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$C^2$ estimates for $k$-Hessian equations and a rigidity theorem
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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.
Forward citations
Cited by 3 Pith papers
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Pogorelov type interior $C^2$ estimate for Hessian quotient equation and its application
Convex viscosity solutions of the Hessian quotient equation are shown to be C^{3,beta} under sharp C^{1,alpha} or W^{2,p} assumptions, via a new Pogorelov type interior C^2 estimate.
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Second order estimates for $\chi$-semi convex solutions of Hessian equations on Hermitian manifolds
For admissible, chi-semi-convex solutions of complex Hessian equations with gradient terms on compact Hermitian manifolds, the paper proves uniform second-order estimates.
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The Pogorelov estimates for the sum Hessian equation with rigidity theorem and parabolic versions
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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