Soluzioni periodiche di PDEs Hamiltoniane
Berti, Massimiliano
Bollettino dell'Unione Matematica Italiana, Tome 7-A (2004), p. 647-661 / Harvested from Biblioteca Digitale Italiana di Matematica
Access to full text
Full (PDF)
Full (DjVu)

Presentiamo nuovi risultati di esistenza e molteplicità di soluzioni periodiche di piccola ampiezza per equazioni alle derivate parziali Hamiltoniane. Otteniamo soluzioni periodiche di equazioni «completamente risonanti» aventi nonlinearità generali grazie ad una riduzione di tipo Lyapunov-Schmidt variazionale ed usando argomenti di min-max. Per equazioni «non risonanti» dimostriamo l'esistenza di soluzioni periodiche di tipo Birkhoff-Lewis, mediante un'opportuna forma normale di Birkhoff e realizzando nuovamente una riduzione di tipo Lyapunov-Schmidt.

New existence and multiplicity results of small amplitude periodic solutions for nonlinear Hamiltonian PDEs are presented. We obtain periodic solutions of «completely resonant» equations with any general nonlinearity thanks to a Lyapunov-Schmidt reduction, variational in nature, and min-max topological arguments. For «non resonant» equations we prove existence of periodic solutions of Birkhoff-Lewis type, by means of a suitable Birkhoff normal form and implementing again a Lyapunov-Schmidt variational reduction.

Publié le : 2004-10-01
@article{BUMI_2004_8_7B_3_647_0,
     author = {Massimiliano Berti},
     title = {Soluzioni periodiche di PDEs Hamiltoniane},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {7-A},
     year = {2004},
     pages = {647-661},
     zbl = {1182.35165},
     mrnumber = {2101656},
     language = {it},
     url = {http://dml.mathdoc.fr/item/BUMI_2004_8_7B_3_647_0}
}

Berti, Massimiliano. Soluzioni periodiche di PDEs Hamiltoniane. Bollettino dell'Unione Matematica Italiana, Tome 7-A (2004) pp. 647-661. http://gdmltest.u-ga.fr/item/BUMI_2004_8_7B_3_647_0/

[1] Ambrosetti, A.-Coti Zelati, V.-Ekeland, I., Symmetry breaking in Hamiltonian systems, Journal Diff. Equat., 67 (1987), 165-184. | MR 879691 | Zbl 0606.58043

[2] Ambrosetti, A.-Rabinowitz, P., Dual Variational Methods in Critical Point Theory and Applications, Journ. Func. Anal, 14 (1973), 349-381. | MR 370183 | Zbl 0273.49063

[3] Bambusi, D., Lyapunov Center Theorems for some nonlinear PDEs: a simple proof, Ann. Sc. Norm. Sup. di Pisa, Ser. IV, vol. XXIX, fasc. 4, 2000. | Zbl 1008.35003

[4] Bambusi, D.-Berti, M., A Birkhoof-Lewis type Theorem for some Hamiltonian PDEs, preprint SISSA, available at http://www.math.utexas.edu/mp-arc. | MR 2176924 | Zbl 1105.37045

[5] Bambusi, D., Birkhoff normal form for some nonlinear PDEs, Commun. Math. Phys., 234 (2003), 253-285. | MR 1962462 | Zbl 1032.37051

[6] Bambusi, D.-Grebert, B., Forme normale pour NLS en dimension quelconque, C.R. Acad. Sci. Paris Ser., 1, 337 (2003), 409-414. | MR 2015085 | Zbl 1030.35143

[7] Bambusi, D.-Paleari, S., Families of periodic solutions of resonant PDEs, J. Nonlinear Sci., 11 (2001), 69-87. | MR 1819863 | Zbl 0994.37040

[8] Bambusi, D.-Paleari, S., Families of periodic orbits for some PDE's in higher dimensions, Comm. Pure and Appl. Analysis, Vol. 1, n. 4, 2002. | MR 1938615 | Zbl 1034.35081

[9] Biasco, L.-Chierchia, L.-Valdinoci, E., Elliptic two-dimensional invariant tori for the planetary three-body problem, 170, n. 2 (2003), 91-135. | MR 2017886 | Zbl 1036.70006

[10] Berti, M.-Biasco, L.-Valdinoci, E., Periodic orbits close to elliptic tori and applications to the three body problem, to appear on Ann. Sc. Norm. Sup. di Pisa, 2004. | MR 2064969 | Zbl 1121.37047

[11] Berti, M.-Bolle, P., Periodic solutions of Nonlinear wave equations with general nonlineairties, Commun. Math. Phys., 243 (2003), 315-328. | MR 2021909 | Zbl 1072.35015

