We study the solutions of the Dirichlet problem on an open set $\Omega\subset \mathbb{R}^n$ for degenerate elliptic operators $L = -\mathrm{div}(\mathrm{dist}(., \partial\Omega)^{\alpha}\nabla)$, with $-1 < \alpha < 1$. This kind of operators appears naturally when the boundary of the domain has dimension $d < n − 1$, for example. We say that the regularity problem associated to $L$ is solvable when there exists a solution whose gradient satisfies a suitable non-tangential bound. Compared to the case of the Laplacian, the definition of this notion must be modified to account for the degeneracy of the operator. I will present partial results toward establishing the solvability of the regularity problem in the case $\Omega = \mathbb{R}_{+}^n$.