Schauder estimates for ratios of elliptic solutions under zero set inclusion

Abstract

Let $u,v$ be two nontrivial functions that solve $$ \mathrm{div}(A\nabla u)=\mathrm{div}(A\nabla v)=0\qquad\mathrm{in \ }B_r\subset\mathbb R^n,\quad n\geq2,\quad r>0, $$ with $A$ symmetric, uniformly elliptic and Lipschitz continuous, and assume the zero set inclusion $Z(u)\subseteq Z(v)$. We prove that the ratio $w=v/u$ continuously extends, together with its gradient, across $Z(u)=\{u=0\}$. In particular, $w\in C^{1,α}_{\mathrm{loc}}(B_{r})$ for any $α\in(0,1)$. Furthermore, given any integer $k\geq2$ and $α\in(0,1)$, if $A\in C^{k-1,α}(B_r)$, then $w\in C^{k,α}_{\mathrm{loc}}(B_{r})$. The estimates are uniform in classes of coefficients $A$ with prescribed ellipticity and regularity bounds, and in normalized classes of denominators $u$ with uniformly bounded doubling constant.