  We consider the mixed Dirichlet-conormal problem on irregular domains in
$\mathbb{R}^d$. Two types of regularity results will be discussed: the
$W^{1,p}$ regularity and a non-tangential maximal function estimate. The domain
is assumed to be Reifenberg-flat, and the interfacial boundary is either
Reifenberg-flat of co-dimension $2$ or is locally sufficiently close to a
Lipschitz function of $m$ variables, where $m=1,\ldots,d-2$. For the
non-tangential maximal function estimate, we also require the domain to be
Lipschitz.
