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-Poincaré 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 \(\mathcal{E}_{\theta}(u,u) \leq \mathcal{E}_{\theta}(h,h)\) 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 Hölder continuous on \(\Omega\), and satisfies a non-local maximum and strong maximum principle. Part of the results in this paper extends the work of Caffarelli and Silvestre in the Euclidean setting and Franchi and Ferrari in Carnot groups.