Gradient estimates and hopf-Oleinik lemma for quasilinear non-uniformly elliptic operators and applications

Abstract

In this project we plan to explore the basic and classical ingredients such as Hopf-Ole\u{\i}nik Lemma and interior and up to the boundary gradient estimates for solutions to non-uniformly elliptic equations of quasilinear type that violates the $\Delta_2$ Lieberman's condition introduced in the 90s, allowing the ellipticity to blow up. The typical operator under this condition is given by $\mathcal{L}u = div(2e^{|\nabla u|^2}) = 2e^{|\nabla u|^2}(2\Delta_{\infty}u + \Delta u) $. Our method is a geometric one based on the development of suitable barriers for these operators. This also allows us to study viscosity solutions of Bernoulli-type Free Boundary Problems governed by this type of operator. (AU)