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An \(L^p\)-estimate for weak solutions of elliptic equations. (English) Zbl 1250.35074
Summary: We prove an \(L^p\)-a priori bound, \(p > 2\), for solutions of second order linear elliptic partial differential equations in divergence form with discontinuous coefficients in unbounded domains.

MSC:
35J15 Second-order elliptic equations
35D30 Weak solutions to PDEs
35B45 A priori estimates in context of PDEs
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[1] G. Stampacchia, “Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus,” Annales de l’Institut Fourier, vol. 15, no. 1, pp. 189-258, 1965. · Zbl 0151.15401 · doi:10.5802/aif.204 · numdam:AIF_1965__15_1_189_0 · eudml:73861
[2] O. A. Ladyzhenskaja and N. N. Ural’tzeva, Equations aux Derivèes Partielles de Type Elliptique, Dunod, Paris, France, 1966.
[3] C. Miranda, “Alcune osservazioni sulla maggiorazione in Lv delle soluzioni deboli delle equazioni ellittiche del secondo ordine,” Annali di Matematica Pura ed Applicata. Serie Quarta, vol. 61, pp. 151-169, 1963. · Zbl 0134.09102 · doi:10.1007/BF02412852
[4] M. Chicco, “An a priori inequality concerning elliptic second order partial differential equations of variational type,” Le Matematiche, vol. 26, pp. 173-182, 1971. · Zbl 0234.35023
[5] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224, Springer, Berlin, Germany, 2nd edition, 1983. · Zbl 0562.35001
[6] N. S. Trudinger, “Linear elliptic operators with measurable coefficients,” Annali della Scuola Normale Superiore di Pisa 3, vol. 27, pp. 265-308, 1973. · Zbl 0279.35025 · numdam:ASNSP_1973_3_27_2_265_0 · eudml:83635
[7] G. Bottaro and M. E. Marina, “Problema di Dirichlet per equazioni ellittiche di tipo variazionale su insiemi non limitati,” Bollettino Unione Matematica Italiana Serie 4, vol. 8, pp. 46-56, 1973. · Zbl 0291.35021
[8] M. Transirico and M. Troisi, “Equazioni ellittiche del secondo ordine a coefficienti discontinui e di tipo variazionale in aperti non limitati,” Bollettino Unione Matematica Italiana Serie 7, vol. 2, pp. 385-398, 1988. · Zbl 0691.35033
[9] M. Transirico, M. Troisi, and A. Vitolo, “Spaces of Morrey type and elliptic equations in divergence form on unbounded domains,” Bollettino Unione Matematica Italiana Serie 7, vol. 9, no. 1, pp. 153-174, 1995. · Zbl 0881.35031
[10] P. L. Lions, “Remarques sur les équations linéaires elliptiques du second ordre sous forme divergence dans les domaines non bornés,” Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali. Serie 8, vol. 78, no. 5, pp. 205-212, 1985. · Zbl 0651.35023
[11] P. L. Lions, “Remarques sur les équations linéaires elliptiques du second ordre sous forme divergence dans les domaines non bornés. II,” Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali. Serie 8, vol. 79, no. 6, pp. 178-183, 1985. · Zbl 0656.35030
[12] M. Chicco and M. Venturino, “Dirichlet problem for a divergence form elliptic equation with unbounded coefficients in an unbounded domain,” Annali di Matematica Pura ed Applicata. Serie Quarta, vol. 178, pp. 325-338, 2000. · Zbl 1031.35044 · doi:10.1007/BF02505902
[13] G. Stampacchia, Equations Elliptiques du Second Ordre à Coefficients Discontinus, Séminaire de Mathématiques Supérieures, 4e session, été 1965, Les Presses de l’Université de Montréal, Montreal, Que, 1966. · Zbl 0151.15501
[14] L. Caso, P. Cavaliere, and M. Transirico, “Solvability of the Dirichlet problem in W2,p for elliptic equations with discontinuous coefficients in unbounded domains,” Le Matematiche, vol. 57, no. 2, pp. 287-302, 2002. · Zbl 1085.35055
[15] H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, France, 1983. · Zbl 0511.46001
[16] P. Cavaliere, M. Longobardi, and A. Vitolo, “Imbedding estimates and elliptic equations with discontinuous coefficients in unbounded domains,” Le Matematiche, vol. 51, no. 1, pp. 87-104, 1996. · Zbl 0905.35017
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