Elliptic operators in rough sets, and the Dirichlet problem with boundary data in H\"older spaces

Abstract

In this paper we study the Dirichlet problem for real-valued second order divergence form elliptic operators with boundary data in H\"{o}lder spaces. Our context is that of open sets $\Omega \subset \mathbb{R}^{n+1}$, $n \ge 2$, satisfying the capacity density condition, without any further topological assumptions. Our main result states that if $\Omega$ is either bounded, or unbounded with unbounded boundary, then the corresponding Dirichlet boundary value problem is well-posed; when $\Omega$ is unbounded with bounded boundary, we establish that solutions exist, but they fail to be unique in general. These results are optimal in the sense that solvability of the Dirichlet problem in H\"{o}lder spaces is shown to imply the capacity density condition. As a consequence of the main result, we present a characterization of the H\"{o}lder spaces in terms of the boundary traces of solutions, and obtain well-posedness of several related Dirichlet boundary value problems. All the results above are new even for 1-sided chord-arc domains, and can be extended to generalized H\"{o}lder spaces associated with a natural class of growth functions.