Liouville type theorems and regularity of solutions to degenerate or singular problems part II: odd solutions

Abstract

We consider a class of equations in divergence form with a singular/degenerate weight \[ -\mathrm{div}(|y|^a A(x,y)\nabla u)=|y|^a f(x,y)+\textrm{div}(|y|^aF(x,y))\;. \] Under suitable regularity assumptions for the matrix $A$, the forcing term $f$ and the field $F$, we prove Hölder continuity of solutions which are odd in $y\in\mathbb{R}$, and possibly of their derivatives. In addition, we show stability of the $C^{0,\alpha}$ and $C^{1,\alpha}$ a priori bounds for approximating problems in the form \[ -\mathrm{div}((\varepsilon^2+y^2)^{a/2} A(x,y)\nabla u)=(\varepsilon^2+y^2)^{a/2} f(x,y)+\textrm{div}((\varepsilon^2+y^2)^{a/2}F(x,y)) \] as $\varepsilon\to 0$. Our method is based upon blow-up and appropriate Liouville type theorems.

Keywords

• degenerate and singular elliptic equations
• liouville type theorems
• blow-up
• fractional laplacian
• divergence form elliptic operator
• schauder estimates
• boundary harnack
• fermi coordinates