Liouville type theorems and regularity of solutions to degenerate or singular problems part I: even solutions

Abstract

We consider a class of equations in divergence form with a singular/degenerate weight $$-\mathrm{div}(|y{|}^{a}A(x,y)\nabla u)=|y{|}^{a}f(x,y)\text{or}\text{div}(|y{|}^{a}F(x,y)).$$ Under suitable regularity assumptions for the matrix $A$ and $f$ (resp. $F$) we prove Hölder continuity of solutions which are even in $y\in \mathbb{R}$, and possibly of their derivatives up to order two or more (Schauder estimates). In addition, we show stability of the $C^{0,\alpha }$ and $C^{1,\alpha }$ a priori bounds for approximating problems in the form $$-\mathrm{div}(({\epsilon }^2+y^2{)}^{a/2}A(x,y)\nabla u)=({\epsilon }^2+y^2{)}^{a/2}f(x,y)\text{or}\text{div}(({\epsilon }^2+y^2{)}^{a/2}F(x,y))$$ as $\epsilon \to 0$. Finally, we derive $C^{0,\alpha }$ and $C^{1,\alpha }$ bounds for inhomogenous Neumann boundary problems as well. Our method is based upon blow-up and appropriate Liouville type theorems.

47 pages