# Nonlinear Potential Theory on Metric Spaces

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*Anders Björn; Jana Björn*

A publication of the European Mathematical Society

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 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, 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.

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.