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Non-local non-homogeneous phase transitions: regularity of optimal profiles and sharp-interface limit

Elisa Davoli, Emanuele Tasso

TL;DR

The work rigorously derives the sharp-interface limit for a non-local, non-homogeneous diffuse-interface model with moving wells, establishing Gamma-convergence of $F_\varepsilon$ to an anisotropic surface-energy $F_0$ whose density $\sigma$ is defined via a one-dimensional optimal-profile problem. A key novelty is the asymptotic calibration approach and a conjugate-functional framework that yield a robust Gamma-liminf bound and link the regularity of 1D optimal profiles to kernel singularities. The recovery-sequence construction is achieved by gluing 1D profiles along jump sets; when the kernel is integrable this uses an invertibility condition on $Q_x$, while in the non-integrable case a Hölder-selection principle ensures coherent gluing. The paper also develops a comprehensive 1D theory for existence, regularity, and parameter dependence of optimal profiles, and provides a detailed higher-dimensional analysis via compactness, slicing, and a directional blow-up that yields the double-liminf inequality. Together, these results connect kernel singularity, moving wells, and sharp-interface regularity, offering a rigorous framework for non-local, non-homogeneous phase transitions with potential applications to continuum limits of spin systems and pattern formation under spatially varying environments.

Abstract

We provide a novel sharp-interface analysis via Gamma-convergence for a non-local and non-homogeneous diffuse-interface model for phase transitions, featuring an interplay between a non-local interaction kernel and a spatially dependent double-well potential. This interaction requires the development of new strategies both for the Gamma-liminf inequality and for the construction of recovery sequences. A key element of our approach is an asymptotic calibration, used to establish the Gamma-liminf lower bound. The study of the optimality of the lower bound hinges upon a novel analysis of the regularity dependence of one-dimensional optimal profiles on a family of parameters. In particular, we show how such regularity is influenced by the singularity of the interaction kernel at the origin, providing a precise and previously unexplored link between the two. Our results rely solely on the assumption of Hölder continuity for the moving wells, and also account for the compactness of sequences with equibounded energies.

Non-local non-homogeneous phase transitions: regularity of optimal profiles and sharp-interface limit

TL;DR

The work rigorously derives the sharp-interface limit for a non-local, non-homogeneous diffuse-interface model with moving wells, establishing Gamma-convergence of to an anisotropic surface-energy whose density is defined via a one-dimensional optimal-profile problem. A key novelty is the asymptotic calibration approach and a conjugate-functional framework that yield a robust Gamma-liminf bound and link the regularity of 1D optimal profiles to kernel singularities. The recovery-sequence construction is achieved by gluing 1D profiles along jump sets; when the kernel is integrable this uses an invertibility condition on , while in the non-integrable case a Hölder-selection principle ensures coherent gluing. The paper also develops a comprehensive 1D theory for existence, regularity, and parameter dependence of optimal profiles, and provides a detailed higher-dimensional analysis via compactness, slicing, and a directional blow-up that yields the double-liminf inequality. Together, these results connect kernel singularity, moving wells, and sharp-interface regularity, offering a rigorous framework for non-local, non-homogeneous phase transitions with potential applications to continuum limits of spin systems and pattern formation under spatially varying environments.

Abstract

We provide a novel sharp-interface analysis via Gamma-convergence for a non-local and non-homogeneous diffuse-interface model for phase transitions, featuring an interplay between a non-local interaction kernel and a spatially dependent double-well potential. This interaction requires the development of new strategies both for the Gamma-liminf inequality and for the construction of recovery sequences. A key element of our approach is an asymptotic calibration, used to establish the Gamma-liminf lower bound. The study of the optimality of the lower bound hinges upon a novel analysis of the regularity dependence of one-dimensional optimal profiles on a family of parameters. In particular, we show how such regularity is influenced by the singularity of the interaction kernel at the origin, providing a precise and previously unexplored link between the two. Our results rely solely on the assumption of Hölder continuity for the moving wells, and also account for the compactness of sequences with equibounded energies.

Paper Structure

This paper contains 29 sections, 37 theorems, 343 equations.

Table of Contents

  1. Introduction
  2. Previous contributions and relevant works.
  3. Main results of the paper
  4. The Gamma-liminf analysis
  5. The construction of recovery sequences
  6. Outline of the paper
  7. Setting of the problem and main results
  8. Notation
  9. Double-well potentials
  10. The sequence of functionals and the surface tension
  11. Admissible interaction kernels
  12. Main results
  13. The 1D-Problem
  14. Existence
  15. Characterization via the conjugate functional
  16. ...and 14 more sections

Key Result

Theorem 2.6

Let $K \subset \Omega\subset \mathbb{R}^m$ be compact, let $J \colon \mathbb{R} \to (0,+\infty)$ be even, and let $W \colon \Omega \times \mathbb{R} \to [0,\infty)$ satisfy H1--H4. Let us further denote for every $x \in K$ Assume further that one of the following conditions is satisfied Then, the class $O_x$ is nonempty for every $x\in K$. Additionally, there exists a weight function $\overline

Theorems & Definitions (92)

  • Remark 2.1: On \ref{['H1']}--\ref{['H5']}
  • Definition 2.2: The space $CP(\Omega;\mathbf{z}(x))$
  • Remark 2.3
  • Definition 2.4: The surface tension $\sigma$
  • Definition 2.5: The class $\mathcal{L}_m(\eta,\lambda,\rho)$
  • Theorem 2.6
  • Theorem 2.7: $\Gamma$-limit with non-integrable interaction kernels
  • Theorem 2.8: $\Gamma$-limit with integrable interaction kernels
  • Proposition 3.1: Existence of minimizers for double well potentials
  • proof : Proof of Proposition \ref{['t:eximin']}
  • ...and 82 more