Pogorelov type $C^2$ estimates for sum Hessian equations

TL;DR

The paper develops Pogorelov-type interior $C^2$ estimates for the Dirichlet problem of sum Hessian equations of the form $\sigma_{k-1}(D^{2}u)+\alpha\sigma_k(D^{2}u)=f(x,u,Du)$, with $u$ vanishing on the boundary. It introduces two key concavity inequalities for $S_k$ and $S_n$ that control mixed second-derivative terms, then constructs a Pogorelov-type test function to derive bounds of the form $(-u)^{\beta}\lambda_{ ext{max}}\le C$, where $\lambda_{ ext{max}}$ is the largest eigenvalue of the Hessian. For $k<n$, the bound holds under the assumption $\sigma_k(D^{2}u)\ge -G$; for $k=n$, this lower-bound condition can be removed, yielding a similar bound for all admissible solutions. These results advance interior $C^2$ estimates for sum Hessian equations, contributing to the broader theory of fully nonlinear PDEs in geometric analysis.

Abstract

In this paper, We establish Pogorelov type $C^2$ estimates for the admissible solutions with $σ_k(D^2u)$ bounded from below of Sum Hessian equations. We also proved the lower bounded condition can be removed when $k = n$.

Theorems & Definitions (9)

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