A stronger constant rank theorem

2023/08/02 by Qinfeng Li, Lu Xu, Li, Qinfeng +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Meromorphic and Entire Functions #Analytic and geometric function theory

paper · pdf · doi:10.48550/arxiv.2308.00940

Abstract

Motivated from one-dimensional rigidity results of entire solutions to Liouville equation, we consider the semilinear equation Δu=G(u) in ℝⁿ,where G>0, G'<0 and GG^''≤ A(G')², with A>0. Let u be a smooth convex solution and σ_k(D² u) be the k-th elementary symmetric polynomial with respect to D²u. We prove stronger constant rank theorems in the following sense. (1) When A≤ 2, if σ₂(D²u) takes a local minimum, then D² u has constant rank 1. (2) When A≤ (n)/(n-1), if σ_n(D² u) takes a local minimum, then σ_n(D² u) is always zero in the domain.

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