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. Under suitable regularity assumptions for the matrix A and f (resp. F) we prove Holder continuity of solutions which are even in y, and possibly of their derivatives up to order two or more (Schauder estimates). In addition, we show stability of the C0,alpha and C1,alpha a priori bounds for approximating problems. Finally, we derive C0,alpha and C1,alpha bounds for inhomogenous Neumann boundary problems as well. Our method is based upon blow-up and appropriate Liouville type theorems.