Pathological solutions to elliptic problems in divergence form with continuous coefficients

Tianling Jin; Vladimir Maz'ya; Jean Van Schaftingen

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

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

Comptes Rendus. Mathématique, Volume 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 :

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

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     author = {Tianling Jin and Vladimir Maz'ya and Jean Van Schaftingen},
     title = {Pathological solutions to elliptic problems in divergence form with continuous coefficients},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {773--778},
     publisher = {Elsevier},
     volume = {347},
     number = {13-14},
     year = {2009},
     doi = {10.1016/j.crma.2009.05.008},
     language = {en},
}

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%A Vladimir Maz'ya
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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/

Bibliographie
Cité par

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[3] E. De Giorgi Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. (3), Volume 3 (1957), pp. 25-43

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[6] V. Kozlov; V. Maz'ya Asymptotic formula for solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients near the boundary, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), Volume 2 (2003) no. 3, pp. 551-600

[7] V. Kozlov; V. Maz'ya Asymptotics of a singular solution to the Dirichlet problem for an elliptic equation with discontinuous coefficients near the boundary, Teistungen, 2001, Birkhäuser, Basel (2003), pp. 75-115

[8] N.G. Meyers An $L^{p}$ -estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa (3), Volume 17 (1963), pp. 189-206

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