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

We consider a class of equations in divergence form with a singular/degenerate weight $$-\mathrm{d}\mathrm{i}\mathrm{v}(\mathrm{\mid }y{\mathrm{\mid }}^{a}A(x,y)\mathrm{\nabla }u)=\mathrm{\mid }y{\mathrm{\mid }}^{a}f(x,y)\text{or div}(\mathrm{\mid }y{\mathrm{\mid }}^{a}F(x,y))\mathrm{.}$$ Under suitable regularity assumptions for the matrix $A$ and $f$ (resp. $F$) we prove H"older 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 $$-\mathrm{d}\mathrm{i}\mathrm{v}(({\epsilon }^2+y^2{)}^{a\mathrm{/}2}A(x,y)\mathrm{\nabla }u)=({\epsilon }^2+y^2{)}^{a\mathrm{/}2}f(x,y)\text{or div}(({\epsilon }^2+y^2{)}^{a\mathrm{/}2}F(x,y))$$ 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.