Sharp Hölder regularity of weak solutions of the Neumann problem and applications to nonlocal PDE in metric measure spaces
Luca Capogna, Ryan Gibara, Riikka Korte, Nageswari Shanmugalingam
TL;DR
This paper establishes sharp Hölder regularity for weak solutions to Neumann problems for the $p$-Laplacian on bounded John domains in doubling metric measure spaces, with measure data on the boundary. Using the Cheeger gradient, trace theory to Besov spaces, Adams–Maz'ya type potential estimates, and Morrey-space data, it proves a Morrey-type gradient bound $|∇u|∈M^{p,(1+α)/(1-p)}$ for $α<0$, yielding $u$ in $C^{(p+α)/(p-1)}$ up to the boundary. The authors then extend these results to nonlocal fractional $p$-Laplacian problems on metric spaces, showing that Besov-energy minimizers regularize with the same sharp Hölder exponents dictated by Morrey data, and connect these findings to the Euclidean sharp results of Caffarelli–Stinga, BLS, and BT. The work thus provides a unified, sharp regularity theory for local and nonlocal PDEs in general doubling metric measure spaces and clarifies how boundary codimension and Morrey decay govern Hölder continuity.
Abstract
We prove global Hölder regularity result for weak solutions $u\in N^{1,p}(Ω, μ)$ to a PDE of $p$-Laplacian type with a measure as non-homogeneous term: \[ -\text{div}\!\left( |\nabla u|^{p-2}\nabla u \right)=\overlineν, \] where $1<p<\infty$ and $\overlineν\in (N^{1,p}(Ω,μ))^*$ is a signed Radon measure supported in $\overline Ω$. Here, $Ω$ is a John domain in a metric measure space satisfying a doubling condition and a $p$-Poincaré 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 Hölder exponent matches with the known sharp result in the Euclidean case \cite{CSt,BLS,BT}.