Doubling construction for O(m)×O(n) invariant solutions to the Allen–Cahn equation. (March 2022)

• Record Type: Journal Article
• Authors: Agudelo, Oscar; Kowalczyk, Michał; Rizzi, Matteo
• Is Part Of: Nonlinear analysis. Volume 216(2022)
• Publication Date: 2022-03
• Subjects: Allen–Cahn equation -- Lawson cones -- Minimal Surfaces -- Jacobi–Toda system; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2021.112705
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We construct new families of two-ended O ( m ) × O ( n ) -invariant solutions to the Allen–Cahn equation Δ u + u − u 3 =0 in R N + 1, with N ≥ 7, whose zero level sets diverge logarithmically from the Lawson cone at infinity. The construction is based on a careful study of the Jacobi–Toda system on a given O ( m ) × O ( n ) -invariant manifold, which is asymptotic to the Lawson cone at infinity.