A priori Hölder estimates for equations degenerating on nodal sets

We prove a priori Hölder bounds for continuous solutions to degenerate equations with variable coefficients of type $$\mathrm{d}\mathrm{i}\mathrm{v}\left(u^{2}A\mathrm{\nabla }w\right)=0\mathrm{i}\mathrm{n}\mathrm{\Omega }\subset {\mathbb{R}}^n,with\mathrm{d}\mathrm{i}\mathrm{v}\left(A\mathrm{\nabla }u\right)=0,$$ where $A$ is a Lipschitz continuous, uniformly elliptic matrix (possibly $u$ has non-trivial singular nodal set). Such estimates are uniform with respect to $u$ in a class of normalized solutions that have a bounded Almgren frequency. As a consequence, a boundary Harnack principle holds for the quotient of two solutions vanishing on a common set. This analysis relies on a detailed study of the associated weighted Sobolev spaces, including integrability of the weight, capacitary properties of the nodal set, and uniform Sobolev inequalities yielding local boundedness of solutions.