Regularity of Solutions to the Fractional Cheeger-Laplacian on Domains in Metric Spaces of Bounded Geometry

TL;DR

This paper investigates the existence, uniqueness, and regularity of solutions to fractional Dirichlet problems in metric measure spaces with bounded geometry, extending classical Euclidean and Carnot group results.

Contribution

It establishes new regularity and maximum principles for fractional Laplacian problems in general metric spaces, broadening the scope of prior Euclidean and Carnot group studies.

Findings

Solutions exist and are unique for the fractional Dirichlet problem.

Solutions are locally Hölder continuous within the domain.

Maximum principles hold for these solutions.

Abstract

We study existence, uniqueness, and regularity properties of the Dirichlet problem related to fractional Dirichlet energy minimizers in a complete doubling metric measure space $(X,d_X,\mu_X)$ satisfying a $2$-Poincar\'e inequality. Given a bounded domain $\Omega\subset X$ with $\mu_X(X\setminus\Omega)>0$, and a function $f$ in the Besov class $B^\theta_{2,2}(X)\cap L^2(X)$, we study the problem of finding a function $u\in B^\theta_{2,2}(X)$ such that $u=f$ in $X\setminus\Omega$ and $\mathcal{E}_\theta(u,u)\le \mathcal{E}_\theta(h,h)$ whenever $h\in B^\theta_{2,2}(X)$ with $h=f$ in $X\setminus\Omega$. We show that such a solution always exists and that this solution is unique. We also show that the solution is locally H\"older continuous on $\Omega$, and satisfies a non-local maximum and strong maximum principle. Part of the results in this paper extend the work of Caffarelli and Silvestre in the Euclidean setting and Franchi and Ferrari in Carnot groups.