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Second-Order $Γ$-Limit for the Cahn-Hilliard Functional with Dirichlet Boundary Conditions, II

Irene Fonseca, Leonard Kreutz, Giovanni Leoni

TL;DR

This work advances the Γ-convergence analysis of the Cahn-Hilliard functional with Dirichlet boundary conditions by removing a previous separation-from-wells assumption on boundary data and identifying a boundary-layer mechanism when no interfaces are present. The authors develop a weighted one-dimensional Γ-convergence framework with a positive weight $\omega$, establishing zeroth, first, and second-order limits and detailed limsup/liminf bounds, including precise curvature-driven contributions. They then translate the 1D insights to the 3D setting, showing that boundary data equal to the lower well on portions of the boundary induces a boundary layer, and deriving the main second-order limit formula in terms of the boundary mean curvature. The results yield an explicit energy expansion with a boundary-curvature term, highlighting the geometric influence on the near-boundary energetics of phase-field models under Dirichlet constraints.

Abstract

This paper continues the study of the asymptotic development of order 2 by $Γ$ -convergence of the Cahn-Hilliard functional with Dirichlet boundary conditions initiated in [8]. While in the first paper, the Dirichlet data are assumed to be well separated from one of the two wells, here this is no longer the case. In the case where there are no interfaces, it is shown that there is a transition layer near the boundary of the domain.

Second-Order $Γ$-Limit for the Cahn-Hilliard Functional with Dirichlet Boundary Conditions, II

TL;DR

This work advances the Γ-convergence analysis of the Cahn-Hilliard functional with Dirichlet boundary conditions by removing a previous separation-from-wells assumption on boundary data and identifying a boundary-layer mechanism when no interfaces are present. The authors develop a weighted one-dimensional Γ-convergence framework with a positive weight , establishing zeroth, first, and second-order limits and detailed limsup/liminf bounds, including precise curvature-driven contributions. They then translate the 1D insights to the 3D setting, showing that boundary data equal to the lower well on portions of the boundary induces a boundary layer, and deriving the main second-order limit formula in terms of the boundary mean curvature. The results yield an explicit energy expansion with a boundary-curvature term, highlighting the geometric influence on the near-boundary energetics of phase-field models under Dirichlet constraints.

Abstract

This paper continues the study of the asymptotic development of order 2 by -convergence of the Cahn-Hilliard functional with Dirichlet boundary conditions initiated in [8]. While in the first paper, the Dirichlet data are assumed to be well separated from one of the two wells, here this is no longer the case. In the case where there are no interfaces, it is shown that there is a transition layer near the boundary of the domain.
Paper Structure (9 sections, 23 theorems, 271 equations)

This paper contains 9 sections, 23 theorems, 271 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A 1D Functional Problem
  4. Zeroth and First-Order $\Gamma$-limit of $G_{\varepsilon}$
  5. Second-Order $\Gamma$-limsup
  6. Properties of Minimizers of $G_{\varepsilon}$
  7. Second-Order $\Gamma$-liminf
  8. Properties of Minimizers of $F_{\varepsilon}$
  9. Second-Order $\Gamma$-Limit

Key Result

Theorem 1.1

Let $\Omega\subset\mathbb{R}^{N}$ be an open, bounded, connected set with a boundary of class $C^{2,d}\ $, $0<d\leq1$. Assume that $W$ satisfies W_Smooth-W' three zeroes and that $g_{\varepsilon}\in H^{1}(\partial\Omega)$ is such that and for all $\varepsilon\in(0,1)$ and for some constants $C>0$ and $\gamma>1$. Suppose also that u0=b holds. Then if $u=b$ and $\mathcal{F}^{(2)}(u)=\infty$ otherw

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Theorem 3.1: Compactness
  • Theorem 3.2
  • Theorem 3.3
  • Corollary 3.4
  • ...and 17 more