Abstract.
We study the Green function associated to second order X-elliptic operators with non-regular coefficients. An existence and uniqueness theorem is given, along with some local estimations and a representation formula for the solution of the Dirichlet problem.
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Bony, J.-M.: Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés. Ann. Inst. Fourier Grenoble 19-1, 277–304 (1969)
Cancelier, C., Xu, C.J.: Fonction de Green pour des opérateurs elliptiques dégénérés a coefficients non réguliers. Preprint
Cancelier, C., Xu, C.J.: Remarques sur les fonctions de Green associées aux opérateurs de Hörmander. C. R. Acad. Sci. Paris, t. 330 (Série I), 433–436 (2000)
Chanillo, S., Wheeden, R.L.: Existence and estimates of Green’s function for degenerate elliptic equations. Ann. Scuola Norm. Sup. Pisa 15, 309–340 (1988)
Franchi, B., Lanconelli, E.: Une métrique associée à une classe d’opérateurs elliptiques dégénérés. In: Proceedings of the meeting “Linear Partial and Pseudo Differential Operators” Torino, Rend. Sem. Mat. Univ. e Politec. Torino, 105–114 (1982)
Franchi, B., Lanconelli, E.: Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10 (4), 523–541 (1983)
Franchi, B., Lanconelli, E.: An embedding theorem for Sobolev Spaces related to non-smooth vector fields and Harnack inequality. Comm. Partial Diff. Eqs. 9 (13), 1237–1264 (1984)
Franchi, B., Serapioni, R., Serra Cassano, F.: Meyers-Serrin type theorems and relaxation of variational integrals depending on vector fields. Houston J. Math. 22 (4), 859–890 (1996)
Franchi, B., Serapioni, R., Serra Cassano, F.: Approximation and imbedding theorems for weighted Sobolev Spaces associated with Lipschitz continuous vector fields. Bollettino U.M.I. 11-B (7), 83–117 (1997)
Friedrichs, K.O.: The identity of weak and strong extensions of differential operators. Trans. Am. Math. Soc. 55, 132–151 (1944)
Garofalo, N., Nhieu, D.-M.: Isoperimetric and Sobolev Inequalities for Carnot-Carathéodory Spaces and the existence of minimal surfaces. Commun. Pure Appl. Math. XLIX, 1081–1144 (1996)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Berlin-Heidelberg-New York: Springer-Verlag, 1998
Gutiérrez, C.E., Lanconelli, E.: Maximum principle, non-homogeneous Harnack inequality, and Liouville theorems for X-elliptic operators. Comm. Partial Diff. Eqs. 28 (11–12), 1833–1862 (2003)
Grüter, C.E., Widman, K.O.: The Green function for uniformly elliptic equations. Manuscripta Math. 37, 303–342 (1982)
John F., Nirenberg, L.: On functions of Bounded Mean Oscillation. Commun. Pure Appl. Math. XIV, 415–426 (1961)
Lanconelli, E.: Alcuni aspetti dell’Analisi Reale negli spazi omogenei. Regolarizzazione delle soluzioni deboli delle equazioni P-ellittiche. Seminario di Analisi Matematica A.A. 1995/96, Dip. Mat. Bologna 53–71 (1996)
Lanconelli, E., Morbidelli, D.: On the Poincarés ineqality for vector fields. Ark. Mat. 38, 327–342 (2000)
Littman, W., Stampacchia, G., Weinberger, H.F.: Regular points for elliptic equations with discontinuous coefficients. Ann. Scuola Norm. Sup. Pisa (III) 17, 43–77 (1963)
Moser, J.: A new proof of de Giorgi’s Theorem concerning the regularity problem for Elliptic Differential Equations. Commun. Pure Appl. Math. XIII, 457–468 (1960)
Moser, J.: On Harnack’s Theorem for Elliptic Differential Equations. Commun. Pure Appl. Math. XIV, 577–591 (1961)
Montanari, A., Morbidelli, D.: Balls defined by nonsmooth vector fields and the Poincarés inequality. Preprint (2003)
Salinas, O.: Harnack inequality and Green function for a certain class of degenerate elliptic differential operators. Revista Matemática Iberoamericana 7 (3), 313–349 (1991)
Acknowledgments.
It is a pleasure to thank Ermanno Lanconelli for his encouragement to my work on this problem.
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Mathematics Subject Classification (2000): 34B27, 35H99, 35J70, 35C15
Revised version: 28 June 2004
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Mazzoni, G. Green function for X-elliptic operators. manuscripta math. 115, 207–238 (2004). https://doi.org/10.1007/s00229-004-0494-5
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DOI: https://doi.org/10.1007/s00229-004-0494-5