Asymptotics for two-dimensional vectorial Allen-Cahn systems

TL;DR

This paper develops new analytical methods to study interface formation in two-dimensional vectorial Allen-Cahn systems, overcoming the lack of existing monotonicity formulas and extending scalar case results.

Contribution

Introduces novel PDE analysis techniques and a new discrepancy relation to establish a monotonicity formula for vectorial Allen-Cahn systems in two dimensions.

Findings

Extended scalar Allen-Cahn results to vectorial systems in 2D

Developed a new monotonicity formula based on discrepancy relations

Identified specific features unique to the vectorial case

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 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 extend related to the absence of known appropriate monotonicity formula. In this paper, we focus on the \emph{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 \emph{new monotonicity formula} which relies on a \emph{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 \emph{specific features} of the vectorial case.