Potential Theory And Degenerate Partial Differential Operators

Recent years have witnessed an increasingly close relationship growing between potential theory, probability and degenerate partial differential operators. The theory of Dirichlet (Markovian) forms on an abstract finite or infinite-dimensional space is common to all three disciplines. This is a fascinating and important subject, central to many of the contributions to the conference on `Potential Theory and Degenerate Partial Differential Operators', held in Parma, Italy, February 1994.

This book provides a detailed introduction to nonlinear potential theory based on supersolutions to certain degenerate elliptic equations of the p-Laplacian type. Recent research has shown that classical notions such as blayage, polar sets, Perron's method, and fine topology have their proper analogues in a nonlinear setting, and this book presents a coherent exposition of this natural extension of classical potential theory. Yet fundamental differences to classical potential theory exist, and in many places a new approach is mandatory. Sometimes new or long-forgotten methods emerge that are applicable to problems in classical potential theory. Quasiregular mappings constitute a natural field of applications, and a careful study of the potential theoretical aspects of these mappings is included. The principle aim of the book is to explore the ground where partial differential equations, harmonic analysis, and function theory meet. The quasilinear equations considered in this book involve a degeneracy condition given in terms of a weight function and therefore most results appear here for the first time in print. The reader interested exclusively in the unweighted theory will find new results, new proofs, and a reorganization of the material as compared to the existing literature. The book is intended for researchers and graduate students in potential theory, variational calculus, partial differential equations, and quasiconformal mappings.

This collection of articles and surveys is devoted to Harmonic Analysis, related Partial Differential Equations and Applications and in particular to the fields of research to which Richard L. Wheeden made profound contributions. The papers deal with Weighted Norm inequalities for classical operators like Singular integrals, fractional integrals and maximal functions that arise in Harmonic Analysis. Other papers deal with applications of Harmonic Analysis to Degenerate Elliptic equations, variational problems, Several Complex variables, Potential theory, free boundaries and boundary behavior of functions.

An examination of the symbiotic and productive relationship between fully nonlinear partial differential equations and generalized potential theories In recent years, there has evolved a symbiotic and productive relationship between fully nonlinear partial differential equations and generalized potential theories. This book examines important aspects of this story. One main purpose is to prove comparison principles for nonlinear potential theories in Euclidian spaces straightforwardly from duality and monotonicity under the weakest possible notion of ellipticity. The book also shows how to deduce comparison principles for nonlinear differential operators, by marrying these two points of view, under the correspondence principle. The authors explain that comparison principles are fundamental in both contexts, since they imply uniqueness for the Dirichlet problem. When combined with appropriate boundary geometries, yielding suitable barrier functions, they also give existence by Perron’s method. There are many opportunities for cross-fertilization and synergy. In potential theory, one is given a constraint set of 2-jets that determines its subharmonic functions. The constraint set also determines a family of compatible differential operators. Because there are many such operators, potential theory strengthens and simplifies the operator theory. Conversely, the set of operators associated with the constraint can influence the potential theory.

This volume presents current trends in analysis and partial differential equations from researchers in developing countries. The fruit of the project 'Analysis in Developing Countries', whose aim was to bring together researchers from around the world, the volume also includes some contributions from researchers from developed countries. Focusing on topics in analysis related to partial differential equations, this volume contains selected contributions from the activities of the project at Imperial College London, namely the conference on Analysis and Partial Differential Equations held in September 2016 and the subsequent Official Development Assistance Week held in November 2016. Topics represented include Fourier analysis, pseudo-differential operators, integral equations, as well as related topics from numerical analysis and bifurcation theory, and the countries represented range from Burkina Faso and Ghana to Armenia, Kyrgyzstan and Tajikistan, including contributions from Brazil, Colombia and Cuba, as well as India and China. Suitable for postgraduate students and beyond, this volume offers the reader a broader, global perspective of contemporary research in analysis.

A self-contained treatment appropriate for advanced undergraduate and graduate students, this volume offers a detailed development of the necessary background for its survey of the nonlinear potential theory of superharmonic functions. Starting with the theory of weighted Sobolev spaces, the text advances to the theory of weighted variational capacity. Succeeding chapters investigate solutions and supersolutions of equations, with emphasis on refined Sobolev spaces, variational integrals, and harmonic functions. Chapter 7 defines superharmonic functions via the comparison principle, and chapters 8 through 14 form the core of the nonlinear potential theory of superharmonic functions. Topics include balayage; Perron's method, barriers, and resolutivity; polar sets; harmonic measure; fine topology; harmonic morphisms; and quasiregular mappings. The book concludes with explorations of axiomatic nonlinear potential theory and helpful appendixes.

The ?rst six chapters of this book are revised versions of the same chapters in the author’s 1969 book, Introduction to Potential Theory. Atthetimeof the writing of that book, I had access to excellent articles,books, and lecture notes by M. Brelot. The clarity of these works made the task of collating them into a single body much easier. Unfortunately, there is not a similar collection relevant to more recent developments in potential theory. A n- comer to the subject will ?nd the journal literature to be a maze of excellent papers and papers that never should have been published as presented. In the Opinion Column of the August, 2008, issue of the Notices of the Am- ican Mathematical Society, M. Nathanson of Lehman College (CUNY) and (CUNY) Graduate Center said it best “. . . When I read a journal article, I often ?nd mistakes. Whether I can ?x them is irrelevant. The literature is unreliable. ” From time to time, someone must try to ?nd a path through the maze. In planning this book, it became apparent that a de?ciency in the 1969 book would have to be corrected to include a discussion of the Neumann problem, not only in preparation for a discussion of the oblique derivative boundary value problem but also to improve the basic part of the subject matter for the end users, engineers, physicists, etc.