Second-Order $\Gamma$-limit for the Cahn-Hilliard Functional

TL;DR

This paper advances the understanding of the Cahn-Hilliard functional by establishing its second-order $ ext{Gamma}$-limit, using a new rearrangement method that avoids Dirichlet boundary conditions, addressing a long-standing open problem.

Contribution

It introduces a novel rearrangement technique to determine the second-order $ ext{Gamma}$-limit of the mass-constrained Cahn-Hilliard functional without relying on Dirichlet boundary conditions.

Findings

Established the second-order $ ext{Gamma}$-limit for the Cahn-Hilliard functional.

Developed a new rearrangement technique applicable without boundary conditions.

Solved a long-standing open problem in the asymptotic analysis of phase transition models.

Abstract

The goal of this paper is to solve a long standing open problem, namely, the asymptotic development of order $2$ by $\Gamma$-convergence of the mass-constrained Cahn-Hilliard functional. This is achieved by introducing a novel rearrangement technique, which works without Dirichlet boundary conditions.