Stability of the saddle solutions for the Allen-Cahn equation

2020/01/21 by Yong Liu, Kelei Wang, Liu, Yong +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2001.07356

openalex publication_date 2020/01/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We are concerned with the saddle solutions of the Allen-Cahn equation constructed by Cabré and Terra \citeC,C2 in ℝ^2m% =ℝ^m×ℝ^m. These solutions vanish precisely on the Simons cone. The existence and uniqueness of saddle solution are shown in \citeC,C2,C1. Regarding the stability, Schatzman \citeSch proved that the saddle solution is unstable for m=1, Cabré \citeC1 showed the instability for m=2,3 and stability for m≥7. This has left open the case of m=4,5,6. In this paper we show that the saddle solutions are stable when m=4,5,6, thereby confirming Cabré's conjecture in \citeC1. The conjecture that saddle solutions in dimensions 2m≥8 should be global minimizers of the energy functional remains open.

Citations

• Minimal Varieties in Riemannian Manifolds

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