Interior estimates of derivatives and a Liouville type theorem for Parabolic k-Hessian equations

2022/09/22 by J. G. Bao, Bao, Jiguang, Jiechen Qiang +5
Mathematics · #35B08 #35B45 #35K55 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2209.10776

openalex publication_date 2022/09/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we establish the gradient and Pogorelov estimates for k-convex-monotone solutions to parabolic k-Hessian equations of the form -u_tσ_k(λ(D²u))=ψ(x,t,u). We also apply such estimates to obtain a Liouville type result, which states that any k-convex-monotone and C^4,2 solution u to -u_tσ_k(λ(D²u))=1 in ℝⁿ×(-∞,0] must be a linear function of t plus a quadratic polynomial of x, under some growth assumptions on u.

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