Partial Differential Equations
Pathological solutions to elliptic problems in divergence form with continuous coefficients
Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 773-778.

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).

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).

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2009.05.008
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
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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, Volume 347 (2009) no. 13-14, pp. 773-778. doi : 10.1016/j.crma.2009.05.008. http://www.numdam.org/articles/10.1016/j.crma.2009.05.008/

[1] Ancona, A. Elliptic operators, conormal derivatives and positive parts of functions, with an appendix by H. Brezis, J. Funct. Anal. (2009) | DOI

[2] Brezis, H. On a conjecture of J. Serrin, Rend. Lincei Mat. Appl., Volume 19 (2008), pp. 335-338

[3] De Giorgi, E. 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] Hager, R.A.; Ross, J. A regularity theorem for linear second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa (3), Volume 26 (1972), pp. 283-290

[5] John, F.; Nirenberg, L. On functions of bounded mean oscillation, Comm. Pure Appl. Math., Volume 14 (1961), pp. 415-426

[6] Kozlov, V.; Maz'ya, V. 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] Kozlov, V.; Maz'ya, V. 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] Meyers, N.G. An Lp-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] Serrin, J. Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa (3), Volume 18 (1964), pp. 385-387

[10] Stein, E.M. Note on the class LlogL, Studia Math., Volume 32 (1969), pp. 305-310

[11] Stein, E.M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993

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