Lecture Notes of 12 the Unione Matematica Italiana

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1 Lecture Notes of 12 the Unione Matematica Italiana For further volumes:

2 Editorial Board Franco Brezzi (Editor in Chief) IMATI-CNR Via Ferrata 5a Pavia, Italy John M. Ball Mathematical Institute St Giles Oxford OX1 3LB United Kingdom Alberto Bressan Department of Mathematics Penn State University University Park State College PA , USA Fabrizio Catanese Mathematisches Institut Universitatstraße Bayreuth, Germany Carlo Cercignani Dipartimento di Matematica Politecnico di Milano Piazza Leonardo da Vinci Milano, Italy Corrado De Concini Dipartimento di Matematica Universitá di Roma La Sapienza Piazzale Aldo Moro Roma, Italy deconcini@mat.uniroma1.it Persi Diaconis Department of Statistics Stanford University 450 Serra Mall Stanford, CA , USA diaconis@math.stanford.edu, cgates@stat.stanford.edu Nicola Fusco Dipartimento di Matematica e Applicazioni Universitá di Napoli Federico II, via Cintia Complesso Universitario di Monte S. Angelo Napoli, Italy nfusco@unina.it Carlos E. Kenig Department of Mathematics University of Chicago 5734 University Avenue Chicago, IL USA cek@math.uchicago.edu Fulvio Ricci Scuola Normale Superiore di Pisa Piazza dei Cavalieri Pisa, Italy fricci@sns.it Gerard Van der Geer Korteweg-de Vries Instituut Universiteit van Amsterdam Plantage Muidergracht TV Amsterdam, The Netherlands geer@science.uva.nl Cédric Villani Institut Henri Poincaré 11 rue Pierre et Marie Curie Paris Cedex 05 France cedric.villani@upmc.fr The Editorial Policy can be found at the back of the volume.

3 Victor Anandam Harmonic Functions and Potentials on Finite or Infinite Networks 123

4 Victor Anandam University of Madras Ramanujan Inst. for Advanced Study in Mathematics Department of Mathematics Chepauk Chennai Tamil Nadu India ISSN ISBN e-isbn DOI / Springer Heidelberg Dordrecht London New York Library of Congress Control Number: Mathematics Subject Classification (2000): 31C20; 31D05; 30F20; 31A30; 15A09 c Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 Bhickoo, Anahita, Parvez

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7 Preface The aim of the current work is to present an autonomous theory of harmonic functions on infinite networks akin to potential theory on locally compact spaces as developed primarily by Brelot (without sanctioning any explicit role for the derivatives of functions defined on the space). Though random walks and electrical networks are two important sources for the advancement of the present theory, neither probabilistic methods nor energy integral techniques are used here to prove the results in an infinite network. The relevance of this study is partly because in many infinite networks (like homogeneous trees, for example), any real-valued function defined on the network is a difference of two superharmonic functions. We consider principally the classification theory of infinite networks based on the existence of Green functions, bounded harmonic functions etc., and then balayage, equilibrium principle, domination principle, Schrödinger operators, polyharmonic functions and the Riesz-Martin compactification of the network. An important feature is the study of parabolic networks. These are the networks on which no positive potentials exist or equivalently, these are the networks on which the Green function cannot be defined. On parabolic networks we investigate the properties of pseudo-potentials (analogous to logarithmic potentials on the complex plane) introduced via a development of a notion of flux. My sincere thanks go to the referees who read the manuscript and made valuable comments. Ramanujan Institute for Advanced Study in Mathematics University of Madras March 2011 Victor Anandam vii

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9 Contents 1 Laplace Operators on Networks and Trees Introduction Preliminaries Examples of Superharmonic Functions on Networks Green s Formulas Minimum Principle Infinite Trees Potential Theory on Finite Networks Incidence Matrix, Kirchhoff s Problem Dirichlet-Poisson Equations in Finite Networks Dirichlet Semi-Norm Schrödinger Operators on Finite Networks Harmonic Function Theory on Infinite Networks Infinite Networks and the Laplace Operator Classification of Infinite Networks Harmonic Measure at Infinity of a Section Positive Harmonic Functions on a Network Integral Representation of Positive Harmonic Functions Hyperbolic Networks Parabolic Networks Flux at Infinity Pseudo-Potentials Schrödinger Operators and Subordinate Structures on Infinite Networks Local Properties of q-superharmonic Functions Classification of q-harmonic Networks Subordinate Structures ix

10 x Contents 5 Polyharmonic Functions on Trees Polyharmonic Functions on Infinite Trees Polyharmonic Functions with Point Singularity Riesz-Martin Representation for Positive m-superharmonic Functions References Index