Degenerate Elliptic Equations and Morrey Spaces
Borrello, Francesco
Bollettino dell'Unione Matematica Italiana, Tome 10-A (2007), p. 989-1011 / Harvested from Biblioteca Digitale Italiana di Matematica
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In this paper we study local regularity for the generalized solution to the Dirichlet problem related to the equation L⁢u≡Xi*⁢(ai⁢j⁢Xj⁢u)=f. where X1,X2,…,Xm are vector fields satisfying Hörmander condition and ai⁢j∈L∞. We give a representation formula for the generalized solution in terms of the Green function and thanks to suitable estimates we achieve our goal. In the case f≥0 we are able to give necessary condition too.

In questo articolo viene studiata la regolarità locale per la soluzione generalizzata del problema di Dirichlet relativo all'equazione L⁢u≡Xi*⁢(ai⁢j⁢Xj⁢u)=f. dove X1,X2,…,Xm sono campi vettoriali soddisfacenti la condizione di Hörmander e ai⁢j∈L∞. Viene data una formula di rappresentazione per la soluzione generalizzata in termini di funzione di Green. I risultati sono ottenuti grazie a opportune stime di quest'ultima. Nel caso in cui f≥0 i teoremi provati sono invertibili.

Publié le : 2007-10-01
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     author = {Francesco Borrello},
     title = {Degenerate Elliptic Equations and Morrey Spaces},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {10-A},
     year = {2007},
     pages = {989-1011},
     zbl = {1184.35143},
     mrnumber = {2507910},
     language = {en},
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Borrello, Francesco. Degenerate Elliptic Equations and Morrey Spaces. Bollettino dell'Unione Matematica Italiana, Tome 10-A (2007) pp. 989-1011. http://gdmltest.u-ga.fr/item/BUMI_2007_8_10B_3_989_0/

[1] Baldi, A., A non-existence problem for degenerate elliptic PDE's, Comm. Partial Differential Equations, Communications in Partial Differential Equations, 25, 7-8 (2000), 1371-1398. | MR 1765139

[2] Cancelier, C. - Xu, C-J., Remarques sur les fonctions de Green associées aux opérateurs de Hörmander, C. R. Acad. Sci. Paris Sér. I Math., 330, 6 (2000), 433-436. | MR 1756954 | Zbl 0949.35023

[3] Capogna, L. - Danielli, D. - Garofalo, N., An embedding theorem and the Harnack inequality for nonlinear subelliptic equations, Comm. Partial Differential Equations, 18, 9-10 (1993), 1765-1794. | MR 1239930 | Zbl 0802.35024

[4] Chen, Y., Regularity of solutions to the Dirichlet problem for degenerate elliptic equation, Chinese Ann. Math. Ser. B, 24, 4 (2003), 529-540. | MR 2024992 | Zbl 1046.35019

[5] Chow, W.-L., Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann., 117 (1939), 98-105. | MR 1880 | Zbl 65.0398.01

[6] Citti, G. - Garofalo, N. - Lanconelli, E., Harnack's inequality for sum of squares of vector fields plus a potential, Amer. J. Math., 115, 3 (1993), 699-734. | MR 1221840 | Zbl 0795.35018

[7] Danielli, D., A Fefferman-Phong type inequality and applications to quasilinear subelliptic equations, Potential Analysis, 11 (1999), 387-413. | MR 1719837 | Zbl 0940.35057

[8] Di Fazio, G., Poisson equations and Morrey spaces, J. Math. Anal. Appl., 163, 1 (1992), 157-167. | MR 1144713 | Zbl 0780.35020

[9] Di Fazio, G., Dirichlet problem characterization of regularity, Manuscripta Math., 84, 1 (1994), 47-56. | MR 1283326 | Zbl 0816.35009

[10] Di Fazio, G. - Zamboni, P., Hölder continuity for quasilinear subelliptic equations in Carnot Carathéodory spaces, Math. Nachr., 272 (2004), 3-10. | MR 2079757 | Zbl 1149.35347

[11] 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. (4), 10, 4 (1983), 523-541. | MR 753153 | Zbl 0552.35032

[12] Franchi, B. - Serapioni, R. - Serra Cassano, F., Approximation and imbedding theorems for weighted Sobolev spaces associated with Lipschitz continuous vector fields, Boll. Un. Mat. Ital. B (7), 11, 1 (1997), 83-117. | MR 1448000 | Zbl 0952.49010

[13] Franchi, B. - Wheeden, R. L., Compensation couples and isoperimetric estimates for vector fields, Colloq. Math., 74, 1 (1997), 9-27. | MR 1455453 | Zbl 0915.46028

[14] Grüter, M. - Widman, K.-O., The Green function for uniformly elliptic equations, Manuscripta Math., 37, 3 (1982), 303-342. | MR 657523 | Zbl 0485.35031

[15] Kohn, J. J. - Nirenberg, L., Non-coercive boundary value problems, Comm. Pure Appl. Math., 18 (1965), 443-492. | MR 181815 | Zbl 0125.33302

[16] Littman, W. - Stampacchia, G. - Weinberger, H. F., Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa (3), 17 (1963), 43-77. | MR 161019 | Zbl 0116.30302

[17] Lu, G., Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander's condition and applications, Rev. Mat. Iberoamericana, 8, 3 (1992), 367-439. | MR 1202416 | Zbl 0804.35015

[18] Nagel, A. - Stein, E. M. - Wainger, S., Balls and metrics defined by vector fields. I. Basic properties, Acta Math., 155, 1-2 (1985), 103-147. | MR 793239 | Zbl 0578.32044

[19] Sánchez-Calle, A., Fundamental solutions and geometry of the sum of squares of vector fields, Invent. Math., 78, 1 (1984), 143-160. | MR 762360

[20] Stampacchia, G., Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15, 1 (1965), 189-258. | MR 192177 | Zbl 0151.15401