Schauder estimates for elliptic equations degenerating on lower dimensional manifolds

Abstract

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 $$ -\mathrm{div}(|y|^aA(x,y)\nabla u)=|y|^af+\mathrm{div}(|y|^aF)\qquad\mathrm{in \ } B_1\subset\mathbb R^d, $$ where $z=(x,y)\in\mathbb R^{d-n}\times\mathbb R^n$, $2\leq n\leq d$ 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 $\Sigma_0=\{|y|=0\}$ of dimension $0\leq d-n\leq d-2$, having $$\lambda|y|^a|\xi|^2\leq |y|^aA(x,y)\xi\cdot\xi\leq\Lambda|y|^a|\xi|^2.$$ Whenever $a+n>0$, the weak solutions with a homogeneous conormal boundary condition at $\Sigma_0$ are provided to be $C^{0,\alpha}$ or even $C^{1,\alpha}$ regular up to $\Sigma_0$. 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.