The Liouville theorem for k-Hessian equations in the half space

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发表日期：

2025年6月28日

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Xiaobiao Jia,
Shanshan Ma

摘要

In this paper, we consider the Liouville theorem for k-Hessian equations in the half space. We prove that any k-convex solution u of k-Hessian equation \(S_k(D^2u)=C_n^k\) with an appropriate boundary condition on \(\{x_n=0\}\) must be a quadratic polynomial, provided that u satisfies a double quadratic growth condition and \(S_{k+1}(D^2u)\ge -\tilde{A}S_k(D^2u)\) for some positive constant \(\tilde{A}\) . Moreover, if u is convex, the double quadratic growth condition can be weakened.