Pathological solutions to elliptic problems in divergence form with continuous coefficients

Jin, Tianling; Maz'ya, Vladimir; Van Schaftingen, Jean

doi:10.1016/j.crma.2009.05.008

Partial Differential Equations

Pathological solutions to elliptic problems in divergence form with continuous coefficients
[Solutions pathologiques de problèmes elliptiques sous forme divergence à coefficients continus]

Présenté par : Haïm Brezis

Jin, Tianling ¹ ; Maz'ya, Vladimir ^2,³ ; Van Schaftingen, Jean ⁴

¹ Rutgers University, Department of Mathematics, 110, Frelinghuysen Road, Piscataway, NJ 08854-8019, USA
² University of Liverpool, Department of Mathematical Sciences, Liverpool L69 3BX, UK
³ Linköping University, Department of Mathematics, 581 83 Linköping, Sweden
⁴ Université catholique de Louvain, département de mathématique, chemin du cyclotron 2, B-1348 Louvain-la-Neuve, Belgium

Comptes Rendus. Mathématique, Tome 347 (2009) no. 13-14, pp. 773-778.

Résumé
Abstract

Nous construisons une fonction $u \in W_{loc}^{1, 1} (B (0, 1))$ , solution de $div (A \nabla u) = 0$ au sens des distributions, où A est continu et $u \notin W_{loc}^{1, p} (B (0, 1))$ pour $p > 1$ . Nous donnons aussi une fonction $u \in W_{loc}^{1, 1} (B (0, 1))$ telle que $u \in W_{loc}^{1, p} (B (0, 1))$ pour tout $p < \infty$ , u satisfait $div (A \nabla u) = 0$ avec A continu mais $u \notin W_{loc}^{1, \infty} (B (0, 1))$ . Ceci répond à des questions souleveées par H. Brezis (On a conjecture of J. Serrin, Rend. Lincei Mat. Appl. 19 (2008) 335–338).

We construct a function $u \in W_{loc}^{1, 1} (B (0, 1))$ which is a solution to $div (A \nabla u) = 0$ in the sense of distributions, where A is continuous and $u \notin W_{loc}^{1, p} (B (0, 1))$ for $p > 1$ . We also give a function $u \in W_{loc}^{1, 1} (B (0, 1))$ such that $u \in W_{loc}^{1, p} (B (0, 1))$ for every $p < \infty$ , u satisfies $div (A \nabla u) = 0$ with A continuous but $u \notin W_{loc}^{1, \infty} (B (0, 1))$ . This answers questions raised by H. Brezis (On a conjecture of J. Serrin, Rend. Lincei Mat. Appl. 19 (2008) 335–338).

Reçu le : 2009-04-06
Accepté le : 2009-05-11
Publié le : 2009-06-05

DOI : 10.1016/j.crma.2009.05.008

Affiliations des auteurs :

Jin, Tianling ¹ ; Maz'ya, Vladimir ^{2, 3} ; Van Schaftingen, Jean ⁴

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     title = {Pathological solutions to elliptic problems in divergence form with continuous coefficients},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {773--778},
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Jin, Tianling; Maz'ya, Vladimir; Van Schaftingen, Jean. Pathological solutions to elliptic problems in divergence form with continuous coefficients. Comptes Rendus. Mathématique, Tome 347 (2009) no. 13-14, pp. 773-778. doi : 10.1016/j.crma.2009.05.008. http://archive.numdam.org/articles/10.1016/j.crma.2009.05.008/

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