Pogorelov interior estimates for general sum-type Hessian equations

TL;DR

This paper develops Pogorelov interior estimates for sum-type Hessian equations using concavity properties, leading to Liouville-type results and characterizations of solutions with quadratic growth.

Contribution

It introduces new Pogorelov estimates for general sum-type Hessian equations under semi-convexity, extending previous results to broader classes of equations.

Findings

Pogorelov estimates derived for sum-type Hessian equations

Liouville-type theorems established for entire solutions

Quadratic growth solutions are quadratic polynomials

Abstract

In this paper, we exploit the concavity of sums of Hessian operators to derive Pogorelov estimates for corresponding equations under the dynamic semi-convexity assumption, and we further obtain several Liouville-type results. Moreover, when k=n-1 and k=n we establish Pogorelov estimates in the admissible cone. As an application, we prove that any entire admissible solution in $\mathbb{R}^n$ with quadratic growth must be a quadratic polynomial.