Haoyang Sheng | Searcharxiv

arXiv2019

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

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 $x$.

math.AP↗

arXiv2019

A Liouville type theorem to $2$-Hessian equations

In this paper, we proved that any 2-convex solution $u$ of $σ_2(D^2u)=1$ with a quadratic growth must be a quadratic polynomial in $\mathbb{R}^n\ (n\geq 3 )$ by using a Pogorelov estimate and the global gradient estimate. And we give a positive answer to the unresolved issue in \cite{CX}.

math.AP↗