Sharp Hölder regularity of weak solutions of the Neumann problem and applications to nonlocal PDE in metric measure spaces

We prove global Hölder regularity result for weak solutions $u\in N^{1,p}(\mathrm{\Omega },\mu )$ to a PDE of $p$-Laplacian type with a measure as non-homogeneous term: $$-\text{div}\text{\hspace{0.17em}⁣}\left(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }u\right)=\bar{\nu },$$ where $1<p<\mathrm{\infty }$ and $\bar{\nu }\in (N^{1,p}(\mathrm{\Omega },\mu ){)}^{\ast }$ is a signed Radon measure supported in $\bar{\mathrm{\Omega }}$. Here, $\mathrm{\Omega }$ is a John domain in a metric measure space satisfying a doubling condition and a $p$-Poincaré inequality, and $\mathrm{\nabla }u$ 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}.