An Interior Gradient Estimate for a class of Second Order Partial Differential Inequalities

TL;DR

This paper establishes a uniform gradient estimate for functions satisfying certain second-order partial differential inequalities, under specific structural conditions, with applications to degenerate elliptic PDEs.

Contribution

It provides a new gradient estimate for a class of second-order PDE inequalities with explicit structure conditions, extending previous results to degenerate elliptic equations.

Findings

Established a uniform gradient bound for solutions

Derived structure conditions for coefficients of PDEs

Applied results to degenerate elliptic equations

Abstract

A uniform gradient for functions u which satisfy a system of N second-order partial differential inequalities is given in this paper. Some structure conditions are given for the coefficients of the matrices of second-order terms and of first-order terms. This result can be applied to study the gradient estimate of a class of second-order degenerate elliptic partial differential equations.