The Dirichlet problem on lower dimensional boundaries: Schauder estimates via perforated domains

In this paper, we investigate the Dirichlet problem on lower dimensional manifolds for a class of weighted elliptic equations with coefficients that are singular on such sets. Specifically, we study the problem $$\{\begin{array}{l}-\mathrm{d}\mathrm{i}\mathrm{v}(\mathrm{\mid }y{\mathrm{\mid }}^{a}A(x,y)\mathrm{\nabla }u)=\mathrm{\mid }y{\mathrm{\mid }}^{a}f+\mathrm{d}\mathrm{i}\mathrm{v}(\mathrm{\mid }y{\mathrm{\mid }}^{a}F),\\ u=\psi ,\text{ on }{\mathrm{\Sigma }}_0,\end{array}$$ where $(x,y)\in {\mathbb{R}}^{d-n}\times {\mathbb{R}}^n$, $2\le n\le d$, $a+n\in (0,2)$, and ${\mathrm{\Sigma }}_0=\{\mathrm{\mid }y\mathrm{\mid }=0\}$ is the lower dimensional manifold where the equation loses uniform ellipticity. Our primary objective is to establish $C^{0,\alpha }$ and $C^{1,\alpha }$ regularity estimates up to ${\mathrm{\Sigma }}_0$, under suitable assumptions on the coefficients and the data. Our approach combines perforated domain approximations, Liouville-type theorems and a fine blow-up argument.