Fractional $p$-Laplacians via Neumann problems in unbounded metric measure spaces

TL;DR

This paper establishes well-posedness, regularity, and Harnack inequalities for solutions to fractional p-Laplace equations in unbounded metric measure spaces, extending previous methods without requiring Poincaré inequalities.

Contribution

It introduces a novel approach using Neumann problems on hyperbolic fillings to analyze fractional p-Laplace equations in general metric spaces.

Findings

Proved well-posedness of fractional p-Laplace equations in unbounded spaces.

Established sharp regularity and Harnack inequalities for solutions.

Extended techniques to spaces lacking Poincaré inequalities.

Abstract

We prove well-posedness, Harnack inequality and sharp regularity of solutions to a fractional $p$-Laplace non-homogeneous equation $(-\Delta_p)^su =f$, with $0<s<1$, $1<p<\infty$, for data $f$ satisfying a weighted $L^{p'}$ condition in a doubling metric measure space $(Z,d_Z,\nu)$ that is possibly unbounded. Our approach is inspired by the work of Caffarelli and Silvestre \cite{CS} (see also Mol{\v{c}}anov and Ostrovski{\u{i}} \cite{MO}), and extends the techniques developed in \cite{CKKSS}, where the bounded case is studied. Unlike in \cite{EbGKSS}, we do not assume that $Z$ supports a Poincar\'e inequality. The proof is based on the well-posedness of the Neumann problem on a Gromov hyperbolic space $(X,d_X, \mu)$ that arises as an hyperbolic filling of $Z$.