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arXiv · 2602.14064

Interior Hessian estimates for Hessian quotient equations in dimension three

Heming Jiao·Zhenan Sui
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Abstract

In this paper, we establish the interior Hessian estimates for $2$-convex solutions to $\frac{\sigma_2}{\sigma_1} (D^2 u) = \psi (x,u)$ in dimension three. In higher dimensions ($n \geq 4$), we prove the interior Hessian estimates for semi-convex solutions. We provide a new method to prove the doubling inequality for smooth solutions in dimensions three and four. In higher dimensions ($n\geq 5$) the doubling inequality is proved under an additional dynamic semi-convexity condition which is the same to that in \cite{SY2025}. The method also applies to the equation $\sigma_2 (D^2 u) = \psi (x, u, \nabla u)$.

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Heming Jiao, Zhenan Sui. 2026-02-15. Interior Hessian estimates for Hessian quotient equations in dimension three. https://arxiv.org/abs/2602.14064

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