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$Γ$-convergence for nonlocal phase transitions involving the $H^{1/2}$ norm and surfactants

Giuliana Fusco, Tim Heilmann

Abstract

We study functionals \begin{equation*} F_\varepsilon (u,ρ) := \frac{1}{\varepsilon} \int_ΩW(u) \, dx + \frac{1}{|\ln(\varepsilon)|} \int_Ω\int_Ω \frac{(u(y) - u(x))^2}{|y - x|^{N+1}} \, dy \,dx + \frac{1}{|\ln(\varepsilon)|} \int_Ω\left| \int_Ω \frac{(u(y) - u(x))^2}{|y - x|^{N+1}} \, dy - ρ(x) \right| \,dx \end{equation*} for a double-well potential $W$ and a nonlocal, critically scaled gradient-like term, together with a surfactant term. We show compactness in the space of $BV$ functions on $Ω$ and the $Γ$-convergence to an energy given as local perimeter-type functional, depending also on the limit density of surfactant on the interface, plus the total variation of the surfactant measure away from the interface.

$Γ$-convergence for nonlocal phase transitions involving the $H^{1/2}$ norm and surfactants

Abstract

We study functionals \begin{equation*} F_\varepsilon (u,ρ) := \frac{1}{\varepsilon} \int_ΩW(u) \, dx + \frac{1}{|\ln(\varepsilon)|} \int_Ω\int_Ω \frac{(u(y) - u(x))^2}{|y - x|^{N+1}} \, dy \,dx + \frac{1}{|\ln(\varepsilon)|} \int_Ω\left| \int_Ω \frac{(u(y) - u(x))^2}{|y - x|^{N+1}} \, dy - ρ(x) \right| \,dx \end{equation*} for a double-well potential and a nonlocal, critically scaled gradient-like term, together with a surfactant term. We show compactness in the space of functions on and the -convergence to an energy given as local perimeter-type functional, depending also on the limit density of surfactant on the interface, plus the total variation of the surfactant measure away from the interface.
Paper Structure (8 sections, 11 theorems, 102 equations, 2 figures)

This paper contains 8 sections, 11 theorems, 102 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Compactness and $\Gamma$-convergence
  3. Notation
  4. The energy functional
  5. The main results
  6. Preliminaries
  7. $\Gamma$-liminf inequality
  8. $\Gamma$-limsup inequality

Key Result

Theorem 2.1

Let $F_{\varepsilon_h}$ be the functional defined in (energy). Given sequences $\varepsilon_h \rightarrow 0^+$, $u_h \in L^1(\Omega)$ and $\rho_h \in L^1(\Omega, [0,\infty))$ such that $\sup_h \left\| \frac{\rho_h}{\ln \varepsilon_h} \right\|_{L^1(\Omega)} < \infty$ and $\sup_h F_{\varepsilon_h}(u_

Figures (2)

  • Figure 1: Sketch of the situation of Proposition \ref{['limsupPolyhedral']}; the gray region between $Z_{1,\varepsilon}$ and $Z_{2,\varepsilon}$ is the set not belonging to $L$ where $v_\varepsilon$ is not constant.
  • Figure 2: Sketch of the situation of Proposition \ref{['temp8']} in the case $N=2$. The thick black lines represents the projection of $\Sigma_g$ on the interfaces.

Theorems & Definitions (18)

  • Theorem 2.1: Compactness
  • Theorem 2.2: $\Gamma$-convergence
  • Remark 2.3
  • Lemma 2.4: h, Lemma 2.4, Estimate of $G$ for a cylinder and its complement inside a fattened hyperplane
  • Lemma 2.5: h, Remark 2.5, Estimate of $G$ for a cylinder and the complement of a cone inside a fattened hyperplane
  • Corollary 2.6: h, Corollary 2.8, Lower bound of $G$ on a special cylinder
  • Lemma 2.7: h, Lemma 2.10, Energy upper bound for a special sequence
  • Lemma 2.8
  • proof
  • Proposition 2.9: $\Gamma$-liminf inequality
  • ...and 8 more