Schauder estimates for elliptic equations degenerating on lower dimensional manifolds

In this paper we begin exploring a local regularity theory for elliptic equations having coefficients which are degenerate or singular on some lower dimensional manifold $$-\mathrm{d}\mathrm{i}\mathrm{v}(\mathrm{\mid }y{\mathrm{\mid }}^{a}A(x,y)\mathrm{\nabla }u)=\mathrm{\mid }y{\mathrm{\mid }}^{a}f+\mathrm{d}\mathrm{i}\mathrm{v}(\mathrm{\mid }y{\mathrm{\mid }}^{a}F)\mathrm{i}\mathrm{n}{B}_1\subset {\mathbb{R}}^d,$$ where $z=(x,y)\in {\mathbb{R}}^{d-n}\times {\mathbb{R}}^n$, $2\le n\le d$ are two integers and $a\in \mathbb{R}$. Such equations are a prototypical example of elliptic equations spoiling their uniform ellipticity on the (possibly very) thin characteristic manifold ${\mathrm{\Sigma }}_0=\{\mathrm{\mid }y\mathrm{\mid }=0\}$ of dimension $0\le d-n\le d-2$, having $$\lambda \mathrm{\mid }y{\mathrm{\mid }}^a\mathrm{\mid }\xi {\mathrm{\mid }}^2\le \mathrm{\mid }y{\mathrm{\mid }}^{a}A(x,y)\xi \cdot \xi \le \mathrm{\Lambda }\mathrm{\mid }y{\mathrm{\mid }}^a\mathrm{\mid }\xi {\mathrm{\mid }}^2\mathrm{.}$$ Whenever $a+n>0$, the weak solutions with a homogeneous conormal boundary condition at ${\mathrm{\Sigma }}_0$ are provided to be $C^{0,\alpha }$ or even $C^{1,\alpha }$ regular up to ${\mathrm{\Sigma }}_0$. Our approach relies on a regularization-approximation scheme which employs domain perforation, very fine blow-up procedures, and a new Liouville theorem in the perforated space. Our theory extends to the case of equations degenerating on suitably smooth curved manifolds.