### Abstract

Given a bounded domain $\Omega \subset \mathbb R^n$, one says that the $L^p$-regularity problem is solvable for the Laplace equation in $\Omega$ if, given any continuous function $f$ defined in $\partial \Omega$ and the harmonic extension $u$ of $f$ to $\Omega$, the non-tangential maximal function of the gradient of $u$ can be controlled in $L^p$ norm by the tangential derivative of $f$ in $\partial\Omega$. In my talk I will review a joint result with Mourgoglou where we proved the $L^p$-regularity for more general domains, and a more recent result (with the additional collaboration of Poggi) where we extend this result to elliptic PDE’s in divergence form.