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

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