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 $$-\mathrm{div}(|y{|}^{a}A(x,y)\nabla u)=|y{|}^{a}f+\mathrm{div}(|y{|}^{a}F)\mathrm{in}{B}_1\subset {\mathbb{R}}^d,$$ 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 $\hat{I}{\pounds }_0=\{|y|=0\}$ of dimension $0\le d-n\le d-2$, having $$\hat{I}»|y{|}^a|\hat{I}¾{|}^2\le |y{|}^{a}A(x,y)\hat{I}¾\cdot \hat{I}¾\le \hat{I}›|y{|}^a|\hat{I}¾{|}^2.$$ Whenever $a+n>0$, the weak solutions with a homogeneous conormal boundary condition at $\hat{I}{\pounds }_0$ are provided to be $C^{0,\hat{I}\pm }$ or even $C^{1,\hat{I}\pm }$ regular up to $\hat{I}{\pounds }_0$. 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.

80 pages, 2 figures. The original version of the work has been split into the present paper and another titled "Remarks on elliptic equations degenerating on lower dimensional manifolds"