Singular sets of Sobolev functions

Darko Žubrinić

doi:10.1016/S1631-073X(02)02316-6

Singular sets of Sobolev functions
[Ensembles singuliers des fonctions de Sobolev]

Darko Žubrinić ¹

¹ Department of Applied Mathematics, Faculty of Electrical Engineering, Unska 3, 10000 Zagreb, Croatia

Comptes Rendus. Mathématique, Volume 334 (2002) no. 7, pp. 539-544.

Abstract (VO)
Résumé

We are interested in finding Sobolev functions with “large” singular sets. Given $N, k \in ℕ$ , 1<p<∞, kp<N, for any compact subset A of $ℝ^{N}$ , such that its upper box dimension is less than N−kp, we construct a Sobolev function $u \in W^{k, p} (ℝ^{N})$ which is singular precisely on A. We introduce the notions of lower and upper singular dimensions of Sobolev space, and show that both are equal to N−kp.

Nous sommes intéressés à trouver des fonctions de Sobolev dont l'ensemble des singularités est « grand ». Étant donné $N, k \in ℕ$ , 1<p<∞, kp<N, pour chaque sous-ensemble A compact de $ℝ^{N}$ , dont la « box-dimension » supérieure est plus petite que N−kp, nous construisons une fonction de Sobolev $u \in W^{k, p} (ℝ^{N})$ qui est singulière précisément sur A. Nous introduisons les notions de dimensions singulières inférieure et supérieure de l'espace de Sobolev, et montrons que ses valeurs sont N−kp.

Reçu le : 2001-12-21
Accepté le : 2002-02-04
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02316-6

Affiliations des auteurs :

Darko Žubrinić ¹

¹ Department of Applied Mathematics, Faculty of Electrical Engineering, Unska 3, 10000 Zagreb, Croatia

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Darko Žubrinić. Singular sets of Sobolev functions. Comptes Rendus. Mathématique, Volume 334 (2002) no. 7, pp. 539-544. doi : 10.1016/S1631-073X(02)02316-6. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02316-6/

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