Neumann problems for $p$-harmonic functions, and induced nonlocal operators in metric measure spaces

Abstract

Following ideas of Caffarelli and Silvestre in~\cite{CS}, and using recent progress in hyperbolic fillings, we define fractional $p$-Laplacians $(-\Delta_p)^\theta$ with $0<\theta<1$ on any compact, doubling metric measure space $(Z,d,\nu)$, and prove existence, regularity and stability for the non-homogenous non-local equation $(-\Delta_p)^\theta u =f.$ These results, in turn, rest on the new existence, global H\"older regularity and stability theorems that we prove for the Neumann problem for $p$-Laplacians $\Delta_p$, $1<p<\infty$, in bounded domains of measure metric spaces endowed with a doubling measure that supports a Poincar\'e inequality. Our work also extends beyond the compact setting, and includes as special cases much of the previous results by other authors in the Euclidean, Riemannian and Carnot group settings. Unlike other recent contributions in the metric measure spaces context, our work does not rely on the assumption that $(Z,d,\nu)$ supports a Poincar\'e inequality.