Classical and Modern Potential Theory and Applications PDF
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Classical and Modern Potential Theory and Applications NATO ASI Series Advanced Science Institutes Series A Series presenting the results of activities sponsored by the NA TO Science Committee, which aims at the dissemination of advanced scientific and technological knowledge, with a view to strengthening links between scientific communities. The Series is published by an international board of publishers in conjunction with the NATO Scientific Affairs Division A Life Sciences Plenum Publishing Corporation B Physics London and New York C Mathematical Kluwer Academic Publishers and Physical Sciences Dordrecht, Boston and London D Behavioural and Social Sciences E Applied Sciences F Computer and Systems Sciences Springer-Verlag G Ecological Sciences Berlin, Heidelberg, New York, London, H Cell Biology Paris and Tokyo I Global Environmental Change NATO-PCO-DATA BASE The electronic index to the NATO ASI Series provides full bibliographical references (with keywords and/or abstracts) to more than 30000 contributions from international scientists published in all sections of the NATO ASI Series. Access to the NATO-PCO-DATA BASE is possible in two ways: - via online FILE 128 (NATO-PCO-DATA BASE) hosted by ESRIN, Via Gafileo Gafilei, 1-00044 Frascati, Italy. - via CD-ROM "NATO-PCO-DATA BASE" with user-friendly retrieval software in English, French and German (©WTV GmbH and DATAWARE Technologies Inc. 1989). The CD-ROM can be ordered through any member of the Board of Publishers or through NATO-PCO, Overijse, Belgium. Series C: Mathematical and Physical Sciences -Vol. 430 Classical and Modern Potential Theory and Applications edited by K. GowriSankaran Department of Mathematics and Statistics, McGiII University, Montreal, Quebec, Canada J. Bliedtner Department of Mathematics, J.w. Goethe University, Frankfurt, Germany D. Feyel Department of Mathematics, University Evry-Essone, Evry, France M. Goldstein Department of Mathematics, Arizona State University, Tempe, Arizona, U.S.A. W. K. Hayman Department of Mathematics, University of York, Heslington, York, U.K. and 1. Netuka Faculty of Mathematics and Physics, Charles University, Praha, Republic of Czechia Springer-Science+Business Media, BV. Proceedings of the NATD Advanced Research Workshop on Classical and Modem Potential Theory and Applications Chateau de Bonas, France July 25-31, 1993 A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-94-010-4498-1 ISBN 978-94-011-1138-6 (eBook) DOI 10.1007/978-94-011-1138-6 Printed on acid-free paper AII Rights Reserved © 1994 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1994 Softcover reprint of the hardcover 1s t edition 1994 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photo copying, recording or by any information storage and retrieval system, without written permission from the copyright owner. Contents Preface ix List of Participants xi D. R. Adams Nonlinear PDE and the Wiener Test 1 O. Akin k-Superharmonic Functions and L.Kelvin's Theorem 11 O. Akin and H. Leutwiler On the Invariance of the Solutions of the Weinstein Equation under Mobius Transformations 19 D.H. Armitage Radial Limiting Behaviour of Harmonic and Superharmonic Functions 31 J. Bauer Multiparameter Processes associated with Ornstein-Uhlenbeck Semi-groups 41 A.D. Bendikov On the Problem of Hypoellipticity on the Infinite Dimensional Torus 57 E.M.J. Bertin L'equation de Monge-Ampere dans un espace de Banach 61 L. Beznea and N. Boboc Excessive Functions and Excessive Measures. Hunt's Theorem on Balayages, Quasi-continuity 77 M. Biroli The Wiener Test for Poincare-Dirichlet Forms 93 J. Bliedtner and P.A. Loeb The Best Approach for Boundary Limits 105 N. Boboc Fine Behaviour of BaJayages in Potential Theory 113 N. Bouleau Some Results about Sequential Integration on Wiener Space 125 B. Burgeth Schwarz Lemma type Inequalities for Harmonic Functions in the Ball 133 S.-1. Eriksson-Bique Duality of H-Cones 149 vi D. Feyel Regularite et Integrabilite des Fonctionnelles de Wiener 159 B. Fuglede Poincare Inequalities in Ll-norm for the Sphere and a Strong Isoperimetric Inequality in R:' 165 S.J. Gardiner Uniform and Tangential Harmonic Approximation 185 J. Glover and M. Rao Inversion and Reflecting Brownian Motion 199 J. Glover, M. Roo, H. Sikic and R. Song r-Potentials 217 K. GowriSankaran Fatou-Doob Limits and the Best Filters 233 A. Grigor'yan Gaussian Upper Bounds for the Heat Kernel and its Derivatives on a Riemannian Manifold 237 R.R. Hall and W.K. Hayman Integrals of Analytic Functions along 2 Curves 253 W. Hansen and N. N adirashvili On the Restricted Mean Value Property for Measurable Functions 267 W. Hengartner and J. Rostand A Constructive Method for Univalent Logharmonic Mappings 273 K. Janssen and H.-H. Muller Choquet Type Integral Representation of Polyexcessive Functions 293 P.A. Loeb and H. Osswald Refining the Local Uniform Convergence Topology 315 T. Murai Daily Rheological Phenomena 317 T. Murazawa Convergence Property and Superharmonic Functions on Balayage Spaces 353 I. Netuka and J. Vesely Mean Value Property of Harmonic Functions 359 F. Perez-Gonzalez and R. Trujillo-Gonzalez Farrell and Mergelyan Sets for the Space of Bounded Harmonic Functions 399 vii A. de la Pradelle Methodes Analytiques en Dimension Infinie 413 S. Song Construction d'un Processus it deux Parametres it partir d'un Semi-groupe it un Parametre 419 J.