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A threshold for higher-order asymptotic development of genuinely nonlocal phase transition energies

Serena Dipierro, Enrico Valdinoci, Mary Vaughan

Abstract

We study the higher-order asymptotic development of a nonlocal phase transition energy in bounded domains and with prescribed external boundary conditions. The energy under consideration has fractional order $2s \in (0,1)$ and a first-order asymptotic development in the $Γ$-sense as described by the fractional perimeter functional. We prove that there is no meaningful second-order asymptotic expansion and, in fact, no asymptotic expansion of fractional order $μ> 2-2s$. In view of this range value for $μ$, it would be interesting to develop a new asymptotic development for the $Γ$-convergence of our energy functional which takes into account fractional orders. The results obtained here are also valid in every space dimension and with mild assumptions on the exterior data.

A threshold for higher-order asymptotic development of genuinely nonlocal phase transition energies

Abstract

We study the higher-order asymptotic development of a nonlocal phase transition energy in bounded domains and with prescribed external boundary conditions. The energy under consideration has fractional order and a first-order asymptotic development in the -sense as described by the fractional perimeter functional. We prove that there is no meaningful second-order asymptotic expansion and, in fact, no asymptotic expansion of fractional order . In view of this range value for , it would be interesting to develop a new asymptotic development for the -convergence of our energy functional which takes into account fractional orders. The results obtained here are also valid in every space dimension and with mild assumptions on the exterior data.

Paper Structure

This paper contains 9 sections, 7 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Paper organization
  4. Almost minimizers
  5. Energy estimates
  6. Covering lemma
  7. Proof of Theorem \ref{['MSA:TH']}
  8. Proof of Theorem \ref{['lem:trivial']}
  9. Refinement of Proposition \ref{['prop:anydim65']} in dimension $1$

Key Result

Theorem 1.1

Suppose $m_1 \not=0$ and let $\bar{u} \in \mathcal{U}_1$ be such that $\bar{u} |_{\Omega} = \chi_E - \chi_{E^c}$ for some measurable set $E \subset \mathbb{R}^n$. Assume at least one of the following holds: Then, there exists a sequence $v_\varepsilon \in X_g$ such that $v_\varepsilon \to \bar{u}$ as $\varepsilon \searrow 0$ and, for any $\mu>1-2s$, In particular,

Theorems & Definitions (13)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Proposition 3.1
  • proof
  • Lemma 4.1
  • proof
  • proof : Proof of Theorem \ref{['MSA:TH']}
  • ...and 3 more