Potential Theory PDF
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Eckmann 408 John Wermer Brown University, Providence, RI/USA Potential Theory Springer-Verlag Berlin Heidelberg GmbH 1974 Library of Congress Cataloging in Publication Data Wermer, John. Potential theory. (Lecture notes in mathematics, 408) Bibliography: p. 1. Potential, Theory of •. I. Title. II. Series: Lecture notes in mathematics (Berlin) 408. QA3.L28 no. 408 [QA331] 510'.8s [515'.7] 74-14663 AMS Subject Classifications (1970): 31-XX,31 8XX,31 805,31 810, 31815,31820 ISBN 978-3-540-06857-0 ISBN 978-3-662-12727-8 (eBook) DOI 10.1007/978-3-662-12727-8 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 photo copying 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 1974 Originally published by Springer-Verlag Berlin' Heidelberg· New York in 1974 POTENTIAL THEORY John Wermer CONTENTS 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 Function 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. Appendix 138 20. References 141 21. Bibliography 144 22. Index 146 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 considerations. 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 aS$umed as known. For convenience we have listed these facts in the Appendix. Some notation is also explained there. Essentially all the proofs we give in the Notes are for Euclidean 3-space R3 and Newtonian potentials J ~. Ix-yl In Section 4 we discuss the situation in Rn, n ~ 3. The modifications needed to go from R3 to Rn, n > 3, are merely technical, so we state most results for Rn, (n ~ 3). VIII When n = 2, there are significant differences. Potential theory in R2 is treated in many books concerned with analytic functions. A detailed treatment can be found in the book, "Potential Theory in Modern Function Theory" by M. Tsuji, TOkyo, 1959. The classical treatise on Potential Theory, which carries the subject up to 1930, is O. Kellogg "Foundations of Potential Theory", Springer, 1929. The present Notes make use of this book and also of the following works: O. Frostman, "Potentiel d'Equilibre et Capacite des Ensembles", Dissertation, Lund, 1935. Lennart Carleson, "Selected Problems on Exceptional Sets", Van Nostrand Mathematical Studies, #13, 1967. L. L. Helms, "Introduction to Potential Theory", Wiley-Interscience, 1969. These four books give some historical discussion of Potential Theory and many references to original papers. We have collected our References in Section 20 of the Notes. I want to express my thanks to Richard Basener, Yuji Ito and Charles Stanton for their help in preparing these Notes. I am also indebted to Lars Hormander for valuable conversations. I am grateful to Miss Sandra Spinacci for typinq the manuscript. This work was partially supported by NSF Grant GP-28574. August 1972 John Wermer 2. Electrostatics We shall consider electric charges distributed in space and the force fields which these charges produce. Consider an electrically charged body, say negatively charged If we look at a charged test particle in the presence of this body, we find that a force acts on it. This force depends on the position (x,y,z) of the particle, and on its charge el. We have -:. Force on particle c el • ~(x,y,z). Here ~~( x,y,z) is a vector independent of the particle. Thus, associated with our ...J. charged body, there is a vector field ~ defined everywhere in space outside the ~ body. ~ is called the electric field due to the body. If we have several charged bodies, with electric fields then we find that the force on a test particle of charge el now is el Thus, the electric field due to several bodies is the sum of their individual fields. If the body is idealized to be a point charge e located at a point A then the electric field at each point B has direction AB and magnitude t (Coulomb's Law), where r = distance (A,B» ~ e.:> O. 2 B o/'--_-x- In vector notation, a pOint charge e at x induces a field ~~ such that -4 e (2.1) ~(x) = 2" r ....:. for all x € 1R3, Fhere r = Ix -xl. Note that ~ is singular at x. At the end of the 18th century it was observed by Lagrange that ~ a scalar function ~ with ... ~ = -grad~, where ~ (x) = I x:xl (Verify) -0~ In general, a force field is said to have a potential function U if ~ J = - grad U, i.e., ...... } = (Fl,F2,F3), Fl = -Ux' F2 = -Uy' F3 -U z . (The minus sign is convention.) Some force fields have potential functions and others don't. ->. ..>. Consider a force field \j that does: -;-y = -grad U. Fix two pOints P,Q. and a 3 path ~: x = xes), y = yes), z = z(s), from P to Q, where the parameter s is arc length. It The work W done by the field in mOving a particle from P to Q is de- fined by where F t = tangential component of B' • The unit tangent vector to ~ is Hence -U ~ - U ~ - U ~ x ds Y ds Z ds - ad s U(x(s),y(s),z(s». Hence W = -J Q (d- - U)ds -(U(Q) - U(p». P ds Thus: the work done by the field in moving the particle from P to Q = -change in potential functions from P to Q. Suppose now we are given n point charges is located at i = 1,2, ••. n. What is the field ~~ they produce? Using (2.1), we get n e. (2.2) q>(x) = L ~. i=l Ix-xii
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