Potential Theory
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408 John Wermer
Potential Theo ry Second Edition
Springer-Verlag Berlin Heidelberg NewYork 1981
Author John Wermer Department of Mathematics Brown University Providence, RI 02912 U.S.A.
AMS Subject Classifications (1980); 31-02, 31BXX, 31B05. 31BlO, 31B15, 31B20 ISBN 3-540-10276-0 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-10276-0 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved. whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin Heidelberg 1981 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
POTENTIAL THEORY John Wermer
CONTENTS
1. Introduction
VII
2. Electrostatics 3. Poisson's Equation
11
4. Fundamental Solutions
17
5. Capacity
26
6. Energy
34
7. Existence of the Equilibrium Potential
41
8. Maximum Principle for Potentials
50
9. Uniqueness of the Equilibrium Potential
56
10. The Cone Condition
60
11. Singularities of Bounded Harmonic Functions
66
12. Green's Functions
74
13. The Kelvin Transform
84
14. Perron's Method
91
15. Barriers
100
16. Kellogg's Theorem
108
17. The Riesz Decomposition Theorem
114
18. Applications of the Riesz Decomposition
129
19. Wiener's Criterion
138
Appendix
158
References
161
Bib liography
164
Index
166
1.
Introduction
Potential theory grew out of mathematical physics, in particular out of the theory of gravitation and the theory of electrostatics. Mathematical physicists such as Poisson and Green introduced some of the central ideas of the subject. A mathematician with a general knowledge of analysis may find it useful to begin his study of classical potential theory by looking at its physical origins.
Sections 2, 5 and 6 of these Notes give in part
heuristic arguments based on physical
These heuristic
arguments suggest mathematical theorems and provide the mathematician with the problem of finding the proper hypotheses and mathematical proofs. These Notes are based on a one-semester course given by the author at Brown University in 1971.
On the part of the reader, they assume a
knowledge of Real Function Theory to the extent of a first year graduate course.
In addition some elementary facts regarding harmonic functions
are assumed as known. the Appendix.
For convenience we have listed these facts in
Some notation is also explained there.
Essentially all the proofs we give in the Notes are for Euclidean 3-space
R3
and Newtonian potentials
Ix-YI In Section 4 we discuss the situation in needed to go from most results for
to n, R
(n > 3).
n f 3.
The modifications
n > 3, are merely technical, so we state
VIII
W
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