Limiting behaviour of intrinsic seminorms in fractional order Sobolev spaces

Rémi Arcangéli¹ ; Juan José Torrens²

¹ Retired Professor Route de Barat 31160 Arbas, France
² Departamento de Ingeniería Matemática e Informática Universidad Pública de Navarra Campus de Arrosadía 31006 Pamplona, Spain

Studia Mathematica, Tome 214 (2013) no. 2, pp. 101-120

Résumé

We collect and extend results on the limit of $\sigma^{1-k}(1-\sigma)^{k}|v|_{l+\sigma,p,\varOmega}^p$ as $\sigma\to0^{+}$ or $\sigma\to1^{-}$, where $\varOmega$ is $\mathbb{R}^n$ or a smooth bounded domain, $k\in\{0,1\}$, $l\in\mathbb{N}$, $p\in[1,\infty)$, and $|\,\cdot\,|_{l+\sigma,p,\varOmega}$ is the intrinsic seminorm of order $l+\sigma$ in the Sobolev space $W^{l+\sigma,p}(\varOmega)$. In general, the above limit is equal to $c[v]^p$, where $c$ and $[\,\cdot\,]$ are, respectively, a~constant and a seminorm that we explicitly provide. The particular case $p=2$ for $\varOmega=\mathbb{R}^n$ is also examined and the results are then proved by using the Fourier transform.

Détail

DOI : 10.4064/sm214-2-1

Keywords: collect extend results limit sigma k sigma sigma varomega sigma sigma where varomega mathbb smooth bounded domain mathbb infty cdot sigma varomega intrinsic seminorm order sigma sobolev space sigma varomega general above limit equal where cdot respectively constant seminorm explicitly provide particular varomega mathbb examined results proved using fourier transform