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A note on the recovery sequence in the double gradient model for phase transitions

Jakob Deutsch

TL;DR

The work advances the understanding of the limsup construction in the vector-valued double-gradient Modica–Mortola model in 2D by linking the optimal profile constant to a periodic cell formula under a bound involving the geodesic distance between wells. By developing a two-dimensional glueing framework that uses traces and transitional maps, the authors remove symmetry restrictions and show that, when $K^*<3d_W(A,B)$, the limiting interfacial energy can be realized by periodic gradients on the unit cube, i.e. $K^*=K^*_{\mathrm{per}}$. The key innovations are horizontal boundary modifications, optimal trace interpolation, and a careful combination of these steps to establish $K^*_{\mathrm{per}}\le K^*$, along with a geodesic-distance bound that applies to a broad class of near-quadratic perturbations of the quadratic two-well potential. This provides a practical criterion to compute the interfacial energy density via a cell formula in solid-solid phase-transition models and broadens applicability beyond symmetric potentials.

Abstract

We investigate the $\limsup$ inequality in the double gradient model for phase transitions governed by a Modica--Mortola functional with a double-well potential in two dimensions. Specifically, we consider energy functionals of the form \[ E_\varepsilon(u, Ω) = \int_Ω\left( \frac{1}{\varepsilon} W(\nabla u) + \varepsilon |\nabla^2 u|^2 \right) dx \] for maps $ u \in H^2(Ω; \mathbb{R}^2) $, where $ W $ vanishes only at two wells. Assuming a bound on the optimal profile constant -- namely the cell problem on the unit cube -- in terms of the geodesic distance between the two wells, we characterise the limiting interfacial energy via periodic recovery sequences as $\varepsilon \to 0^+$.

A note on the recovery sequence in the double gradient model for phase transitions

TL;DR

The work advances the understanding of the limsup construction in the vector-valued double-gradient Modica–Mortola model in 2D by linking the optimal profile constant to a periodic cell formula under a bound involving the geodesic distance between wells. By developing a two-dimensional glueing framework that uses traces and transitional maps, the authors remove symmetry restrictions and show that, when , the limiting interfacial energy can be realized by periodic gradients on the unit cube, i.e. . The key innovations are horizontal boundary modifications, optimal trace interpolation, and a careful combination of these steps to establish , along with a geodesic-distance bound that applies to a broad class of near-quadratic perturbations of the quadratic two-well potential. This provides a practical criterion to compute the interfacial energy density via a cell formula in solid-solid phase-transition models and broadens applicability beyond symmetric potentials.

Abstract

We investigate the inequality in the double gradient model for phase transitions governed by a Modica--Mortola functional with a double-well potential in two dimensions. Specifically, we consider energy functionals of the form for maps , where vanishes only at two wells. Assuming a bound on the optimal profile constant -- namely the cell problem on the unit cube -- in terms of the geodesic distance between the two wells, we characterise the limiting interfacial energy via periodic recovery sequences as .

Paper Structure

This paper contains 9 sections, 16 theorems, 170 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setting and Preliminaries
  3. Notation & Setup
  4. Discussion on geodesic distances
  5. Main Result
  6. Horizontal modification of optimal profiles
  7. Optimal interpolation of traces
  8. The proof of $K^*_\rm{per} \leq K^*$
  9. The geodesic distance bound

Key Result

Lemma 2.5

Let $h > 0$. Suppose $(u_n, \varepsilon_n) \subset H^2(\omega_h, \mathbb{R}^2) \times (0,1)$ is an optimal profile sequence with respect to $\omega \subset \mathbb{R}$. Then, it is also an optimal profile sequence with respect to any open set $\tilde{\omega} \subset \omega$ with $|\partial \tilde{\

Figures (2)

  • Figure 1: The modification from Lemma \ref{['lemma:horizontal_boundary_condition']}. A De Giorgi type argument allows one to choose suitable $s_n^1, s_n^2$ such that the energy along the trace of $u_n$ is bounded along ${s^i_n} \times (-1/2, 1/2)$ in a suitable way whilst keeping the energy of the modification $w_n$ in the light blue area proportional to $\delta$.
  • Figure 2: A visualisation of the glueing procedure used in the proof of Theorem \ref{['thm:horizontal_modification']}. The map $\tilde{v}$ and $\tilde{w}$ are given by Lemma \ref{['lemma:trace_translation_glueing']} and Lemma \ref{['lemma:trace_the_same_midpoint_glueing']}.

Theorems & Definitions (36)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • ...and 26 more