A constant rank theorem for partial convex solutions of partial differential equations

TL;DR

This paper establishes a constant rank theorem for partial convex solutions of certain PDEs, extending previous results and providing a microscopic perspective on partial convexity principles.

Contribution

It introduces a new constant rank theorem for partial convex solutions, generalizing prior work and utilizing the Bian-Guan test function approach.

Findings

Proves a constant rank property for partial convex solutions.

Extends the partial convexity principle to a microscopic level.

Generalizes previous results in the field.

Abstract

Thanks to the test function of Bian-Guan[2], we successfully obtain a constant rank theorem for partial convex solutions of a class partial differential equations. This is the microscopic version of the macroscopic partial convexity principle in [1], and also is a generalization of the result in [2].