*preprint*

**Inserted:** 11 apr 2024

**Last Updated:** 11 apr 2024

**Year:** 2024

**Abstract:**

We prove $\textit{a priori}$ and $\textit{a posteriori}$ Holder bounds and
Schauder $C^{1,\alpha}$ estimates for continuous solutions to singular-degenerate equations with variable coefficients of type
\[
\mathrm{div}\left(

u

^a A\nabla w\right)=0\qquad\mathrm{in \
}\Omega\subset\mathbb{R}^n,
\]
where the weight $u$ solves an elliptic equation
of type $\mathrm{div}\left(A\nabla u\right)=0$ with a Lipschitz-continuous and uniformly elliptic matrix $A$ and has a nontrivial, possibly singular, nodal
set. Such estimates are uniform with respect to $u$ in a class of normalized
solutions having bounded Almgren's frequency. More precisely, we provide
$\textit{a priori}$ Holder bounds in any space dimension, and Schauder
estimates when $n=2$. When $a=2$, the results apply to the ratios of two solutions to the same PDE sharing their zero sets. Then, one can infer higher
order boundary Harnack principles on nodal domains by applying the Schauder
estimates for solutions to the auxiliary degenerate equation. The results are
based upon a fine blow-up argument, Liouville theorems and quasiconformal maps.