A quantitative Hopf-Oleinik lemma for degenerate fully nonlinear operators and applications to free boundary problems

TL;DR

This paper establishes a quantitative Hopf-Oleinik lemma for degenerate fully nonlinear operators, providing boundary growth estimates and applications to free boundary problems and flame propagation models.

Contribution

It introduces a new quantitative Hopf-Oleinik lemma for degenerate operators and applies it to free boundary problems and flame propagation models.

Findings

Linear boundary growth with universal constants

Lipschitz regularity for free boundary solutions

Uniform Lipschitz bounds for flame propagation

Abstract

We prove a quantitative inhomogeneous Hopf-Oleinik lemma for viscosity solutions of $$|\nabla u|^{\alpha}F(D^{2}u)=f $$ and, more generally, for viscosity supersolutions of $|\nabla u|^{\alpha}\,{M}^-_{\lambda,\Lambda}(D^{2}u)\le f$. The result yields linear boundary growth with universal constants depending only on the structural data. We also exhibit a counterexample showing that the Hopf lemma fails for equations that act only in the large-gradient regime (in the sense of Imbert and Silvestre), thereby delineating the scope of our theorem. As applications, we obtain Lipschitz regularity for viscosity solutions of one-phase Bernoulli free boundary problems driven by these degenerate fully nonlinear operators and derive $\varepsilon$-uniform Lipschitz bounds for a one-phase flame propagation model.