A Quantitative Hopf-Oleinik Lemma for Degenerate Fully Nonlinear Operators and Applications to Free Boundary Problems

We prove a quantitative inhomogeneous Hopf-Oleinik lemma for viscosity solutions of |∇ u|αF(D²u)=f | ∇ u | α F ( D 2 u ) = f and, more generally, for viscosity supersolutions of |∇ u|α\,M⁻λ ,Λ(D²u)≤ f | ∇ u | α M λ , Λ - ( D 2 u ) ≤ f . The result yields linear boundary growth with universal constants depending only on the structural data. 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 ε ε –uniform Lipschitz bounds for a one–phase flame propagation model.