On stable solutions to the Allen-Cahn equation with bounded energy density in ℝ⁴

2025/09/02 by Enric Florit-Simon, Joaquim Serra, Florit-Simon, Enric +1
Materials Science · Mathematics · #35J20 #35J61 #49Q05 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Solidification and crystal growth phenomena

paper · pdf · doi:10.48550/arxiv.2509.02739

openalex publication_date 2025/09/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that stable solutions u:ℝ⁴→ (-1,1) to the Allen-Cahn equation with bounded energy density (or equivalently, with cubic energy growth) are one-dimensional. This is known to entail important geometric consequences, such as robust curvature estimates for stable phase transitions, and the multiplicity one and Morse index conjectures of Marques-Neves for Allen-Cahn approximations of minimal hypersurfaces in closed 4-manifolds.

Related

• The Allen-Cahn equation on closed manifolds
• Some remarks on the structure of finite Morse index solutions to the Allen-Cahn equation in ℝ²
• On the Allen-Cahn equation in the Grushin plane: a monotone entire solution that is not one-dimensional
• Stability of the saddle solutions for the Allen-Cahn equation
• Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones