A priori regularity estimates for equations degenerating on nodal sets

We prove a priori and a posteriori Hölder bounds and Schauder $C^{1,\alpha }$ estimates for continuous solutions of degenerate elliptic equations with variable coefficients of the form $$\mathrm{d}\mathrm{i}\mathrm{v}\left(\mathrm{\mid }u{\mathrm{\mid }}^{a}A\mathrm{\nabla }w\right)=0\mathrm{i}\mathrm{n}\mathrm{\Omega }\subset {\mathbb{R}}^2,a\in \mathbb{R},$$ where the weight $u$ is itself a solution to an elliptic equation of the type $\mathrm{d}\mathrm{i}\mathrm{v}(A\mathrm{\nabla }u)=0$, with $A$ a Lipschitz-continuous, uniformly elliptic matrix. The function $u$ is allowed to have a nontrivial, possibly singular nodal set. The estimates are uniform with respect to $u$ within a class of normalized solutions having bounded Almgren frequency. In the special case $a=2$, our results apply to the ratio of two solutions to the same elliptic equation sharing a common zero set. Precisely, we prove higher-order boundary Harnack principles on nodal domains, via the derived Schauder estimates for the associated degenerate equations. The results are based upon a fine blow-up argument, a Liouville theorem, and quasiconformal maps.