Foundations of Potential Theory
The present volume gives a systematic treatment of potential functions. It takes its origin in two courses, one elementary and one advanced, which the author has given at intervals during the last ten years, and has a two-fold purpose: first, to serve as
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MATHEMATISCHEN WISSENSCHAFTEN IN EINZELDARSTELLUNGEN MIT BESONDERER BEROCKSICHTIGUNG DER ANWENDUNGSGEBIETE GEMEINSAM MIT
W. BLASCHKE
M. BORN
Co RUNGE
HAMBURG
G{}TTINGEN
G{}TTINGEN
HERAUSGEGEBEN VON
R. COURANT G{}TTINGEN
BAND XXXI
FOUNDATIONS OF POTENTIAL THEORY BY
OLIVER DIMON KELLOGG
SPRINGER-VERLAG
BERLIN· HEIDELBERG· NEW YORK
1967
FOUNDATIONS OF
POTENTIAL THEORY BY
OLIVER DIMON KELLOGG PROFESSOR OF MATHEMATICS IN HARVARD UNIVERSITY CAMBRIDGE· MASSACHUSETTS· U. S.A.
WITH 30 FIGURES
REPRINT FROM THE FIRST EDITION OF 1929
SPRINGER-VERLAG BERLIN· HEIDELBERG· NEW YORK 1967
Aile Rechte, insbesondere das der Ubersetzung in fremde Sprachen, vorbehalten. Ohne ausdriickliehe Genehmigung des Verlages ist es auch nicht gestattet, dieses Buch oder Teile daraus auf photomechanischem Wege (Photokopie, Mikrokopie) oder auf andere Art zu vervielfaltigen Library of Congress Catalog Card Number 67-29862 Titel Nr. \014
Softcover reprint of the hardcover 1st edition 1967
ISBN-13: 978-3-642-86750-7 001: 10.1007/978-3-642-86748-4
e-ISBN-I3: 978-3-642-86748-4
Preface. The present volume gives a systematic treatment of potential functions. It takes its origin in two courses, one elementary and one advanced, which the author has given at intervals during the last ten years, and has a two-fold purpose: first, to serve as an introduction for students whose attainments in the Calculus include some knowledge of partial derivatives and multiple and line integrals; and secondly, to provide the reader with the fundamentals of the subject, so that he may proceed immediately to the applications, or to the periodical literature of the day. It is inherent in the nature of the subject that physical intuition and illustration be appealed to freely, and this has been done. However, in order that the book may present sound ideals to the student, and also serve the mathematician, both for purposes of reference and as a basis for further developments, the proofs have been given by rigorous methods. This has led, at a number of points, to results either not found elsewhere, or not readily accessible. Thus, Chapter IV contains a proof for the general regular region of the divergence theorem (Gauss', or Green's theorem) on the reduction of volume to surface integrals. The treatment of the fundamental existence theorems in Chapter XI by means of integral equations meets squarely the difficulties incident to ·the discontinuity of the kernel, and the same chapter gives an account of the most recent developments with respect to the Dirichlet problem. Exercises are introduced in the conviction that no mastery of a mathematical subject is possible without working with it. They are designed primarily to illustrate or extend the theory, although the desirability of requiring an occasional concrete numerical result has not been lost sight of. Grateful acknowledgements are due to numerous friends on both sides of the Atlantic for their kind interest in the work. It is to my colleague Professor COOLIDGE that lowe the first suggestion to und
