Two-end solutions to the Allen-Cahn equation in ℝ³

2015/02/20 by Changfeng Gui, Yong Liu, Gui, Changfeng +3
Mathematics · #Analysis of PDEs (math.AP) #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Meromorphic and Entire Functions #math.AP #math.DG

paper · pdf · doi:10.48550/arxiv.1502.05963

55 pages; comments welcome

arxiv created 2015/02/20 · openalex publication_date 2015/02/20 · arxiv updated 2015/02/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we consider the Allen-Cahn equation -Δu = u-u³ in \mathbb R³ We prove that for each k∈( √(2),+∞), there exists a solution to the equation which has growth rate k, i.e. ‖ u-H(⋅ -k ln r + c_k) ‖_L^∞ → 0 The main ingredients of our proof consist: (1) compactness of solutions with growth k, (2) moduli space theory of analytical variety of formal dimension one.

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