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Pogorelov type estimates for $(n-1)$-Hessian equations and related rigidity theorems
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abstract
In this paper, we establish Pogorelov type $C^2$ estimates for admissible solutions to the Dirichlet problem of $(n-1)$-Hessian equation based on a concavity inequality, which is inspired by the Lu-Tsai's work on the global curvature estimates for the $n-1$ curvature equation. As an application, we apply such estimates to obtain a rigidity theorems for admissible solutions of $(n-1)$-Hessian equation only under quadratic growth conditions. This result gives a positive answer to a open problem for $k$-Hessian equation, which is proposed by Chang-Yuan, in case $k=n-1$.
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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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