One-sided Rellich inequalities, Regularity problem and uniform rectifiability

Abstract: Let $\Omega\subset \mathbb R^{n+1}$, $n\geq1$, be a bounded open set satisfying the interior corkscrew condition with a uniformly $n$-rectifiable boundary but without any connectivity assumptions. We establish the estimate $$ \Vert \partial_\nu u_f \Vert_{M} \lesssim \Vert \nabla_H f \Vert_{L^1(\partial\Omega)}, \quad \mbox{for all $f\in\operatorname{Lip}(\partial\Omega)$} $$ where $u_f$ is the solution to the Dirichlet problem with boundary data $f$, $\partial_\nu u_f$ is the normal derivative of $u_f$ at the boundary in the weak sense, $\Vert \cdot \Vert_{M}$ denotes the total variation norm and $\nabla_H f$ is the Haj{\l}asz-Sobolev gradient of $f$. Conversely, if $\Omega\subset \mathbb R^{n+1}$ is a corkscrew domain with $n$-Ahlfors regular boundary and the previous inequality holds for solutions to the Dirichlet problem on $\Omega$, then $\partial\Omega$ must satisfy the weak-no-boxes condition introduced by David and Semmes. Hence, in the planar case, the one-sided Rellich inequality characterizes the uniform rectifiability of $\partial\Omega$. We also show solvability of the regularity problem in weak $L^1$ for bounded corkscrew domains with a uniformly $n$-rectifiable boundary, that is $$\Vert N(\nabla u_f) \Vert_{L^{1,\infty}(\partial\Omega)} \lesssim \Vert \nabla_H f\Vert_{L^1(\partial\Omega)},\quad \mbox{for all $f\in\operatorname{Lip}(\partial\Omega)$}$$ where $N$ is the nontangential maximal operator. As an application of our results, we prove that for general elliptic operators, the solvability of the Dirichlet problem does not imply the solvability of the regularity problem.