Submitted Paper

Inserted: 17 feb 2026
Last Updated: 18 feb 2026

Year: 2025

Abstract:

We prove a quantitative inhomogeneous Hopf-Oleinik lemma for viscosity solutions of $ \vert \nabla u \vert ^{\alpha} F(D^2 u)=f$ and, more generally, for viscosity supersolutions of $ \vert\nabla u \vert^{\alpha} \mathcal{M}_{\lambda,\Lambda}^- (D^2 u) \leq 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 -uniform Lipschitz bounds for a one-phase flame propagation model.