Differential forms in Carnot groups: a variational approach

Annalisa Baldi

Abstract


Carnot groups (connected simply connected nilpotent stratified Lie groups) can be endowed with a complex of ``intrinsic'' differential forms. In this paper we want to provide an evidence of the intrinsic character of Rumin's complex, in the spirit of the Riemannian approximation, like in e.g., the notes of Gromov (Textes Mathématiques 1981) and in Rumin (Geom. Funct. Anal.,2000) . More precisely, we want to show that the intrinsic differential is a limit of suitably weighted usual first order de Rham differentials. As an application, we prove that the L^2-energies associated to classical Maxwell's equations in R^n Gamma-converges to the L^2-energies associated to an ''intrinsic'' Maxwell's equation in a free Carnot group.


Keywords


Carnot groups; differential forms; gamma-convergence

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References


Giovanni Alberti. Variational models for phase transitions, an approach via gamma-convergence. In Calculus of variations and partial dierential equations (Pisa, 1996), pages 95{114. Springer, Berlin, 2000.

Annalisa Baldi and Bruno Franchi. Dierential forms in Carnot groups: a gamma-convergence approach. To appear on Calc. Var. DOI 10.1007/s00526-011-0409-8, 2011.

Annalisa Baldi and Bruno Franchi. A variational approach to maxwell equations in Carnot groups. In preparation, 2011.

Annalisa Baldi, Bruno Franchi, Nicoletta Tchou, and Maria Carla Tesi. Compensated compactness for dierential forms in Carnot groups and applications. Adv. Math., 223(5):1555{1607, 2010.

Andrea Bonglioli, Ermanno Lanconelli, and Francesco Uguzzoni. Stratied Lie groups and potential theory for their sub-Laplacians. Springer Monographs in Mathematics. Springer, Berlin, 2007.

Nicolas Bourbaki. Elements de mathematique. XXVI. Groupes et algebres de Lie. Chapitre 1: Algebres de Lie. Actualites Sci. Ind. No. 1285. Hermann, Paris, 1960.

Gianni Dal Maso. An introduction to gamma-convergence. Progress in Nonlinear Differential Equations and their Applications, 8. Birkhauser Boston Inc., Boston, MA, 1993.

Ennio De Giorgi and Tullio Franzoni. Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 58(6):842{850, 1975.

Herbert Federer. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969.

Gerald B. Folland. Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mat., 13(2):161-207, 1975.

Gerald B. Folland and Elias M. Stein. Hardy spaces on homogeneous groups, volume 28 of Mathematical Notes. Princeton University Press, Princeton, N.J., 1982.

Bruno Franchi and Maria Carla Tesi. Wave and Maxwell's Equations in Carnot Groups. preprint, 2010.

Mariano Giaquinta, Giuseppe Modica, and JirSoucek. Cartesian currents in the calculus of variations. I, volume 37 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 1998.

Matthew Grayson and Robert Grossman. Models for free nilpotent Lie algebras. J. Algebra, 135(1):177-191, 1990.

Mikhael Gromov. Structures metriques pour les varietes riemanniennes, volume 1 of Textes Mathematiques [Mathematical Texts]. CEDIC, Paris, 1981. Edited by J.Lafontaine and P. Pansu.

Luciano Modica and Stefano Mortola. Un esempio di gamma-convergenza. Boll. Un. Mat. Ital. B (5), 14(1):285-299, 1977.

Linda Preiss Rothschild and Elias M. Stein. Hypoelliptic dierential operators and nilpotent groups. Acta Math., 137(3-4):247-320, 1976.

Michel Rumin. Formes dierentielles sur les varietes de contact. J. Dierential Geom., 39(2):281-330, 1994.

Michel Rumin. Dierential geometry on C-C spaces and application to the Novikov-Shubin numbers of nilpotent Lie groups. C. R. Acad. Sci. Paris Ser. I Math., 329(11):985-990, 1999.

Michel Rumin. Sub-Riemannian limit of the dierential form spectrum of contact manifolds. Geom. Funct. Anal., 10(2):407-452, 2000.

Michel Rumin. Around heat decay on forms and relations of nilpotent Lie groups. In Seminaire de Theorie Spectrale et Geometrie, Vol. 19, Annee 2000-2001, volume 19 of Semin. Theor. Spectr. Geom., pages 123-164. Univ. Grenoble I, Saint, 2001.

K^osaku Yosida. Functional analysis, volume 123 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, sixth edition, 1980.




DOI: 10.6092/issn.2240-2829/2664

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