Asymptotics for two-dimensional vectorial Allen-Cahn systems - Sorbonne Université Access content directly
Journal Articles Acta Mathematica Year : 2022

Asymptotics for two-dimensional vectorial Allen-Cahn systems

Fabrice Bethuel
  • Function : Author
  • PersonId : 1197336

Abstract

The formation of codimension-one interfaces for multi-well gradient-driven problems is well-known and established in the scalar case, where the equation is often referred to as the Allen-Cahn equation. The proofs rely for a large part on a monotonicity formula for the energy density, which is itself related to the vanishing of the so-called discrepancy function. The vectorial case in contrast is quite open. This lack of results and insight is to a large extent related to the absence of known appropriate monotonicity formula. In this paper, we focus on the elliptic case in two dimensions, and introduce methods, relying on the analysis of the partial differential equation, which allow to circumvent the lack of monotonicity formula for the energy density. In the last part of the paper, we recover a new monotonicity formula which relies on a new discrepancy relation. These tools allow to extend to the vectorial case in two dimensions most of the results obtained for the scalar case. We emphasize also some specific features of the vectorial case.
Fichier principal
Vignette du fichier
bethuel-vect-allen-cahn.D2V3-21-10.12.pdf (2.31 Mo) Télécharger le fichier
Origin Files produced by the author(s)

Dates and versions

hal-03880999 , version 1 (01-12-2022)

Identifiers

  • HAL Id : hal-03880999 , version 1

Cite

Fabrice Bethuel. Asymptotics for two-dimensional vectorial Allen-Cahn systems. Acta Mathematica, In press. ⟨hal-03880999⟩
17 View
103 Download

Share

Gmail Mastodon Facebook X LinkedIn More