The regularity problem for the Laplace equation in rough domains

Let $\Omega \subset \mathbb{R}^{n+1}$, $n\geq 1$, be a bounded open and connected set satisfying the corkscrew condition. Assume also that its boundary $\partial \Omega$ is uniformly $n$-rectifiable and its measure theoretic boundary agrees with its topological boundary up to a set of $n$-dimensional Hausdorff measure zero. In this paper we study the equivalence between the solvability of $(D_{p'})$, the Dirichlet problem for the Laplacian with boundary data in $L^{p'}(\partial \Omega)$, and $(R_{p})$ (resp. $(\tilde R_{p})$), the regularity problem for the Laplacian with boundary data in the Haj\l asz Sobolev space $W^{1,p}(\partial \Omega)$ (resp. $\tilde W^{1,p}(\partial \Omega)$, the usual Sobolev space in terms of the tangential derivative), where $p \in (1,2+\varepsilon)$ and $1/p+1/p'=1$. In particular, we show that if $(D_{p'})$ is solvable then so is $(R_{p})$, while in the opposite direction, solvability of $(\tilde R_{p})$ implies solvability of $(D_{s})$, for all $s>p'$. Under additional geometric assumptions (two-sided local John condition or weak Poincare inequality on the boundary), we show that $(D_{p'}) \Rightarrow (\tilde R_{p})$ and $(R_{p})\Rightarrow (D_{s})$, for all $s>p'$. In particular, our results show that in chord-arc domains (resp. two-sided chord-arc domains), there exists $p_0 \in (1,2+\varepsilon)$ so that $(R_{p_0})$ (resp. $(\tilde R_{p_0})$) is solvable. We also provide a counterexample of a chord-arc domain $\Omega_0 \subset \mathbb{R}^{n+1}$, $n \geq 3$, so that $(\tilde R_p)$ is not solvable for any $p \in [1, \infty)$.