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 Rigidity theorem for parabolic 2-Hessian equations
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abstract
In this paper, we consider the entire solutions to the parabolic $2$-Hessian equations of the form $-u_t\sigma_2(D^2 u)=1$ in $\mathbb{R}^n\times (-\infty,0]$. We prove some rigidity theorems for the parabolic $2$-Hessian equations in $\mathbb{R}^n\times (-\infty,0]$ by establishing Pogorelov type estimates for $2$-convex-monotone solutions of the parabolic $2$-Hessian equations.
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math.AP 1years
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.