A quantitative version of the Hopf–Oleinik lemma for a quasilinear non-uniformly elliptic operator

Abstract

This paper establishes a quantitative version of the Hopf–Oleinik lemma (HOL) for a quasilinear non-uniformly elliptic operator of the form ${\mathcal{L}}_{\mathrm{\infty }}u:=2{\mathrm{\Delta }}_{\mathrm{\infty }}u+\mathrm{\Delta }u$. One key point in the proof is the passage from non-uniformly elliptic operators to locally uniformly ones via a new, uniform, and, rescaled version of the gradient estimate obtained by Evans and Smart for solutions to a family of non-uniformly quasilinear elliptic operators.