Perron solutions and boundary regularity for nonlocal nonlinear Dirichlet problems

Abstract

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 solutions give rise to two different concepts of regularity for boundary points, namely Sobolev and Perron regularity. We show that these two notions are equivalent and we also provide several characterizations of regular boundary points. Along the way, we give a new definition of Perron solutions, which is applicable to arbitrary exterior Dirichlet data $g: \Omega^c \to [-\infty,\infty]$. We obtain resolutivity results for these Perron solutions, and show that the Sobolev and Perron solutions coincide for a large class of exterior Dirichlet data. This also implies invariance of the Perron solutions under perturbations on sets of zero fractional capacity. A uniqueness result for the Dirichlet problem is also obtained for the class of bounded solutions taking prescribed continuous exterior data quasieverywhere on the boundary.