Related papers: The Dirichlet problem for p-harmonic functions wit…

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 considered here is that of metric spaces, where…

The Perron method for solving the Dirichlet problem for $p$-harmonic functions is extended to unbounded open sets in the setting of a complete metric space with a doubling measure supporting a $p$-Poincar\'e inequality, $1<p<\infty$. The…

In this paper we study the Perron method for solving the p-harmonic Dirichlet problem on the topologist's comb. For functions which are bounded and continuous at the accessible points, we obtain invariance of the Perron solutions under…

We initiate the study of fine $p$-(super)minimizers, associated with $p$-harmonic functions, on finely open sets in metric spaces, where $1 < p < \infty$. After having developed their basic theory, we obtain the $p$-fine continuity of the…

We study the obstacle problem for unbounded sets in a proper metric measure space supporting a (p,p)-Poincare inequality. We prove that there exists a unique solution. We also prove that if the measure is doubling and the obstacle is…

Given a bounded finely open set $V$ and a function $f$ on the fine boundary of $V$, we introduce four types of upper Perron solutions to the nonlinear Dirichlet problem for $p$-energy minimizers, $1<p<\infty$, with $f$ as boundary data.…

In this two papers we deal with the relative homotopy Dirichlet problem for p-harmonic maps from compact manifolds with boundary to manifolds of non-positive sectional curvature. Notably, we give a complete solution to the problem in case…

For nonlinear operators of fractional $p$-Laplace type, we consider two types of solutions to the nonlocal Dirichlet problem: Sobolev solutions based on fractional Sobolev spaces and Perron solutions based on superharmonic functions. These…

Within the setting of metric spaces equipped with a doubling measure and supporting a $p$-Poincar\'e inequality, establishing existence of solutions to Dirichlet problem in a bounded domain in such a metric space is accomplished via direct…

We use sphericalization to study the Dirichlet problem, Perron solutions and boundary regularity for p-harmonic functions on unbounded sets in Ahlfors regular metric spaces. Boundary regularity for the point at infinity is given special…

We study the Dirichlet problem for non-homogeneous equations involving the fractional $p$-Laplacian. We apply Perron's method and prove Wiener's resolutivity theorem.

In this paper we solve the Dirichlet problems for different classes of plurisubharmonic functions on compact sets in $\mathbb C^n$ including continuous, pluriharmonic and maximal functions.

For any compact set $K\subset \mathbb{R}^n$ we develop the theory of Jensen measures and subharmonic peak points, which form the set $\mathcal{O}_K$, to study the Dirichlet problem on $K$. Initially we consider the space $h(K)$ of functions…

Let $p$ be a real number greater than one and let $X$ be a locally compact, noncompact metric measure space that satisfies certain conditions. The $p$-Royden and $p$-harmonic boundaries of $X$ are constructed by using the $p$-Royden algebra…

We characterize those compact sets for which the Dirichlet problem has a solution within the class of continuous $m$-subharmonic functions defined on a compact set, and then within the class of $m$-harmonic functions.

Let $X$ be a noncomplete metric space satisfying the usual (local) assumptions of a doubling property and a Poincar\'e inequality. We study extensions of Newtonian Sobolev functions to the completion $\widehat{X}$ of $X$ and use them to…

We study two notions of Dirichlet problem associated with BV energy minimizers (also called functions of least gradient) in bounded domains in metric measure spaces whose measure is doubling and supports a $(1,1)$-Poincar\'e inequality.…

The aim of this paper is to give a new proof of the complete characterization of measures for which there exist a solution of the Dirichlet problem for the complex Monge-Ampere operator in the set of plurisubharmonic functions with finite…

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain satisfying the uniform exterior cone condition. We establish existence and uniqueness of continuous solutions of the Dirichlet Problem associated to certain intrinsic nonlinear mean value…

The identification between the complex plane and the Riemann sphere preserves holomorphic and harmonic functions and is a classical tool. In this paper we consider a similar mapping from an unbounded metric space $X$ to a bounded space and…