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Nonlocal Phase Transitions with Singular Heterogeneous Kernels

Wes Caldwell

Abstract

In this paper the study of a non-local Cahn-Hilliard-type singularly perturbed family of functionals is undertaken, generalizing known results by Alberti & Bellettini. The kernels considered include those leading to Gagliardo seminorms for fractional Sobolev spaces. The limit energy is computed via $Γ$-convergence and shown to be an anisotropic surface energy on the interface between the two phases.

Nonlocal Phase Transitions with Singular Heterogeneous Kernels

Abstract

In this paper the study of a non-local Cahn-Hilliard-type singularly perturbed family of functionals is undertaken, generalizing known results by Alberti & Bellettini. The kernels considered include those leading to Gagliardo seminorms for fractional Sobolev spaces. The limit energy is computed via -convergence and shown to be an anisotropic surface energy on the interface between the two phases.

Paper Structure

This paper contains 9 sections, 13 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Main Theorem
  3. Discussion of Hypotheses
  4. Sets of Finite Perimeter and Functions of Bounded Variation
  5. Slicing and Integral Geometry
  6. Compactness and the $\Gamma-\liminf$
  7. Modification Theorem
  8. $\Gamma$-Limsup Inequality
  9. Acknowledgments

Key Result

Theorem 1.1

Let $\varepsilon_j \to 0$ as $j \to \infty$. Under the hypotheses (i) and (ii) on $J$ and $W$, the following statements hold:

Theorems & Definitions (28)

  • Theorem 1.1
  • Proposition 1.2
  • proof
  • Definition 1.3
  • Definition 1.4
  • Lemma 1.5: Blaschke-Petkantschin Formula, c.f. leoni_first_2023, schneider_stochastic_2008
  • Lemma 1.6
  • proof
  • proof : Proof of Theorem \ref{['item:compactness']}
  • Lemma 2.1
  • ...and 18 more