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arXiv · 1907.07006

A Pogorelov estimate and a Liouville type theorem to parabolic $k$-Hessian equations

Yan He·Haoyang Sheng·Ni Xiang
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Abstract

We consider Pogorelov type estimates and Liouville type theorems to parabolic $k$-Hessian equations of the form $-u_t σ_k (D^2u) =1$ in $\mathbb{R}^n\times (-\infty, 0]$. We derive that any \textbf{$k+1$-convex-monotone} solution to $-u_t σ_k (D^2u) =1$ when $u(x,0)$ satisfies a quadratic growth and $0<m_1\le -u_t\le m_2$ must be a linear function of $t$ plus a quadratic polynomial of $x$.

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Yan He, Haoyang Sheng, Ni Xiang. 2019-07-16. A Pogorelov estimate and a Liouville type theorem to parabolic $k$-Hessian equations. https://arxiv.org/abs/1907.07006

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