A priori H\"older estimates for equations degenerating on nodal sets

TL;DR

This paper establishes uniform H"older continuity estimates for solutions to certain degenerate elliptic equations with variable coefficients, even when solutions have complex nodal sets, and proves a boundary Harnack principle for their quotients.

Contribution

It introduces a priori H"older bounds for solutions with singular nodal sets in degenerate elliptic equations, extending regularity theory to more general degenerate contexts.

Findings

H"older bounds are uniform for solutions with bounded Almgren frequency.

Boundary Harnack principle holds for ratios of solutions vanishing on the same set.

Weighted Sobolev space analysis underpins the regularity results.

Abstract

We prove a priori H\"older bounds for continuous solutions to degenerate equations with variable coefficients of type $$ \mathrm{div}\left(u^2 A\nabla w\right)=0\quad\mathrm{in \ }\Omega\subset\mathbb R^n,\qquad \mbox{with}\qquad \mathrm{div}\left(A\nabla u\right)=0, $$ 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.