Strutture subriemanniane in alcuni problemi di Analisi
Lanconelli, Ermanno
Bollettino dell'Unione Matematica Italiana, Tome 8-A (2005), p. 273-298 / Harvested from Biblioteca Digitale Italiana di Matematica
Access to full text
Full (PDF)
Full (DjVu)

Vengono presentati alcuni problemi, idee e tecniche sorte nell'ambito della teoria delle equazioni alle derivate parziali del secondo ordine, con forma caratteristica semidefinita positiva e con soggiacenti strutture sub-riemanniane. Se ne traccia lo sviluppo a partire dalla classica teoria delle funzioni armoniche e caloriche, attraverso la teoria del potenziale negli spazi armonici astratti e la teoria della regolarità locale delle soluzioni.

We present some problems, ideas and techniques arising in the theory of Partial Differential Equations of Second Order with non-negative characteristic form and with underlying sub-riemannian structures. We show their development starting from the basic properties of classical harmonic and caloric functions. We stress their relationship with abstract potential theory and local regularity theory of solutions.

Publié le : 2005-06-01
@article{BUMI_2005_8_8B_2_273_0,
     author = {Ermanno Lanconelli},
     title = {Strutture subriemanniane in alcuni problemi di Analisi},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {8-A},
     year = {2005},
     pages = {273-298},
     zbl = {1182.31013},
     mrnumber = {2149385},
     language = {it},
     url = {http://dml.mathdoc.fr/item/BUMI_2005_8_8B_2_273_0}
}

Lanconelli, Ermanno. Strutture subriemanniane in alcuni problemi di Analisi. Bollettino dell'Unione Matematica Italiana, Tome 8-A (2005) pp. 273-298. http://gdmltest.u-ga.fr/item/BUMI_2005_8_8B_2_273_0/

[1] Amano, K., Maximum Principle for degenerate elliptic-parabolic operators, Ind. Univ. Math. J., 28, no. 4 (1979), 545-557. | MR 542943 | Zbl 0423.35023

[2] Bedford, E. - Gaveau, B., Hypersurfaces with bounded Levi form, Ind. Univ. Mat. J, 27, no. 5 (1978), 867-873. | MR 499287 | Zbl 0365.32011

[3] Bony, J. M., Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptic dégénérés, Ann. Inst. Fourier (Grenoble), 19 (1969), 277-304. | MR 262881 | Zbl 0176.09703

[4] Biroli, M. - Mosco, U., Sobolev and isoperimetric inequalities for Dirichlet forms on homogeneous spaces, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 6 (1995), 37-44. | MR 1340280 | Zbl 0837.31006

[5] Costantinescu, C. - Cornea, A., Potential Theory on Harmonic Spaces, Springer-Verlag Berlin Heidelberg New York 1972. | MR 419799 | Zbl 0248.31011

[6] Citti, G., C∞-regularity of solutions of the Levi equation, Ann. Inst. H. Poincaré, Anal. Non Linaire, 15, no. 4 (1998), 517-534. | MR 1632929 | Zbl 0921.35033

[7] Citti, G. - Lanconelli, E. - Montanari, A., Smoothness of Lipschitz continuous graphs with non vanishing Levi curvature, Acta Mathematica, 188 (2002), 87-128. | MR 1947459 | Zbl 1030.35084

[8] Citti, G. - Montanari, A., C∞-regularity of solutions of an equation of Levi’s type in R2⁢N+1, Ann. Mat. Pura Appl., 180 (2001), 27-58. | MR 1848050 | Zbl 1030.35019

[9] De Giorgi, E., Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 3, 6 (1957), 25-43. | MR 93649 | Zbl 0084.31901

[10] Derridj, M., Une problème aux limite pour une classe d’opérateur du second ordre hypoelliptiques, Ann. Inst. Fourier (Grenoble), 21 (1971), 99-148. | MR 601055 | Zbl 0215.45405

[11] Fichera, G., Sulle equazioni differenziali lineari ellitico-paraboliche del secondo ordine, Atti Accad, Naz. Lincei Mem. Cl. Sci. Fis. Mat. Nat., 5 (1956), 1-30. | MR 89348 | Zbl 0075.28102

[12] Folland, G. B., Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161-207. | MR 494315 | Zbl 0312.35026

[13] Franchi, B. - Lanconelli, E., De Giorgi's Theorem for a Class of Strongly Degenerate Elliptic Equations, Atti Accad. Naz. Lincei Rend Cl. Sci. Fis. Mat. Natur., 72, no. 5 (1983), 273-277. | MR 728257 | Zbl 0543.35041

