Solvability of the Poisson-Dirichlet problem in domains with boundaries of mixed dimension

We extend several characterizations of the $L^p$-solvability of the homogeneous Dirichlet problem to (possibly degenerate) elliptic operators $L=-\text{div}(wA\mathrm{\nabla })$ defined on a large class of open sets $\mathrm{\Omega }$ in ${\mathbb{R}}^n$. This framework encompasses not only uniformly elliptic operators in Lipschitz domains, but also Caffarelli-Sylvestre-type operators, and boundaries $\mathrm{\partial }\mathrm{\Omega }$ that are not $(n-1)$-dimensional, for instance. We prove that, for $p\in (1,\mathrm{\infty })$, the $L^p$-solvability of the homogeneous Dirichlet problem is equivalent to the solvability of the Poisson-Dirichlet problem $Lu=wf-\text{div}(wF)$, \emph{i.e.} to the existence of a solution $u$ satisfying a $L^p$ non-tangential estimate whenever $f$ and $F$ belong to suitable weighted $L^p$ tent spaces. Furthermore, it is also equivalent to the solvability of the Poisson-Regularity problem for data in appropriate weighted $L^{p^{\mathrm{\prime }}}$ tent spaces, where estimates are obtained for $\mathrm{\nabla }u$ rather than $u$.