Concentration sets for multiple equal-depth wells potentials in the 2D elliptic case

TL;DR

This paper investigates the formation of interfaces in a 2D multiwell elliptic problem, introducing new methods to analyze concentration phenomena without relying on monotonicity formulas.

Contribution

It presents novel techniques to study codimension-one interfaces in vectorial elliptic problems, overcoming the challenge of missing monotonicity formulas.

Findings

Interfaces concentrate on one-dimensional rectifiable sets

Methods circumvent the need for monotonicity formulas

Results extend understanding of multiwell elliptic problems in 2D

Abstract

The formation of codimension-one interfaces for multiwell 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 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 elliptic case in two dimensions, and introduce some methods which allow to circumvent the lack of monotonicity formula. This methods lead, as expected, to concentration on one-dimensional rectifiable sets.