Local regularity of solutions to quasilinear subelliptic equations in Carnot Caratheodory spaces

Di Fazio, Giuseppe; Zamboni, Pietro

Bollettino dell'Unione Matematica Italiana, Tome 9-A (2006), p. 485-504 / Harvested from Biblioteca Digitale Italiana di Matematica

Access to full text
Full (PDF)
Full (DjVu)

Abstract
Résumé

We prove Harnack inequality for weak solutions to quasilinear subelliptic equation of the following kind ${}\sum_{J=1}^{m}X_{j}^{*}A_{j}(x,u(x),Xu(x))+B(x,u(x),Xu(x))=0,$ where $X_{1},\ldots,X_{m}$ are a system of non commutative locally Lipschitz vector fields. As a consequence, the weak solutions of (*) are continuous.

In questa nota proviamo la disuguaglianza di Harnack per le soluzioni deboli di una equazione sub-ellittica quasilineare del tipo ${}\sum_{J=1}^{m}X_{j}^{*}A_{j}(x,u(x),Xu(x))+B(x,u(x),Xu(x))=0,$ dove $X_{1},\ldots,X_{m}$ denotano un sistema non commutativo di campi vettoriali localmente lipschitziani. Come conseguenza otteniamo la continuità delle soluzioni deboli della (*).

Publié le : 2006-06-01

MR 2233147 | Zbl 1178.35163

@article{BUMI_2006_8_9B_2_485_0,
     author = {Giuseppe Di Fazio and Pietro Zamboni},
     title = {Local regularity of solutions to quasilinear subelliptic equations in Carnot Caratheodory spaces},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {9-A},
     year = {2006},
     pages = {485-504},
     zbl = {1178.35163},
     mrnumber = {2233147},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2006_8_9B_2_485_0}
}

Di Fazio, Giuseppe; Zamboni, Pietro. Local regularity of solutions to quasilinear subelliptic equations in Carnot Caratheodory spaces. Bollettino dell'Unione Matematica Italiana, Tome 9-A (2006) pp. 485-504. http://gdmltest.u-ga.fr/item/BUMI_2006_8_9B_2_485_0/

Bibliographie

[1] Buckley, S., Inequalities of John Nirenberg type in doubling spaces, J. Anal. Math., 79 (1999), 215-240. | MR 1749313 | Zbl 0990.46019

[2] Capogna, L. - Danielli, D. - Garofalo, N., An embedding theorem and the Harnack inequality for nonlinear subelliptic equations, Comm. P.D.E., 18 (1993), 1765-1794. | MR 1239930 | Zbl 0802.35024

[3] Chiarenza, F., Regularity for solutions of quasilinear elliptic equations under minimal assumptions, Potential Analysis, 4 (1995), 325-334. | MR 1354887 | Zbl 0838.35022

[4] Chiarenza, F. - Fabes, E. - Garofalo, N., Harnack's inequality for Schrödinger operators and continuity of solutions, Proc. A.M.S., 98 (1986), 415-425. | MR 857933 | Zbl 0626.35022

[5] Danielli, D., A Fefferman-Phong type inequality and applications to quasilinear subelliptic equations, Potential Analysis, 115 (1999), 387-413. | MR 1719837 | Zbl 0940.35057

[6] Danielli, D. - Garofalo, N. - Nhieu, D., Trace inequalities for Carnot-Caratheodory spaces and applications, Ann. Scuola Norm. Sup. Pisa, 4 (1998), 195-252. | MR 1664688 | Zbl 0938.46036

[7] Di Fazio, G. - Zamboni, P., A Fefferman-Poincaré type inequality for Carnot-Carathéodory vector fields, Proc. A.M.S., 130 (2002), 2655-2660. | MR 1900873 | Zbl 1031.46038

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

[9] Ladyzhenskaya, O.A. - Uralceva, N., Linear and quasilinear elliptic equations, Academic Press (1968).

[10] Lieberman, G., Sharp forms of Estimates for Subsolutions and Supersolutions of Quasilinear Elliptic Equations Involving Measures, Comm. P.D.E., 18 (1993), 1191-1212. | MR 1233190 | Zbl 0802.35041

[11] Ragusa, M.A. - Zamboni, P., Local regularity of solutions to quasilinear elliptic equations with general structure, Communications in Applied Analysis, 3 (1999), 131-147. | MR 1669745 | Zbl 0922.35050

[12] Rakotoson, J.M., Quasilinear equations and Spaces of Campanato-Morrey type, Comm. P.D.E., 16 (1991), 1155-1182. | MR 1116857 | Zbl 0827.35021

[13] Rakotoson, J.M. - Ziemer, W.P., Local behavior of solutions of quasilinear elliptic equations with general structure, Trans. A. M. S., 319 (1990), 747-764. | MR 998128 | Zbl 0708.35023

[14] Serrin, J., Local behavior of solutions of quasilinear equations, Acta Math., 111 (1964), 247-302. | MR 170096 | Zbl 0128.09101

[15] Serrin, J. - Zou, H., Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189, n. 1 (2002), 79-142. | MR 1946918 | Zbl 1059.35040

[16] Zamboni, P., Harnack's inequality for quasilinear elliptic equations with coefficients in Morrey spaces, Rend. Sem. Mat. Univ. Padova, 89 (1993), 87-96. | MR 1229045 | Zbl 0802.35043

[17] Zamboni, P., Local boundedness of solutions of quasilinear elliptic equations with coefficients in Morrey spaces, Boll. Un. Mat. It., 8-B (1994), 985-997. | MR 1315830 | Zbl 0827.35040

[18] Zamboni, P., Local behavior of solutions of quasilinear elliptic equations with coefficients in Morrey Spaces, Rendiconti di Matematica, Serie VII, 15 (1995), 251-262. | MR 1339243 | Zbl 0832.35046

[19] Zamboni, P., Unique continuation for non-negative solutions of quasilinear elliptic equations, Bull. Austral. Math. Soc., 64 (2001), 149-156. | MR 1848087 | Zbl 0989.47037

[20] Zamboni, P.The Harnack inequality for quasilinear elliptic equations under minimal assumptions, Manuscripta Math., 102 (2000), 311-323. | MR 1777522 | Zbl 0954.35063