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Comptes Rendus Mathématique
Volume 334, n° 7
pages 539-544 (2002)
Doi : S1631-073X(02)02316-6
Received : 21 December 2001 ;  accepted : 4 February 2002
Singular sets of Sobolev functions
Ensembles singuliers des fonctions de Sobolev
 

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

Note presented by Paul Malliavin

Abstract

We are interested in finding Sobolev functions with “large” singular sets. Given N,kN, 1<p <∞, kp <N , for any compact subset A of RN, such that its upper box dimension is less than Nkp , we construct a Sobolev function uWk,p(RN) 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 . To cite this article: D. Žubrinić, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 539-544.

The full text of this article is available in PDF format.
Résumé

Nous sommes intéressés à trouver des fonctions de Sobolev dont l'ensemble des singularités est « grand ». Étant donné N,kN, 1<p <∞, kp <N , pour chaque sous-ensemble A compact de RN, dont la « box-dimension » supérieure est plus petite que Nkp , nous construisons une fonction de Sobolev uWk,p(RN) 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 . Pour citer cet article : D. Žubrinić, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 539-544.

The full text of this article is available in PDF format.


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