Lema de Hopf-Oleinik não homogêneo para operadores totalmente não lineares com termo hamiltoniano

Abstract

This work establishes a quantitative version of the inhomogeneous Hopf-Oleinik Lemma for supersolutions of fully nonlinear elliptic equations with Hamiltonian terms, and applies this result to obtain optimal regularity estimates for l-semiconvex supersolutions. Specifically, we analyze fully nonlinear equations of the form P_,Λ,W,`,< ( 2D, D, G) = 5 (G), where P_,Λ,W,`,< represents a class of Pucci-type extremal operators with gradient dependence and unbounded coefficients. Our results generalize classical estimates for fully nonlinear PDEs, highlighting the construction of barriers and the strong version of the Harnack inequality, as well as providing fundamental tools and ideas to deal with gradient-dependent structures under different growth regimes. It is worth noting that the Hopf-Oleinik Lemma established here remains valid even in the presence of unbounded nonlinearities, significantly strengthening previous results in this direction.