Anders Björn

Nonlinear Potential Theory on Metric Spaces

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 p-harmonic 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 p-harmonic 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 an interested reader and as a reference text for an active researcher. The presentation is rather self-contained, but the reader is assumed to know measure theory and functional analysis.The first half of the book deals with Sobolev type spaces, so-called Newtonian spaces, based on upper gradients on general metric spaces. In the second half, these spaces are used to study p-harmonic 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.

The Perron method for p-harmonic functions in metric spaces

We use the Perron method to construct and study solutions of the Dirichlet problem for p-harmonic functions in proper metric measure spaces endowed with a doubling Borel measure supporting a weak (1,q)-Poincare inequality (for some 1⩽q<p). The upper and lower Perron solutions are constructed for functions defined on the boundary of a bounded domain and it is shown that these solutions are p-harmonic in the domain. It is also shown that Newtonian (Sobolev) functions and continuous functions are resolutive, i.e. that their upper and lower Perron solutions coincide, and that their Perron solutions are invariant under perturbations of the function on a set of capacity zero. We further study the problem of resolutivity and invariance under perturbations for semicontinuous functions. We also characterize removable sets for bounded p-(super)harmonic functions.

A weak Kellogg property for quasiminimizers

The Kellogg property says that the set of irregular boundary points has capacity zero, i.e. given a bounded open set

Obstacle and Dirichlet problems on arbitrary nonopen sets in metric spaces, and fine topology

\Omega

Sobolev extensions of Hölder continuous and characteristic functions on metric spaces

there is a set

The Dirichlet problem for p-harmonic functions with respect to the Mazurkiewicz boundary, and new capacities

E \subset \partial\Omega

Prime ends for domains in metric spaces

with capacity zero such that for all

Boundary regularity for degenerate and singular parabolic equations

p

Factors of generalized Fermat numbers

-harmonic functions

The Cartan, Choquet and Kellogg properties for the fine topology on metric spaces

u