Published Paper

Inserted: 24 mar 2026
Last Updated: 24 mar 2026

Journal: Transactions of the American Mathematical Society
Year: 2026
Doi: 10.1090/tran/9759

Abstract:

We prove \emph{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.