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Gamma-Convergence of Higher-Order Phase Transition Models

Denis Brazke, Gianna Götzmann, Hans Knüpfer

TL;DR

The paper addresses the small-\(\varepsilon\) behavior of a family of one-dimensional higher-order phase-transition energies $G_\varepsilon^{\lambda,n}$ with $n\ge2$, proving that in the subcritical regime $0<\lambda<\lambda_n$ their $\Gamma$-limit in $L^1(I)$ is a sharp-interface functional $G^{\lambda,n}[u] = C_W^{\lambda,n} \#S(u)$, where $C_W^{\lambda,n}>0$ is determined by an optimal-profile problem on $\mathbb{R}$. A key methodological advance is a nonlinear interpolation inequality of order $n$, derived from Gagliardo–Nirenberg-type estimates, which controls the concave term and yields compactness via reduction to the known $G_\varepsilon^{0,n}$ theory. The main contributions are the precise $\Gamma$-convergence result, the positivity of the optimal-profile constant, and the construction of energetically favored competitors near interfaces that enable the liminf and recovery sequence arguments. Together, these results rigorously justify a sharp-interface description for higher-order phase-transition models and extend classical second-order results to arbitrary order $n\ge2$.

Abstract

We investigate the asymptotic behavior as $\varepsilon \to 0$ of singularly perturbed phase transition models of order $n \geq 2$, given by \begin{align} G_\varepsilon^{λ,n}[u] := \int_I \frac 1\varepsilon W(u) -λ\varepsilon^{2n-3} (u^{(n-1)})^2 + \varepsilon^{2n-1} (u^{(n)})^2 \ dx, \quad u \in W^{n,2}(I), \end{align} where $λ>0$ is fixed, $I \subset \mathbb{R}$ is an open bounded interval, and $W \in C^0(\mathbb{R})$ is a suitable double-well potential. We find that there exists a positive critical parameter depending on $W$ and $n$, such that the $Γ$-limit of $G_\varepsilon^{λ,n}$ with respect to the $L^1$-topology is given by a sharp interface functional in the subcritical regime. The cornerstone for the corresponding compactness property is a novel nonlinear interpolation inequality involving higher-order derivatives, which is based on Gagliardo-Nirenberg type inequalities.

Gamma-Convergence of Higher-Order Phase Transition Models

TL;DR

The paper addresses the small- behavior of a family of one-dimensional higher-order phase-transition energies with , proving that in the subcritical regime their -limit in is a sharp-interface functional , where is determined by an optimal-profile problem on . A key methodological advance is a nonlinear interpolation inequality of order , derived from Gagliardo–Nirenberg-type estimates, which controls the concave term and yields compactness via reduction to the known theory. The main contributions are the precise -convergence result, the positivity of the optimal-profile constant, and the construction of energetically favored competitors near interfaces that enable the liminf and recovery sequence arguments. Together, these results rigorously justify a sharp-interface description for higher-order phase-transition models and extend classical second-order results to arbitrary order .

Abstract

We investigate the asymptotic behavior as of singularly perturbed phase transition models of order , given by \begin{align} G_\varepsilon^{λ,n}[u] := \int_I \frac 1\varepsilon W(u) -λ\varepsilon^{2n-3} (u^{(n-1)})^2 + \varepsilon^{2n-1} (u^{(n)})^2 \ dx, \quad u \in W^{n,2}(I), \end{align} where is fixed, is an open bounded interval, and is a suitable double-well potential. We find that there exists a positive critical parameter depending on and , such that the -limit of with respect to the -topology is given by a sharp interface functional in the subcritical regime. The cornerstone for the corresponding compactness property is a novel nonlinear interpolation inequality involving higher-order derivatives, which is based on Gagliardo-Nirenberg type inequalities.

Paper Structure

This paper contains 10 sections, 15 theorems, 112 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Main Results
  3. Compactness
  4. Nonlinear interpolation
  5. Lower bound and compactness
  6. Asymptotic Analysis
  7. Asymptotic Analysis
  8. Construction of favored competitors
  9. Gamma-convergence
  10. Appendix

Key Result

Theorem 2.1

Let $n \in {\mathbb N}_{\geq2}$ and assume $W$ satisfies (it-W1) -- (it-W3). Then there exists $\lambda_n = \lambda_n(L)>0$ such that for every open bounded interval $I \subset \mathbb{R}$, and every $u \in W^{n,2}(I)$.

Figures (4)

  • Figure 4.1: We construct non-intersecting balls around the points of discontinuity of $u$.
  • Figure 4.2: Away from the discontinuity, the sequence $u_\varepsilon$ (orange) is modified (pink) into a favored competitor sequence $v_\varepsilon$ such that its energy $G_\varepsilon^{\lambda,n}$ decreases.
  • Figure 4.3: As $\varepsilon$ goes to zero, the recovery sequence $u_\varepsilon$ (pink) converges to $u$ (blue).
  • Figure A.1: The aim is to cover the set $E_k$ by open bounded intervals. This enables the application of Lemma \ref{['IntLem']}.

Theorems & Definitions (33)

  • Theorem 2.1: Nonlinear interpolation for higher orders
  • Theorem 2.1: Nonlinear interpolation for higher orders
  • Remark 2.2
  • Theorem 2.3: Compactness
  • Theorem 2.4: $\Gamma$-convergence
  • Remark 2.5
  • Remark 2.6
  • Lemma 3.1
  • proof
  • proof : Proof of Theorem \ref{['Theo:NonLinInt_n']}.
  • ...and 23 more