In this talk, we present the second-order asymptotic development of the Cahn-Hilliard functional under Dirichlet boundary conditions via Γ-convergence. We begin by reviewing results from the literature on the asymptotic expansion of the Cahn-Hilliard functional. Subsequently, we discuss our ongoing research focused on the Cahn-Hilliard functional with Dirichlet boundary conditions. In particular, we examine the case where no interior interfaces are present and highlight several open questions for future investigation. This seminar is based on work in collaboration with Irene Fonseca and Giovanni Leoni.

This seminar is part of the activities of the Excellence Department Project CUP B83C23001390001 and it is funded by the European Union – Next Generation EU and  of the project "Variational methods for stationary and evolution problems with singularities and interfaces", PRIN 2022 J4FYNJ - CUP B53D23009320006 and it is funded by the European Union – Next Generation EU.