[12] Berti, M.-Bolle, P., Multiplicity of periodic solutions of Nonlinear wave equations, Nonlinear Analysis, TMA, 56 n. 7 (2004), 1011-1046. | MR 2038735 | Zbl 1064.35119

[13] Birkhoof, G. D.-Lewis, D. C., On the periodic motions near a given periodic motion of a dynamical system, Ann. Mat., 12 (1934), 117-133. | JFM 59.0733.05 | MR 1553217

[14] Bourgain, J., Construction of periodic solutions of nonlinear wave equations in higher dimension, Geom. and Func. Anal., vol. 5, n. 4, 1995. | MR 1345016 | Zbl 0834.35083

[15] Bourgain, J., Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations, Ann. of Math., 148 (1998), 363-439. | MR 1668547 | Zbl 0928.35161

[16] Chierchia, L.-You, J., KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Comm. Math. Phys., 211, no. 2 (2000), 497-525. | MR 1754527 | Zbl 0956.37054

[17] Conley, C.-Zehnder, E., An index theory for periodic solutions of a Hamiltonian system, Lecture Notes in Mathematics1007, Springer, 1983, 132-145. | MR 730268 | Zbl 0528.34043

[18] Craig, W., Problèmes de petits diviseurs dans les équations aux dérivées partielles, Panoramas et Synthèses, 9, Société Mathématique de France, Paris, 2000. | MR 1804420 | Zbl 0977.35014

[19] Craig, W.-Wayne, E., Newton's method and periodic solutions of nonlinear wave equation, Comm. Pure and Appl. Math, vol. XLVI (1993), 1409-1498. | MR 1239318 | Zbl 0794.35104

[20] Craig, W.-Wayne, E., Nonlinear waves and the 1:1:2 resonance, Singular limits of dispersive waves (Lyon, 1991), 297-313, NATO Adv. Sci. Inst. Ser. B Phys., 320, Plenum, New York, 1994. | MR 1321211 | Zbl 0849.35133

[21] Fadell, E. R.-Rabinowitz, P., Generalized cohomological index theories for the group actions with an application to bifurcations question for Hamiltonian systems, Inv. Math., 45 (1978), 139-174. | MR 478189 | Zbl 0403.57001

[22] Gentile, G.-Mastropietro, V., Construction of periodic solutions of the nonlinear wave equation with Dirichlet boundary conditions by the Lindstedt series method, to appear on Journal Math. Pures Appl. | MR 2082491 | Zbl 1065.35028

[23] Lewis, D. C., Sulle oscillazioni periodiche di un sistema dinamico, Atti Acc. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat., 19 (1934), 234-237. | Zbl 0009.08903

[24] Lyapunov, A. M., Problème général de la stabilité du mouvement, Ann. Sc. Fac. Toulouse, 2 (1907), 203-474. | MR 21186

[25] Lidskij, B. V.-Shulman, E. I., Periodic solutions of the equation ut⁢t-ux⁢x+u3=0, Funct. Anal. Appl., 22 (1980), 332-333. | Zbl 0837.35012

[26] Kuksin, S. B., Perturbation of conditionally periodic solutions of infinite-dimensional Hamiltonian systems, Izv. Akad. Nauk SSSR, Ser. Mat.52, no. 1 (1988), 41-63. | MR 936522 | Zbl 0662.58036

[27] Moser, J., Periodic orbits near an Equilibrium and a Theorem by Alan Weinstein, Comm. on Pure and Appl. Math., vol. XXIX, 1976. | MR 426052 | Zbl 0346.34024

[28] Moser, J., Proof of a generalized form of a fixed point theorem due to G. D. Birkhoof, Geometry and Topology, Lectures Notes in Math., 597 (1977), 464-494. | MR 494305 | Zbl 0358.58009

[29] Poincaré, H., Les Méthodes nouvelles de la Mécanique Céleste, Gauthier Villars, Paris, 1892. | JFM 30.0834.08

[30] Pöschel, J., A KAM-Theorem for some nonlinear PDEs, Ann. Scuola Norm. Sup. Pisa, Cl. Sci., 23 (1996), 119-148. | MR 1401420 | Zbl 0870.34060

[31] Pöschel, J., On the construction of almost periodic solutions for a nonlinear Schrödinger equation, Ergodic Theory Dynam. Systems, 22 (2002), 1537-1549. | MR 1934149 | Zbl 1020.37044

[32] Rabinowitz, P., Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, 65. | MR 845785 | Zbl 0609.58002

[33] Wayne, E., Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Commun. Math. Phys., 127, no. 3 (1998), 479-528. | MR 1040892 | Zbl 0708.35087

[34] Weinstein, A., Normal modes for Nonlinear Hamiltonian Systems, Inv. Math, 20 (1973), 47-57. | MR 328222 | Zbl 0264.70020