[14] Franchi, B. - Lanconelli, E., Une métrique associée à une classe d’opérateurs elliptiques dégénerés, Proceedings of the meeting «Linear Partial and Pseudo Differential Operators», Torino (1982), Rend. Sem. Mat. Univ. e Politec. Special Issue (1984), 105-114. | MR 745979 | Zbl 0553.35033

[15] 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, no. 6 (1983), 523-541. | MR 753153 | Zbl 0552.35032

[16] Franchi, B. - Lu, G. - Wheeden, R., A relationship between Poincaré-type inequalities and representation formulas in spaces of homogeneous type, Internat. Math. Res. Notices (1996), 1-14. | MR 1383947 | Zbl 0856.43006

[17] Grigor'Yan, A., The heat equation on non-compact Riemannian maniflods, Matem. Sbornik, 182 (1991), 55-87. Engl. Transl. Math USSR Sb., 72 (1992), 47-77. | Zbl 0776.58035

[18] Garofalo, N. - Nhieu, D. H., Isoperimetric and Sobolev inequalty for Carnot-Carathéodory spaces and the existence of minimal sufaces, Comm. Pure Appl. Math., 10 (1996), 1153-1196. | Zbl 0880.35032

[19] Hadamard, J., Extension à l'èquation de la chaleur d'un théorème de A. Harnack, Rend. Circ. Mat. Palermo, 3, no. 2 (1954), 337-346. | MR 68713 | Zbl 0058.32201

[20] Hajlasz, P. - Koskela, P., Sobolev met Poincaré, Memoirs Amer. Math. Soc., 688 (2000). | MR 1683160 | Zbl 0954.46022

[21] Hormander, L., Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171. | MR 222474 | Zbl 0156.10701

[22] Howe, R., On the rôle of the Heisenberg group in harmonic analysis, Bull. Amer. Math. Soc., 3, no. 2 (1980), 821-843. | MR 578375 | Zbl 0442.43002

[23] Jerison, D., The Poincaré inequality for vector fields satisfying Hormander condition, Duke Math. J., 53 (1986), 503-523. | MR 850547 | Zbl 0614.35066

[24] Jerison, D. - Lee, J. M., Intrinsic CR normal coordinates and the CR Yamabe problem, J. Amer. Math. Soc., 1 (1988), 1-13. | MR 982177 | Zbl 0634.32016

[25] Jerison, D. - Lee, J. M., Extremals of the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Diff. Geom., 29 (1989), 303-343. | MR 982177 | Zbl 0671.32016

[26] Kolmogorov, A., Zur Theorie der Brownschen Bewegung, Ann. Math., II Ser., 35 (1934), 116-117. | MR 1503147 | Zbl 0008.39906

[27] Lanconelli, E., Regolaritá hölderiana delle soluzioni deboli di certe equazioni ellittiche fortemente degeneri, Seminario di Analisi Matematica Ist. Mat. Univ. Bologna (1981-1982), IX.1.-IX.27.

[28] Lanconelli, E. - Kogoj, A. E., X-Elliptic Operators and X-Control Distances, Ricerche di Matematica, Napoli, II Special Issue in Memory of Ennio De Giorgi, 38 (2000), 223-243. | MR 1826225 | Zbl 1029.35102

[29] Lanconelli, E. - Morbidelli, D., On the Poincaré inequality for vector fields, Arkiv för Matematik, 38 (2000), 327-342. | MR 1785405 | Zbl 1131.46304

[30] Lanconelli, E. - Polidoro, S., On a class of hypoelliptic evolution operators, Rend. Sem. Mat. Univ. Pol. Torino, 52, no. 1 (1994), 29-63. | MR 1289901 | Zbl 0811.35018

[31] Lanconelli, E. - Pascucci, A. - Polidoro, S., Linear and nonlinear ultraparabolic equations of Kolmogorov type arising in diffusion theory and in finance, in «Nonlinear Problems in Mathematical Physics and Related Topics Vol. II In Honor of Professor O. A. Ladyzhenskaya» International Mathematical Series Kluwer Ed. (2002), 243-265. | MR 1972000 | Zbl 1032.35114

[32] Lascialfari, F. - Montanari, A., Smooth regularity for solutions of the Levi Monge-Ampère equation, Rend. Mat. Acc. Naz. Lincei, 12 (2001), 115-123. | MR 1898454 | Zbl 1097.35061

[33] Malchiodi, A. - Uguzzoni, F., A perturbation result for the Webster scalar curvature problem on the CR sphere, J. Math. Pures Appl., 81 (2002), 983-997. | MR 1946912 | Zbl 1042.53025

[34] Montanari, A. - Lanconelli, E., Pseudoconvex Fully Nonlinear Partial Differential Operators. Strong Comparison Theorems, J. Differential Equations, 202 (2004), 306-331. | MR 2068443 | Zbl 1161.35414

