Analysis of some injection bounds for Sobolev spaces by wavelet decomposition

Silvia Bertoluzza; Silvia Falletta

doi:10.1016/j.crma.2011.02.015

Harmonic Analysis/Functional Analysis

Analysis of some injection bounds for Sobolev spaces by wavelet decomposition
[Analyse de quelques bornes dʼinjection des espaces de Sobolev, en utilisant la décomposition par ondelettes]

Présenté par : Yves Meyer

Silvia Bertoluzza ¹ ; Silvia Falletta ²

¹ IMATI-CNR, V. Ferrata 1, 27100 Pavia, Italy
² Dip. Matematica, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino, Italy

Comptes Rendus. Mathématique, Volume 349 (2011) no. 7-8, pp. 421-423.

Résumé
Abstract

On considère les espaces de Sobolev $H^{s} (Ω)$ et $H_{0}^{s} (Ω)$ , et lʼespace de Besov $B_{2, \infty}^{1 / 2} (Ω)$ , ou Ω est un domaine suffisamment régulier (voir Lemme 2) de $R^{2}$ . On sait que pour des valeurs de $s \in [0, 1 / 2)$ les deux espaces de Sobolev coïncident, avec équivalence des normes, et quʼon a lʼinclusion $B_{2, \infty}^{1 / 2} (Ω) \subset H^{s} (Ω)$ . Cet article donne une analyse explicite des constantes qui apparaissent dans les bornes dʼinclusion $H^{s} (Ω) ↪ H_{0}^{s} (Ω)$ and $B_{2, \infty}^{1 / 2} (Ω) ↪ H^{s} (Ω)$ et, plus précisément, de leur dépendance du paramètre de régularité s. On utilise pour cela la caractérisation par ondelettes des normes correspondantes.

We consider the Sobolev spaces $H^{s} (Ω)$ and $H_{0}^{s} (Ω)$ and the Besov spaces $B_{2, \infty}^{1 / 2} (Ω)$ , where Ω is a sufficiently regular (see Lemma 2) subdomain of $R^{2}$ . It is well known that for the values of $s \in [0, 1 / 2)$ the two Sobolev spaces coincide, with equivalence of the norms, and that the inclusion $B_{2, \infty}^{1 / 2} (Ω) \subset H^{s} (Ω)$ holds. The Note is concerned with the explicit analysis of the constants appearing in the continuity bounds for the injections $H^{s} (Ω) ↪ H_{0}^{s} (Ω)$ and $B_{2, \infty}^{1 / 2} (Ω) ↪ H^{s} (Ω)$ and of their dependence on the regularity s of the spaces. The analysis is carried out by using the wavelet characterization of the corresponding norms.

Reçu le : 2010-12-22
Accepté le : 2011-02-14
Publié le : 2011-02-25

DOI : 10.1016/j.crma.2011.02.015

Affiliations des auteurs :

Silvia Bertoluzza ¹ ; Silvia Falletta ²

¹ IMATI-CNR, V. Ferrata 1, 27100 Pavia, Italy
² Dip. Matematica, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino, Italy

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     author = {Silvia Bertoluzza and Silvia Falletta},
     title = {Analysis of some injection bounds for {Sobolev} spaces by wavelet decomposition},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {421--423},
     publisher = {Elsevier},
     volume = {349},
     number = {7-8},
     year = {2011},
     doi = {10.1016/j.crma.2011.02.015},
     language = {en},
}

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%A Silvia Falletta
%T Analysis of some injection bounds for Sobolev spaces by wavelet decomposition
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Silvia Bertoluzza; Silvia Falletta. Analysis of some injection bounds for Sobolev spaces by wavelet decomposition. Comptes Rendus. Mathématique, Volume 349 (2011) no. 7-8, pp. 421-423. doi : 10.1016/j.crma.2011.02.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.02.015/

Bibliographie
Cité par

[1] S. Bertoluzza Substructuring preconditioners for the three fields domain decomposition method, Math. Comp., Volume 73 (2003) no. 246, pp. 659-689

[2] S. Bertoluzza; F. Brezzi; G. Sangalli The method of mothers for non-overlapping non-matching DDM, Numer. Math., Volume 107 (2007) no. 3, pp. 397-431

[3] S. Bertoluzza, S. Falletta, Analysis of some injection bounds for Besov spaces by using wavelet schemes, Technical report, 2011.

[4] A. Cohen Numerical Analysis of Wavelet Methods, Studies in Mathematics and Its Applications, vol. 32, North-Holland, Amsterdam, 2003

[5] S. Falletta, Analysis of the mortar method with approximate integration: the effect of cross-points, Technical report 1298, I.A.N., 2002.

[6] J.L. Lions; E. Magenes Non Homogeneous Boundary Value Problems and Applications, vol. I, Springer, 1972

[7] H. Triebel Interpolation Theory, Function Spaces, Differential Operators, Heidelberg, 1995

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