A priori regularity estimates for equations degenerating on nodal sets

We prove a priori and a posteriori H\"older bounds and Schauder $C^{1,\alpha}$ estimates for continuous solutions of degenerate elliptic equations with variable coefficients of the form $$\mathrm{div}\left(|u|^a A\nabla w\right)=0\qquad\mathrm{in \ }\Omega\subset\mathbb R^2,\quad a\in\mathbb R,$$ where the weight $u$ is itself a solution to an elliptic equation of the type $\mathrm{div}(A \nabla u) = 0$, 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.