Helms L.L. Potential Theory

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Lester L. Helms

Potential Theory

123

Lester L. Helms

University of Illinois

Department of Mathematics

1409 W. Green St.

Urbana IL 61801, USA

l-helms@math.uiuc.edu

Editorial board:

Sheldon Axler, San Francisco State University

Vincenzo Capasso, Università degli Studi di Milano

Carles Casacuberta, Universitat de Barcelona

Angus MacIntyre, Queen Mary, University of London

Kenneth Ribert, University of California, Berkeley

Claude Sabbah, CNRS, École Polytechnique

Endre Süli, University of Oxford

Wojbor Woyczy

nski, Case Western Reserve University

ISBN 978-1-84882-318-1 e-ISBN 978-1-84882-319-8

DOI 10.1007/978-1-84882-319-8

Springer Dordrecht Heidelberg London New York

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

Library of Congress Control Number: 2009926475

Mathematics Subject Classiﬁcation (2000): 33E30, 35C15, 35J05, 35J15, 35J40, 35J55, 35J67

° Springer-Verlag London Limited 2009

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Springer is part of Springer Science+Business Media (www.springer.com)

Dedicated to my wife Dorothea Helms

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Preface

The ﬁrst six chapters of this book are revised versions of the same chapters

in the author’s 1969 book, Introduction to Potential Theory.Atthetimeof

the writing of that book, I had access to excellent articles, books, and lecture

notes by M. Brelot. The clarity of these works made the task of collating

them into a single body much easier. Unfortunately, there is not a similar

collection relevant to more recent developments in potential theory. A new-

comer to the subject will ﬁnd the journal literature to be a maze of excellent

papers and papers that never should have been published as presented. In

the Opinion Column of the August, 2008, issue of the Notices of the Amer-

ican Mathematical Society, M. Nathanson of Lehman College (CUNY) and

(CUNY) Graduate Center said it best “...When I read a journal article, I

often ﬁnd mistakes. Whether I can ﬁx them is irrelevant. The literature is

unreliable.” From time to time, someone must try to ﬁnd a path through the

maze.

In planning this book, it became apparent that a deﬁciency in the 1969

book would have to be corrected to include a discussion of the Neumann

problem, not only in preparation for a discussion of the oblique derivative

boundary value problem but also to improve the basic part of the subject

matter for the end users, engineers, physicists, etc. Generally speaking, the

choice of topics was intended to make the book more pragmatic and less

esoteric while exposing the reader to the major accomplishments by some

of the most prominent mathematicians of the eighteenth through twentieth

centuries. Most of these accomplishments had their origin in practical mat-

ters as, for example, Green’s assumption that there is a function, which he

called a potential function, which could be used to solve problems related to

electromagnetic ﬁelds.

This book is targeted primarily at students with a background in a senior

or graduate level course in real analysis which includes basic material on

topology, measure theory, and Banach spaces. I have tried to present a clear

path from the calculus to classic potential theory and then to recent work

vii

viii Preface

on elliptic partial diﬀerential equations using potential theory methods. The

goal has been to move the reader into a fertile area of mathematical research

as quickly as possible.

The author is indebted to L. H¨ormander and K. Miller for their prompt

responses to queries about the details of some of their proofs. The author also

thanks Karen Borthwick of Springer, UK, for guiding the author through the

publication process.

Urbana, Illinois Lester L. Helms

September, 2008

Contents

0 Preliminaries ............................................. 1

0.1 Notation .............................................. 1

0.2 Useful Theorems ....................................... 4

1 Laplace’s Equation ....................................... 7

1.1 Introduction ........................................... 7

1.2 Green’sTheorem ....................................... 8

1.3 Fundamental Harmonic Function . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 TheMeanValue Property ............................... 10

1.5 PoissonIntegralFormula ................................ 14

1.6 Gauss’AveragingPrinciple .............................. 21

1.7 TheDirichletProblemfor a Ball ......................... 24

1.8 KelvinTransformation .................................. 32

1.9 PoissonIntegralforHalf-space ........................... 33

1.10 Neumann Problemfor a Disk ............................ 39

1.11 NeumannProblemfortheBall........................... 42

1.12 Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2 The Dirichlet Problem .................................... 53

2.1 Introduction ........................................... 53

2.2 Sequences ofHarmonicFunctions......................... 54

2.3 SuperharmonicFunctions................................ 59

2.4 PropertiesofSuperharmonicFunctions.................... 65

2.5 Approximationof SuperharmonicFunctions................ 71

2.6 Perron-WienerMethod.................................. 75

2.7 TheRadialLimit Theorem .............................. 88

2.8 Nontangential Boundary Limit Theorem . . . . . . . . . . . . . . . . . . . 92

2.9 HarmonicMeasure...................................... 100