Singular sets of Sobolev functions
[Ensembles singuliers des fonctions de Sobolev]
Comptes Rendus. Mathématique, Tome 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, 1<p<∞, kp<N, pour chaque sous-ensemble A compact de N , dont la « box-dimension » supérieure est plus petite que Nkp, nous construisons une fonction de Sobolev uW k,p ( 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 Nkp.

We are interested in finding Sobolev functions with “large” singular sets. Given N,k, 1<p<∞, kp<N, for any compact subset A of N , such that its upper box dimension is less than Nkp, we construct a Sobolev function uW k,p ( 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 Nkp.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02316-6
Žubrinić, Darko 1

1 Department of Applied Mathematics, Faculty of Electrical Engineering, Unska 3, 10000 Zagreb, Croatia
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Žubrinić, Darko. Singular sets of Sobolev functions. Comptes Rendus. Mathématique, Tome 334 (2002) no. 7, pp. 539-544. doi : 10.1016/S1631-073X(02)02316-6. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02316-6/

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