Well-posedness of Dirichlet boundary value problems for reflected fractional $p$-Laplace-type inhomogeneous equations in compact doubling metric measure spaces

Abstract

In this paper we consider the setting of a locally compact, non-complete metric measure space $(Z,d,\nu)$ equipped with a doubling measure $\nu$, under the condition that the boundary $\partial Z:=\overline{Z}\setminus Z$ (obtained by considering the completion of $Z$) supports a Radon measure $\pi$ which is in a $\sigma$-codimensional relationship to $\nu$ for some $\sigma>0$. We explore existence, uniqueness, comparison property, and stability properties of solutions to inhomogeneous Dirichlet problems associated with certain nonlinear nonlocal operators on $Z$. We also establish interior regularity of solutions when the inhomogeneity data is in an $L^q$-class for sufficiently large $q>1$, and verify a Kellogg-type property when the inhomogeneity data vanishes and the Dirichlet data is continuous.