The regularity problem for the Laplace equation in rough domains

Abstract

Let $\Omega \subset \mathbb{R}^{n+1}$, $n\geq 2$, be a bounded open and connected set satisfying the corkscrew condition with uniformly $n$-rectifiable boundary. In this paper we study the connection 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 $1/p+1/p'=1$. Our main result shows that $(D_{p'})$ is solvable if and only if so is $(R_{p})$. Under additional geometric assumptions (two-sided local John condition or weak Poincar\'e inequality on the boundary), we prove that $(D_{p'}) \Rightarrow (\tilde R_{p})$. In particular, we deduce that in bounded 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 extend the results to unbounded domains with compact boundary and show that in two-sided corkscrew domains with $n$-Ahlfors-David regular boundaries the single layer potential operator is invertible from $L^p(\partial \Omega)$ to the inhomogeneous Sobolev space $ W^{1,p}(\partial \Omega)$. Finally, we 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 $p \in [1, \infty)$.