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

TL;DR

This paper investigates the well-posedness and regularity of solutions to nonlinear nonlocal Dirichlet boundary value problems for fractional p-Laplace-type equations in complex metric measure spaces with boundary measures.

Contribution

It establishes existence, uniqueness, and stability of solutions, along with interior regularity and boundary properties in a general metric space setting, extending classical PDE results.

Findings

Proved existence and uniqueness of solutions in the metric measure space setting.

Established interior regularity for solutions with sufficiently integrable data.

Verified boundary properties like the Kellogg-type theorem for solutions.

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.