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Dimension reduction of fractional Sobolev seminorms on thin domains

Andrea Braides, Andrea Pinamonti, Margherita Solci

Abstract

We study the asymptotic behaviour of Gagliardo seminorms in $H^s$ defined on thin films $Ω_\e=ω\times(0,\e)$. The first relevant order is $\e^{1-2s}$, at which the corresponding limit captures the vertical fractional oscillations through one-dimensional sections. The second relevant order produces dimension-reduction regimes that undergo a qualitative transition at the critical exponent $s=\tfrac12$. For $s<\tfrac12$, the dominant contribution is driven by interactions at finite planar distance, and the dimension-reduction scale is $\e^2$. In this regime, the limit is a lower-dimensional \emph{fractional} energy with an effective gain of $\tfrac12$ in the differentiability index. At the critical exponent $s=1/2$, the dimension-reduction scale is $\e^{2}|\log\e|$, and the limit is {\em local}, with dominant interactions at scales between $\e$ and $1$, producing a Dirichlet-type limit on $ω$. For $s>\tfrac12$, the dominant contribution is instead driven by interactions at distances of order $\varepsilon$, the dimension-reduction scale is $\e^{3-2s}$, and the second-order $Γ$-limit is still local. We also study the case $s=s_\e\to 1^-$, showing a Bourgain--Brezis--Mironescu-type result.

Dimension reduction of fractional Sobolev seminorms on thin domains

Abstract

We study the asymptotic behaviour of Gagliardo seminorms in defined on thin films . The first relevant order is , at which the corresponding limit captures the vertical fractional oscillations through one-dimensional sections. The second relevant order produces dimension-reduction regimes that undergo a qualitative transition at the critical exponent . For , the dominant contribution is driven by interactions at finite planar distance, and the dimension-reduction scale is . In this regime, the limit is a lower-dimensional \emph{fractional} energy with an effective gain of in the differentiability index. At the critical exponent , the dimension-reduction scale is , and the limit is {\em local}, with dominant interactions at scales between and , producing a Dirichlet-type limit on . For , the dominant contribution is instead driven by interactions at distances of order , the dimension-reduction scale is , and the second-order -limit is still local. We also study the case , showing a Bourgain--Brezis--Mironescu-type result.
Paper Structure (23 sections, 20 theorems, 222 equations)

This paper contains 23 sections, 20 theorems, 222 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Dimension-reduction convergence
  4. Fractional Sobolev spaces
  5. $\Gamma$-limit in the vertical direction
  6. The dimension-reduction regimes
  7. Determination of the scalings
  8. $\Gamma$-limit in the high-integrability case
  9. Strong dimension-reduction compactness
  10. A Bourgain--Brezis--Mironescu-type result for $s\to 1^-$
  11. Convergence for $s_0\in(1/2,1)$
  12. The critical scaling $s=1/2$
  13. $\Gamma$-limit in the low-integrability case
  14. A weak dimension-reduction compactness result
  15. Increase of integrability in dimension reduction
  16. ...and 8 more sections

Key Result

Lemma 2.2

Let $\{u_\varepsilon\}$ be a sequence with $u_\varepsilon\in H^1(\Omega_\varepsilon)$, and suppose that Then, up to the addition of constants, $u_\varepsilon$ is precompact; that is, there exists $c_\varepsilon$ such that $u_\varepsilon+c_\varepsilon$ is precompact, with respect to the di-men-sion-reduction convergence in $L^2$, the limit $u$ belongs to $H^1(\omega)$ and

Theorems & Definitions (46)

  • Definition 2.1: Dimension-reduction convergence
  • Lemma 2.2: Dimension-reduction compactness for local functionals
  • Theorem 2.3: Bourgain--Brezis--Mironescu limit theorem BBMponce
  • Lemma 3.1: Estimate of the seminorm in the thin direction
  • proof
  • Proposition 3.2: Alternative expression for $C_{s,d}$
  • Theorem 3.3: Gamma-limit at the first scaling
  • proof
  • Theorem 4.1: Non-local dimension-reduction compactness
  • proof
  • ...and 36 more