Pogorelov interior estimates for sum-of-Hessians equations

Weisong Dong; Sirui Xu; Ruijia Zhang

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arXiv · 2603.15345

Pogorelov interior estimates for sum-of-Hessians equations

Weisong Dong·Sirui Xu·Ruijia Zhang

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Abstract

In this paper, we develop a new approach to sum-of-Hessians equations involving multiple $k$-Hessian operators of distinct orders. By exploiting the concavity of sums of Hessian operators, we derive Pogorelov estimates for the corresponding equations under the dynamic semi-convexity condition. Moreover, when the highest order is $n-1$ or $n$, we establish such estimates for admissible solutions without relying on this condition, and they are therefore optimal. As an application, when the right-hand side is identically $1$, we prove that every entire admissible solution in $\mathbb{R}^n$ with quadratic growth is necessarily a quadratic polynomial.

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Weisong Dong, Sirui Xu, Ruijia Zhang. 2026-03-16. Pogorelov interior estimates for sum-of-Hessians equations. https://arxiv.org/abs/2603.15345

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