[35] Montanari, A. - Morbidelli, D., Balls defined by nonsmooth vector fields and the Poincaré inequality, Annales Inst. Fourier (Grenoble), 54, 2 (2004), 327-339. | MR 2073841 | Zbl 1069.46504

[36] Moser, J., On Harnack’s Theoreme for Elliptic Differential equations, Comm. Pure Appl. Math., 24, 6 (1961), 577-591. | MR 159138 | Zbl 0111.09302

[37] Moser, J., On a pointwise Estimate for Parabolic Differential Equations, Comm. Pure Appl. Math., 24 (1971), 727-740. | MR 288405 | Zbl 0227.35016

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

[39] Nirenberg, L., A strong maximum principle for parabolic equations, Comm. Pure Appl. Math., 6, no. 6 (1953), 167-177. | MR 55544 | Zbl 0050.09601

[40] Oleinik, O. A. - Radkevic, E. V., Second Order Equations With Nonnegative Characteristic Form, American Mathematical Society, Providence, Rhode Island (1973). Translated from Itogi Nauki-Serija Matematika (1971). | MR 457907

[41] Picone, M., Maggiorazioni degli integrali di equazioni lineari ellittico-paraboliche alle derivate parziali del secondo ordine, Rend. Accad. Naz. Lincei., 5, no. 6 (1927), 138-143.

[42] Picone, M., Maggiorazione degli integrali delle equazioni totalmente paraboliche del secodno ordine, Annali Mat. Pura Appl., 74 (1929), 145-192. | MR 1553141

[43] Picone, M., Nuove formule di maggiorazione per gli integrali delle equazioni lineari alle derivate parziali del secondo ordine ellittico-paraboliche, Atti. Accad. Naz. Lincei, 28, no. 6 (1938), 331-338. | Zbl 0020.35903

[44] Pini, B., Sulla soluzione generalizzata di Wiener per il primo problema di valori al contorno nel caso parabolico, Rend. Sem. Mat. Univ. Padova, 23 (1954), 422-434. | MR 65794 | Zbl 0057.32801

[45] Phillips, R. - Sarason, L., Elliptic-parabolic equations of the second order, J. Math. Mech., 17 (1967/1968), 891-917. | MR 219868 | Zbl 0163.34402

[46] Polidoro, S. - Pascucci, A., The Moser iterative method for a class of ultraparabolic equations, Commun. Contemp. Math., 6, 2 (2004), 1-23. | MR 2068847 | Zbl 1096.35080

[47] Rampazzo, F. - Sussmann, H. J., Set-valued differentials and a nonsmooth version of Chow’s theorem, Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, Florida, 2001, IEEE Publications, New York, 3 (2001), 2613- 2618.

[48] Rothschild, L. P. - Stein, E. M., Hypoelliptic differential operators and nilpotent groups, Acta Math., 137 (1976), 247-320. | MR 436223 | Zbl 0346.35030

[49] Saloff-Coste, L., A note on Poincaré, Sobolev and Harnack inequality, Internat. Math. Res. Notices (1992), 27-38. | MR 1150597 | Zbl 0769.58054

[50] Saloff-Coste, L., Aspects of Sobolev-type inequalities, London Math. Society, Lectures Note Series289, Cambridge University Press (2002). | MR 1872526 | Zbl 0991.35002

[51] Slodkowki, Z. - Tomassini, G., Weak solutions for the Levi equation and envelope of holomorphy, Ind. Univ. Mat. J., 101, no. 2 (1991), 392-407. | MR 1136942 | Zbl 0744.35015

[52] Slodkowski, Z. - Tomassini, G., The Levi equation in higher dimension and relationships to the envelope of holomorphy, Amer. J. of Math., 116 (1994), 479- 499. | MR 1269612 | Zbl 0802.35050

[53] Stein, E. M., Harmonic Analisys: real variable methods, orthogonality and oscillatory integral, Priceton Universty Press, Princeton, N.J (1978). | Zbl 0821.42001

[54] Stampacchia, G., Le Probléme de Dirichlet pour les équations elliptiqes du second ordre á coefficients discontinues, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258. | MR 192177 | Zbl 0151.15401

[55] Tomassini, G., Geometric properties of the solutions of the Levi equation, Rend. Sem. Mat. Fis. Milano, 57 (1987), 103-108. | MR 1017922 | Zbl 0688.32012

[56] Tomassini, G., Geometric properties of solutions of the Levi Equation, Ann. Mat. Pura Appl., 27, no. 5 (1988), 331-334. | MR 980986 | Zbl 0681.35017

[57] Tomassini, G., Nonlinear equations related to the Levi form, Rend. Circ. Mat. Palermo, 40 (1991), 281-297. | MR 1151589 | Zbl 0836.35056