentialTin( 5{13 ).Theeigenfunctionsof( 4{7 )aretheharmonicextensionsoftheeigenfunctionsofT,andifisaneigenvalueofT,then=1=2isthecorrespondingeigenvalueof( 4{7 ).2H(seeProposition 1 )istheonlyeigenfunctioninHcorrespondingtotheeigenvalue0.Alltheremaining 52

PAGE 53

4{7 )associatedwitheigenfunctionsinHarecontainedinthehalf-openinterval[0;1),theeigenvaluesofthedoublelayerpotentialTin( 5{13 )arecontainedin[1=2;1=2).Proposition 3 givesalowerboundforthepositiveeigenvaluesofthegeneralizedeigenproblem,orequivalently,alowerboundforthegapbetweenthelargestandthesecondlargesteigenvalueofT.BasedonthefactthatthedoublelayerpotentialTisselfadjointandcompactrelativetotheinnerproduct( 5{28 ),asestablishedinLemma 4 ,weconcludethatanyf2H10()canbeexpressedasalinearcombinationoforthogonaleigenfunctionsfor( 4{7 ).ThepotentialimmediatelyafterthelightningdischargeiscomputedinChapter6byexpandingthepotentialusingtheorthonormaleigenfunctionsof( 4{7 )andstudyingthelimitsastendstoinnityandttendstozerotocomputethesolutionto( 8{1 ).Wendthatthepotentialimmediatelyafterthelightningdischargeisconstantthroughoutthelightningdomainandtheconstantvaluedependsontheinitialpotentialandtheeigenfunctionof( 4{7 ).Outsidethelightningdomain,thechangeinthepotentialisthesolutiontotheproblem=0inLc;=0on@;=L0on@L: 53

PAGE 54

[1] [2] [3] [4] [5] [6] ,InverseAcousticandElectromagneticScatteringTheory,Springer-Verlag,Berlin,1998. [7] [8] [9] [10] [11] ,Adiscretemodelforthelightningdischarge,J.Comput.Phys.,144(1998),pp.137{150. [12] [13] [14] [15] [16] ,Anumericalmodelingstudyofamontanathunderstorm:2.modelresultsversusobservationsinvolvingelectricalaspects,J.Geophys.Res.,92(1987),pp.5661{5675. 54

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[18] [19] [20] [21] [22] [23] [24] [25] [26] [27] ,Bestimmungmaximalerentladungs-stromstarkenausihrermagnetisirendenwirkung,Ann.Phys.Chem.,65(1898),pp.458{475. [28] ,Uberdieblizentlandungenerreichtstronstarke,Phys.Z.,2(1900),pp.307{307. [29] [30] [31] [32] 55

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,Investigationsonlightningdischargesandontheelectriceldofthunderstorms,Phil.Trans.R.Soc.,Ser.A,221(1920),pp.73{115. [34] 56

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BeyzaCalskanAslanwasborninKutahya,Turkey,in1977.ShewasawardedaBachelorofSciencedegreeinmathematicsin1999fromMiddleEastTechnicalUniversity(METU),Ankara,Turkey.In2000,shestartedhergraduatestudyinmathematicsattheUniversityofFlorida,fromwhichshereceivedherM.S.inmathematicsin2003andherPh.D.inmathematicsin2007. 57


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F20101207_AABRBX aslan_b_Page_27.txt
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