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Low regularity potentials in heterogeneous Cahn--Hilliard functionals

Riccardo Cristoferi, Jakob Deutsch, Luca Pignatelli

Abstract

In this paper, we study the prototypical model of liquid-liquid phase separation, the Cahn-Hilliard functional, in a highly irregular setting. Specifically, we analyze potentials with low regularity vanishing on space-dependent wells. Under remarkably weak hypotheses, we establish a robust compactness result. Strengthening the regularity of the wells and of the growth of the potential close to the wells only slightly, we completely characterize the asymptotic behavior of the associated family of functionals through a $Γ$-convergence analysis. As a notable technical result, we prove the existence of geodesics for a degenerate metric and establish a uniform bound on their Euclidean length.

Low regularity potentials in heterogeneous Cahn--Hilliard functionals

Abstract

In this paper, we study the prototypical model of liquid-liquid phase separation, the Cahn-Hilliard functional, in a highly irregular setting. Specifically, we analyze potentials with low regularity vanishing on space-dependent wells. Under remarkably weak hypotheses, we establish a robust compactness result. Strengthening the regularity of the wells and of the growth of the potential close to the wells only slightly, we completely characterize the asymptotic behavior of the associated family of functionals through a -convergence analysis. As a notable technical result, we prove the existence of geodesics for a degenerate metric and establish a uniform bound on their Euclidean length.

Paper Structure

This paper contains 13 sections, 26 theorems, 353 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Assumptions and main results
  3. Preliminaries
  4. Functions of bounded variation and sets of finite perimeter
  5. $\Gamma$-convergence
  6. Young measures
  7. Technical results
  8. Properties of the geodesic distance
  9. An approximation result for sets of finite perimeter
  10. Compactness
  11. Liminf inequality
  12. Limsup inequality
  13. Auxiliary results

Key Result

Theorem 2.16

Assume H1-H5 hold. Let $\{\varepsilon_n\}_n$ be an infinitesimal sequence. Let $\{u_n\}_n \subset H^1(\Omega;\mathbb R^M)$ be such that Then, there exists a subsequence (not relabeled) such that $u_{n}\to u$ strongly in $L^1(\Omega;\mathbb R^M)$, for some $u\in BV(\Omega; \{a,b\})$.

Figures (5)

  • Figure 1: Example of slicing.
  • Figure 2: The substitution of $\tilde{\varphi}_m$ in Theorem \ref{['existence-of-geodesics-with-bounded-path-length']}.
  • Figure 3: The idea for the construction of potentials $W$ in Example \ref{['ex:geodesics_infinite_Euclidean_length']}. Heuristically, we create obstacles in the energy landscape such that geodesics connecting points in $C_1$ to points in $C_{1/2}$ cannot pass through.
  • Figure 4: Partition of the tubular neighborhood.
  • Figure :

Theorems & Definitions (71)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Example 2.9
  • Example 2.10
  • ...and 61 more