Singular sets of Sobolev functions

Žubrinić, Darko

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 $\mathrm{N},\mathrm{k}\in \mathbb{N}$, 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 $\mathrm{u}\in {\mathrm{W}}^{\mathrm{k},\mathrm{p}}\left({ℝ}^N\right)$ 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 $\mathrm{N},\mathrm{k}\in \mathbb{N}$, 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 $\mathrm{u}\in {\mathrm{W}}^{\mathrm{k},\mathrm{p}}\left({ℝ}^N\right)$ 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.

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