Schauder estimates for ratios of elliptic solutions under zero set inclusion

Let $u,v$ be two nontrivial functions that solve $$\mathrm{d}\mathrm{i}\mathrm{v}(A\mathrm{\nabla }u)=\mathrm{d}\mathrm{i}\mathrm{v}(A\mathrm{\nabla }v)=0\mathrm{i}\mathrm{n}{B}_r\subset {\mathbb{R}}^n,n\ge 2,r>0,$$ with $A$ symmetric, uniformly elliptic and Lipschitz continuous, and assume the zero set inclusion $Z(u)\subseteq Z(v)$. We prove that the ratio $w=v\mathrm{/}u$ continuously extends, together with its gradient, across $Z(u)=\{u=0\}$. In particular, $w\in C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,\alpha }(B_r)$ for any $\alpha \in (0,1)$. Furthermore, given any integer $k\ge 2$ and $\alpha \in (0,1)$, if $A\in C^{k-1,\alpha }(B_r)$, then $w\in C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{k,\alpha }(B_r)$. 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.