We construct a function which is a solution to in the sense of distributions, where A is continuous and for . We also give a function such that for every , u satisfies with A continuous but . This answers questions raised by H. Brezis (On a conjecture of J. Serrin, Rend. Lincei Mat. Appl. 19 (2008) 335–338).
Nous construisons une fonction , solution de au sens des distributions, où A est continu et pour . Nous donnons aussi une fonction telle que pour tout , u satisfait avec A continu mais . Ceci répond à des questions souleveées par H. Brezis (On a conjecture of J. Serrin, Rend. Lincei Mat. Appl. 19 (2008) 335–338).
Accepted:
Published online:
Tianling Jin 1; Vladimir Maz'ya 2, 3; Jean Van Schaftingen 4
@article{CRMATH_2009__347_13-14_773_0, 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}, }
TY - JOUR AU - Tianling Jin AU - Vladimir Maz'ya AU - Jean Van Schaftingen TI - Pathological solutions to elliptic problems in divergence form with continuous coefficients JO - Comptes Rendus. Mathématique PY - 2009 SP - 773 EP - 778 VL - 347 IS - 13-14 PB - Elsevier DO - 10.1016/j.crma.2009.05.008 LA - en ID - CRMATH_2009__347_13-14_773_0 ER -
%0 Journal Article %A Tianling Jin %A Vladimir Maz'ya %A Jean Van Schaftingen %T Pathological solutions to elliptic problems in divergence form with continuous coefficients %J Comptes Rendus. Mathématique %D 2009 %P 773-778 %V 347 %N 13-14 %I Elsevier %R 10.1016/j.crma.2009.05.008 %G en %F CRMATH_2009__347_13-14_773_0
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/
[1] Elliptic operators, conormal derivatives and positive parts of functions, with an appendix by H. Brezis, J. Funct. Anal. (2009) | DOI
[2] On a conjecture of J. Serrin, Rend. Lincei Mat. Appl., Volume 19 (2008), pp. 335-338
[3] 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
[4] A regularity theorem for linear second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa (3), Volume 26 (1972), pp. 283-290
[5] On functions of bounded mean oscillation, Comm. Pure Appl. Math., Volume 14 (1961), pp. 415-426
[6] 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] 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] An -estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa (3), Volume 17 (1963), pp. 189-206
[9] Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa (3), Volume 18 (1964), pp. 385-387
[10] Note on the class , Studia Math., Volume 32 (1969), pp. 305-310
[11] Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993
Cited by Sources:
Comments - Policy