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

Nous construisons une fonction uWloc1,1(B(0,1)), solution de div(Au)=0 au sens des distributions, où A est continu et uWloc1,p(B(0,1)) pour p>1. Nous donnons aussi une fonction uWloc1,1(B(0,1)) telle que uWloc1,p(B(0,1)) pour tout p<, u satisfait div(Au)=0 avec A continu mais uWloc1,(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 uWloc1,1(B(0,1)) which is a solution to div(Au)=0 in the sense of distributions, where A is continuous and uWloc1,p(B(0,1)) for p>1. We also give a function uWloc1,1(B(0,1)) such that uWloc1,p(B(0,1)) for every p<, u satisfies div(Au)=0 with A continuous but uWloc1,(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 :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.05.008

Tianling Jin 1 ; Vladimir Maz'ya 2, 3 ; Jean Van Schaftingen 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
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     title = {Pathological solutions to elliptic problems in divergence form with continuous coefficients},
     journal = {Comptes Rendus. Math\'ematique},
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Tianling Jin; Vladimir Maz'ya; Jean Van Schaftingen. Pathological solutions to elliptic problems in divergence form with continuous coefficients. Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 773-778. doi : 10.1016/j.crma.2009.05.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.05.008/

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