Any k+1-convex-monotone solution to -u_t σ_k(D²u)=1 with quadratic growth on u(x,0) and 0<m1≤-u_t≤m2 is linear in t plus quadratic in x.
A Liouville type theorem to $2$-Hessian equations
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abstract
In this paper, we proved that any 2-convex solution $u$ of $\sigma_2(D^2u)=1$ 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}.
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2019 1verdicts
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A Pogorelov estimate and a Liouville type theorem to parabolic $k$-Hessian equations
Any k+1-convex-monotone solution to -u_t σ_k(D²u)=1 with quadratic growth on u(x,0) and 0<m1≤-u_t≤m2 is linear in t plus quadratic in x.