-M. Wu Capacities and Harmonic Measures for Uniformly Elliptic Operators of non-Divergence Form 453 Problems 461 PREFACE A NATO Advanced Research Workshop on Classical and Modern Potential The ory and Applications was held at the Chateau de Bonas, France, during the last week of July 1993. The workshop was organized by the Co-Directors M. Goldstein (Ari zona) and K. GowriSankaran (Montreal). The other members of the organizing committee were J. Bliedtner (Frankfurt), D. Feyel (Paris), W.K. Hayman (York, England) and I. Netuka (Praha). The objective of the workshop was to bring to gether the researchers at the forefront of the aspects of the Potential Theory for a meaningful dialogue and for positive interaction amongst the mathematicians prac tising different aspects of the theory and its applications. Fifty one mathematicians participated in the workshop. The workshop covered a fair representation of the classical aspects of the theory covering topics such as approximations, radial be haviour, value distributions of meromorphic functions and the modern Potential theory including axiomatic developments, probabilistic theories, studies on infinite dimensional Wiener spaces, solutions of powers of Laplacian and other second order partial differential equations. There were keynote addresses delivered by D. Armitage (Belfast), N. Bouleau (Paris), A. Eremenko (Purdue), S.J. Gardiner (Dublin), W. Hansen (Bielefeld), W. Hengartner (Laval U., Quebec), K. Janssen (Dusseldorf), T. Murai (Nagoya), A. de la Pradelle (Paris) and J.M. Wu (Urbana). There were thirty six other invited talks of one half hour duration each. The lectures covered quite a range of topics evenly distributed between the classical and the modern aspects of the subject. We want to thank all the speakers for their extremely interesting and stimulating talks. We also want to thank all the participants for such interesting scientific discussions and for the excellent research atmosphere that prevailed during the entire week of the workshop. A lot of credit goes to the staff at the Chateau who looked so well after the participants and their various needs giving us all the opportunity to engage in scientific activities. Our sincere thanks are due to them. This volume consists of the Proceedings of the workshop. They include the con tributions based on talks given at the workshop as well as a some of the research which culminated from the discussions that took place during that time. All the contributions were put through the normal refereeing procedure and only the pa pers accepted by the referees appear in this volume. The volume also includes the outcome of an afternoon session on the discussion of state of the art problems in the subject. Our thanks are due to Mr. Xun Zhang (Montreal) for his help in editing this part of the Proceedings. This workshop was financed mostly by the Scientific Affairs Division of NATO. Additional support was provided by the Ministere de la Defense, Gouvernment de France. We express our sincere thanks to all of them for the generous support. ix x Frau C. Belz (Frankfurt) helped us in the preparation of the abstracts and the other material before the workshop and our thanks are due to her. We want to express our very special gratitude to the administrative staff in the Department of Mathematics and Statistics of McGill University, and in particular to Ms. Valerie McConnell, Elaine Swan and Elaine Tremblay who did so much work concerning the workshop and later with the editorial work of the Proceedings. Also Ms. Heather MacAuliffe and Elaine Swan retyped some of the manuscripts and reformatted many other manuscripts and our thanks are due to them for tremendously efficient work in such a short time framework. We would like to thank Ms. Luna Zaru and Mr. Xun Zhang for proofreading. Last but not the least, we would like to express our thanks to the Kluwer Academic Publishers, Dordrecht, The Netherlands, and in particular to Mrs. Odet Kiers-Pols for such efficient and helpful attention. K. GowriSankaran and J. Bliedtner D. Feyel M. Goldstein W.K. Hayman 1. Netuka LIST OF PARTICIPANTS David R. Adams, Dept. of Mathematics, University of Kentucky, Lexington, KY, 40506, USA ([email protected]). Orner Akin, Dept. of Mathematics, Faculty of Science, University of Ankara, 06100 Tandogan, Ankara, Turkey. Ivan Andronov, Dept. of Mathematical Physics, State University of St. Petersburg, Russia ([email protected]). David Armitage, Dept. of Mathematics, The Queen's University of Belfast, Belfast BT7 INN, Northern Ireland, UK ([email protected]) Line Baribeau, Departement de Mathematiques et Statistiques, Universite Laval, Quebec, Canada, GIK 7P4 ([email protected]). Jiirgen Bauer, Dept. of Mathematics, Johann Wolfgang Goethe-Universitiit, 600054 Frankfurt am Main, Germany. Alexander Bendikov, Mathematisches Institut, Universitat Erlangen-Niirnberg, t, Bismarckstrasse 1 91054 Erlangen, Germany. Christian Berg, Kobenhavns Univ. Mat. Inst., Universitetsparken 5, DK-2100 Kobenhavn, Denmark ([email protected]). Emile Bertin, Mathematisch Istituut, Rijksuniversiteit, Utrecht, The Netherlands ([email protected]). Lucien Beznea, Mathematisch-Geographische Fakultat, Katholische Universitat Eichstatt, Ostenstrasse 26-28,85071 Eichstatt, Germany. Marco Biroli, Dipartimento di Mathematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy. Jiirgen Bliedtner, Fachbereich Mathematik, Johann Wolfgang Goethe-Universitat, 60054 Frankfurt am Main, Germany ([email protected]) Nicu Boboc, Faculty of Mathematics, University of Bucarest, Str. Academiei 14, Ro-70109 Bucharest, Romania. Nicolas Bouleau, Directeur du Lab de Math. Appliquees, U.A. CNRS 1502, Ecole Nationale des Ponts et Chausees, La Courtine, 93167 Noisy Le Grand, France ([email protected]) xi
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