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Asymptotic analysis of fractional Sobolev spaces on thin films in the low-integrability regime

Andrea Braides, Andrea Pinamonti, Margherita Solci

Abstract

We study the behaviour of fractional Sobolev spaces $H^s(Ω_\varepsilon)$ with $s\in(0,1/2)$ defined on ``thin films'' $Ω_\varepsilon=ω\times (0,\varepsilon)$ in $\mathbb R^d$, and prove that they tend to the space $H^{s+\frac12}(ω)$ as $\varepsilon\to 0$. This is made precise by using a notion of dimension-reduction convergence, with respect to which suitably scaled Gagliardo seminorms define equicoercive functionals. Asymptotic results are proved for $s\to 0^+$ and $s\to 1/2^-$.

Asymptotic analysis of fractional Sobolev spaces on thin films in the low-integrability regime

Abstract

We study the behaviour of fractional Sobolev spaces with defined on ``thin films'' in , and prove that they tend to the space as . This is made precise by using a notion of dimension-reduction convergence, with respect to which suitably scaled Gagliardo seminorms define equicoercive functionals. Asymptotic results are proved for and .

Paper Structure

This paper contains 7 sections, 10 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Dimension-reduction compactness
  4. Analysis at scale $\varepsilon^2$
  5. Dimensional-reduction regimes
  6. Limit energies defined on full-dimensional sets
  7. Analysis at the critical scale for pointwise convergence

Key Result

Lemma 3.1

There exists $C>0$ such that for all $\varepsilon>0$, $s\in(0,1)$, and $u\in H^s(\Omega_\varepsilon)$ we have where $v\colon\omega\times (0,1)\to \mathbb R$ is the scaled function given by $v(x^\prime, t)=u(x^\prime,\varepsilon t)$.

Theorems & Definitions (22)

  • Lemma 3.1: slicing on the "thin" direction
  • proof
  • Remark 3.2: rigidity
  • Definition 3.3: (weak) dimension-reduction convergence
  • Lemma 3.4: dimensional-reduction compactness -- i
  • proof
  • Lemma 3.5: dimensional-reduction compactness -- ii
  • proof
  • Theorem 4.1
  • proof
  • ...and 12 more