Comptes Rendus
no. 7
Singular sets of Sobolev functions
[Ensembles singuliers des fonctions de Sobolev]
Darko Žubrinić1
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.

Nous sommes intéressés à trouver des fonctions de Sobolev dont l'ensemble des singularités est « grand ». Étant donné N,k∈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∈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.

We are interested in finding Sobolev functions with “large” singular sets. Given N,k∈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∈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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02316-6
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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