Hessian estimate for semiconvex solutions to the sigma-2 equation

TL;DR

This paper establishes interior Hessian estimates for semiconvex solutions to the sigma-2 equation, advancing understanding of their regularity properties through novel inequalities and transformation techniques.

Contribution

It introduces new a priori Hessian estimates for semiconvex solutions, extending prior results from almost convex solutions using innovative inequalities and transformations.

Findings

Derived interior Hessian estimates for semiconvex solutions

Developed a Jacobi inequality and a Legendre-Lewy transform rule

Established a mean value inequality for nonuniformly elliptic equations

Abstract

We derive a priori interior Hessian estimates for semiconvex solutions to the sigma-2 equation. An elusive Jacobi inequality, a transformation rule under the Legendre-Lewy transform, and a mean value inequality for the still nonuniformly elliptic equation without area structure are the key to our arguments. Previously, this result was known for almost convex solutions.