Weak Harnack inequalities for eigenvalues and constant rank theorems

TL;DR

This paper establishes a weak Harnack inequality for the eigenvalues of the Hessian of convex solutions to certain nonlinear elliptic equations, providing a quantitative enhancement of the constant rank theorem.

Contribution

It introduces a weak Harnack inequality for eigenvalues of the Hessian, advancing the understanding of convex solutions under Bian-Guan structure conditions.

Findings

Proved a weak Harnack inequality for eigenvalues.

Quantitative version of the constant rank theorem.

Applicable to convex solutions of nonlinear elliptic equations.

Abstract

We consider convex solutions of nonlinear elliptic equations which satisfy the structure condition of Bian-Guan. We prove a weak Harnack inequality for the eigenvalues of the Hessian of these solutions. This can be viewed as a quantitative version of the constant rank theorem.