  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 $p \in [1, \infty)$.
