EMS Tracts in Mathematics 2011; 415 pp; hardcover Volume: 17 ISBN10: 303719099X ISBN13: 9783037190999 List Price: US$84 Member Price: US$67.20 Order Code: EMSTM/17
 The \(p\)Laplace equation is the main prototype for nonlinear elliptic problems and forms a basis for various applications, such as injection moulding of plastics, nonlinear elasticity theory, and image processing. Its solutions, called pharmonic functions, have been studied in various contexts since the 1960s, first on Euclidean spaces and later on Riemannian manifolds, graphs, and Heisenberg groups. Nonlinear potential theory of pharmonic functions on metric spaces has been developing since the 1990s and generalizes and unites these earlier theories. This monograph gives a unified treatment of the subject and covers most of the available results in the field, so far scattered over a large number of research papers. The aim is to serve both as an introduction to the area for interested readers and as a reference text for active researchers. The presentation is rather self contained, but it is assumed that readers know measure theory and functional analysis. The first half of the book deals with Sobolev type spaces, socalled Newtonian spaces, based on upper gradients on general metric spaces. In the second half, these spaces are used to study pharmonic functions on metric spaces, and a nonlinear potential theory is developed under some additional, but natural, assumptions on the underlying metric space. Each chapter contains historical notes with relevant references, and an extensive index is provided at the end of the book. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. Readership Graduate students and researchers interested in measure theory and functional analysis. Table of Contents  Newtonian spaces
 Minimal \(p\)weak upper gradients
 Doubling measures
 Poincaré inequalities
 Properties of Newtonian functions
 Capacities
 Superminimizers
 Interior regularity
 Superharmonic functions
 The Dirichlet problem for \(p\)harmonic functions
 Boundary regularity
 Removable singularities
 Irregular boundary points
 Regular sets and applications thereof
 Appendices
 Bibliography
 Index
