Interior $C^2$ estimate for Hessian quotient equation in general dimension

Siyuan Lu

Interior $C^2$ estimate for Hessian quotient equation in general dimension

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arxiv 2401.12229 v1 pith:2QN3IF2W submitted 2024-01-17 math.AP

Siyuan Lu This is my paper · ORCID

classification math.AP

keywords interiorestimateequationhessianproblemquotientsigmaanswer

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abstract

In this paper, we study the interior $C^2$ regularity problem for the Hessian quotient equation $\left(\frac{\sigma_n}{\sigma_k}\right)(D^2u)=f$. We give a complete answer to this longstanding problem: for $k=n-1,n-2$, we establish an interior $C^2$ estimate; for $k\leq n-3$, we show that interior $C^2$ estimate fails by finding a singular solution.

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Remarks on Hessian quotient equations on Riemannian manifolds
math.DG 2025-01 reject novelty 4.0 of 10

A dimension-two a priori second-order estimate for Hessian quotient equations is claimed via a new test function, but the final contradiction does not actually constrain the largest Hessian eigenvalue.

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