Superposition with subunitary powers in Sobolev spaces

Petru Mironescu

doi:10.1016/j.crma.2015.03.020

Mathematical analysis

Superposition with subunitary powers in Sobolev spaces

Presented by: Haïm Brézis

Petru Mironescu ¹

¹Université de Lyon, Université Lyon-1, CNRS UMR 5208, Institut Camille-Jordan, 43, bd du 11-Novembre-1918, 69622 Villeurbanne cedex, France

Comptes Rendus. Mathématique, Volume 353 (2015) no. 6, pp. 483-487

Abstract (Original language)
Résumé

Let $0 < a < 1$ and set $Φ (t) = {| t |}^{a}$ , $\forall t \in R$ . Let $1 < p < \infty$ and $n \geq 1$ . We prove that the superposition operator $u \mapsto Φ (u)$ maps the Sobolev space $W^{1, p} (R^{n})$ into the fractional Sobolev space $W^{a, p / a} (R^{n})$ . We also investigate the case of more general nonlinearities.

Pour $0 < a < 1$ , soit $Φ (t) = {| t |}^{a}$ , $\forall t \in R$ . Soient $1 < p < \infty$ et $n \geq 1$ . Nous montrons que l'opérateur de superposition $u \mapsto Φ (u)$ envoie l'espace de Sobolev $W^{1, p} (R^{n})$ dans l'espace de Sobolev fractionnaire $W^{a, p / a} (R^{n})$ . Nous examinons aussi la superposition avec des non-linéarités plus générales.

Received: 2014-09-29
Accepted: 2015-03-09
Published online: 2015-04-24

DOI: 10.1016/j.crma.2015.03.020

Author's affiliations:

Petru Mironescu ¹

¹ Université de Lyon, Université Lyon-1, CNRS UMR 5208, Institut Camille-Jordan, 43, bd du 11-Novembre-1918, 69622 Villeurbanne cedex, France

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     author = {Petru Mironescu},
     title = {Superposition with subunitary powers in {Sobolev} spaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {483--487},
     year = {2015},
     publisher = {Elsevier},
     volume = {353},
     number = {6},
     doi = {10.1016/j.crma.2015.03.020},
     language = {en},
}

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Petru Mironescu. Superposition with subunitary powers in Sobolev spaces. Comptes Rendus. Mathématique, Volume 353 (2015) no. 6, pp. 483-487. doi: 10.1016/j.crma.2015.03.020

References
Cited by

[1] G. Bourdaud; Y. Meyer Fonctions qui opèrent sur les espaces de Sobolev, J. Funct. Anal., Volume 97 (1991) no. 2, pp. 351-360

[2] H. Triebel Theory of Function Spaces, Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel, Switzerland, 1983

[3] L. Véron, personal communication.

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