Jana 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.

Fat sets and pointwise boundary estimates forp-harmonic functions in metric spaces

We extend a result of John Lewis [L] by showing that if a doubling metric measure space supports a (1,q0)-Poincaré inequality for some 1<q0<p, then every uniformlyp-fat set is uniformlyq-fat for someq<p. This bootstrap result implies the Hardy inequality for Newtonian functions with zero boundary values for domains whose complements are uniformly fat. While proving this result, we also characterize positive Radon measures in the dual of the Newtonian space using the Wolff potential and obtain an estimate for the oscillation ofp-harmonic functions andp-energy minimizers near a boundary point.

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.

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

We study the double obstacle problem for p-harmonic functions on arbitrary bounded nonopen sets E in quite general metric spaces. The Dirichlet and single obstacle problems are included as special cases. We obtain the Adams criterion for the solubility of the single obstacle problem and establish connections with fine potential theory. We also study when the minimal p-weak upper gradient of a function remains minimal when restricted to a nonopen subset. Many of the results are new even for open E (apart from those which are trivial in this case) and also on Rn.

Sobolev extensions of Hölder continuous and characteristic functions on metric spaces

We study when characteristic and Holder continuous functions are traces of Sobolev functions on doubling metric measure spaces. We provide analytic and geometric conditions sufficient for extending characteristic and Holder continuous functions into globally defined Sobolev functions. ©Canadian Mathematical Society 2007.

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

In this paper we develop the Perron method for solving the Dirichlet problem for the analog of the p-Laplacian, i.e. for p-harmonic functions, with Mazurkiewicz boundary values. The setting conside ...

Prime ends for domains in metric spaces

In this paper we propose a new definition of prime ends for domains in metric spaces under rather general assumptions. We compare our prime ends to those of Caratheodory and Nakki. Modulus ends and ...

Boundary regularity for degenerate and singular parabolic equations

We characterise regular boundary points of the parabolic

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

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