Sharp H\"older regularity of weak solutions of the Neumann problem and applications to nonlocal PDE in metric measure spaces

TL;DR

This paper establishes sharp global H"older regularity for weak solutions to a p-Laplacian type PDE with measure data in metric measure spaces, extending previous results and applying to nonlocal PDEs.

Contribution

It proves optimal H"older regularity for solutions of nonlinear PDEs with measure data in general metric measure spaces, improving prior estimates and connecting to Euclidean sharp results.

Findings

Proves global H"older regularity for weak solutions in metric measure spaces.

Extends regularity results to nonlocal PDEs in doubling metric spaces.

Achieves H"older exponent matching Euclidean sharp results.

Abstract

We prove global H\"older regularity result for weak solutions $u\in N^{1,p}(\Omega, \mu)$ to a PDE of $p$-Laplacian type with a measure as non-homogeneous term: \[ -\text{div}\!\left( |\nabla u|^{p-2}\nabla u \right)=\overline\nu, \] where $1<p<\infty$ and $\overline\nu \in (N^{1,p}(\Omega,\mu))^*$ is a signed Radon measure supported in $\overline \Omega$. Here, $\Omega$ is a John domain in a metric measure space satisfying a doubling condition and a $p$-Poincar\'e inequality, and $\nabla u$ is the Cheeger gradient. The regularity results obtained in this paper improve on earlier estimates proved by the authors in \cite{CGKS} for the study of the Neumann problem, and have applications to the regularity of solutions of nonlocal PDE in doubling metric spaces. Moreover, the obtained H\"older exponent matches with the known sharp result in the Euclidean case \cite{CSt,BLS,BT}.