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$C^2$ estimates for $k$-Hessian equations and a rigidity theorem

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arxiv 2408.10781 v2 pith:JGLFIKHD submitted 2024-08-20 math.AP math.DG

Ruijia Zhang This is my paper · ORCID
classification math.APmath.DG
keywords equationsestimateshessiancurvatureadditionallyapplicationboundaryconcavity
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abstract

We derive a concavity inequality for $k$-Hessian operators under the semi-convexity condition. As an application, we establish interior estimates for semi-convex solutions of the $k$-Hessian equations with vanishing Dirichlet boundary and obtain a Liouville-type result. Additionally, we provide new and simple proofs of Guan-Ren-Wang's results on global curvature estimates for $k$-curvature equations.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Second order estimates for $\chi$-semi convex solutions of Hessian equations on Hermitian manifolds

    math.AP 2025-01 conditional novelty 6.0 of 10

    For admissible, chi-semi-convex solutions of complex Hessian equations with gradient terms on compact Hermitian manifolds, the paper proves uniform second-order estimates.

  2. The Pogorelov estimates for the sum Hessian equation with rigidity theorem and parabolic versions

    math.AP 2024-12 conditional novelty 5.0 of 10

    New Pogorelov-type C^2 estimates and rigidity theorems are proved for (k-1)-convex semi-convex solutions of the elliptic and parabolic sum Hessian equations.

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