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

TL;DR

This paper analyzes how fractional Sobolev spaces on thin films behave as the film thickness approaches zero, revealing convergence to higher-order Sobolev spaces and providing asymptotic results for the fractional parameter.

Contribution

It establishes the limit behavior of fractional Sobolev spaces on thin domains as the thickness tends to zero, introducing a dimension-reduction convergence framework.

Findings

Fractional Sobolev spaces converge to higher-order spaces as thickness vanishes.

Scaled Gagliardo seminorms are equicoercive under the new convergence.

Asymptotic behavior characterized for fractional order approaching zero and one-half.

Abstract

We study the behaviour of fractional Sobolev spaces $H^s(\Omega_\varepsilon)$ with $s\in(0,1/2)$ defined on ``thin films'' $\Omega_\varepsilon=\omega\times (0,\varepsilon)$ in $\mathbb R^d$, and prove that they tend to the space $H^{s+\frac12}(\omega)$ 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 $s\to 1/2^-$.