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 −div(|y|aA(x,y)∇u)=|y|af(x,y) or div(|y|aF(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∈R, and possibly of their derivatives up to order two or more (Schauder estimates). In addition, we show stability of the C0,α and C1,α a priori bounds for approximating problems in the form −div((ε2+y2)a/2A(x,y)∇u)=(ε2+y2)a/2f(x,y) or div((ε2+y2)a/2F(x,y))as ε→0. Finally, we derive C0,α and C1,α bounds for inhomogenous Neumann boundary problems as well. Our method is based upon blow-up and appropriate Liouville type theorems.

Topics & Concepts