Berti, Massimiliano:
Soluzioni periodiche di PDEs Hamiltoniane
Bollettino dell'Unione Matematica Italiana Serie 8 7-B (2004), fasc. n.3, p. 647-661, Unione Matematica Italiana (Italian)
pdf (283 Kb), djvu (231 Kb). | MR2101656 | Zbl 1182.35165
Sunto
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.
Referenze Bibliografiche
[1]
A. AMBROSETTI-
V. COTI ZELATI-
I. EKELAND,
Symmetry breaking in Hamiltonian systems,
Journal Diff. Equat.,
67 (
1987), 165-184. |
MR 879691 |
Zbl 0606.58043[2]
A. AMBROSETTI-
P. RABINOWITZ,
Dual Variational Methods in Critical Point Theory and Applications,
Journ. Func. Anal,
14 (
1973), 349-381. |
MR 370183 |
Zbl 0273.49063[3]
D. BAMBUSI,
Lyapunov Center Theorems for some nonlinear PDEs: a simple proof,
Ann. Sc. Norm. Sup. di Pisa, Ser. IV, vol.
XXIX, fasc. 4,
2000. |
fulltext mini-dml |
Zbl 1008.35003[5]
D. BAMBUSI,
Birkhoff normal form for some nonlinear PDEs,
Commun. Math. Phys.,
234 (
2003), 253-285. |
MR 1962462 |
Zbl 1032.37051[6]
D. BAMBUSI-
B. GREBERT,
Forme normale pour NLS en dimension quelconque,
C.R. Acad. Sci. Paris Ser., 1,
337 (
2003), 409-414. |
MR 2015085 |
Zbl 1030.35143[7]
D. BAMBUSI-
S. PALEARI,
Families of periodic solutions of resonant PDEs,
J. Nonlinear Sci.,
11 (
2001), 69-87. |
MR 1819863 |
Zbl 0994.37040[8]
D. BAMBUSI-
S. PALEARI,
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]
L. BIASCO-
L. CHIERCHIA-
E. VALDINOCI,
Elliptic two-dimensional invariant tori for the planetary three-body problem,
170, n. 2 (
2003), 91-135. |
MR 2017886 |
Zbl 1036.70006[10]
M. BERTI-
L. BIASCO-
E. VALDINOCI,
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]
M. BERTI-
P. BOLLE,
Periodic solutions of Nonlinear wave equations with general nonlineairties,
Commun. Math. Phys.,
243 (
2003), 315-328. |
MR 2021909 |
Zbl 1072.35015[12]
M. BERTI-
P. BOLLE,
Multiplicity of periodic solutions of Nonlinear wave equations,
Nonlinear Analysis, TMA,
56 n. 7 (
2004), 1011-1046. |
MR 2038735 |
Zbl 1064.35119[13]
G. D. BIRKHOOF-
D. C. LEWIS,
On the periodic motions near a given periodic motion of a dynamical system,
Ann. Mat.,
12 (
1934), 117-133. |
MR 1553217 |
Jbk 59.0733.05[14]
J. BOURGAIN,
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]
J. BOURGAIN,
Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations,
Ann. of Math.,
148 (
1998), 363-439. |
MR 1668547 |
Zbl 0928.35161[16]
L. CHIERCHIA-
J. YOU,
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]
C. CONLEY-
E. ZEHNDER,
An index theory for periodic solutions of a Hamiltonian system,
Lecture Notes in Mathematics 1007,
Springer,
1983, 132-145. |
MR 730268 |
Zbl 0528.34043[18]
W. CRAIG,
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]
W. CRAIG-
E. WAYNE,
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]
W. CRAIG-
E. WAYNE,
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]
E. R. FADELL-
P. RABINOWITZ,
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]
G. GENTILE-
V. MASTROPIETRO,
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]
D. C. LEWIS,
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]
A. M. LYAPUNOV,
Problème général de la stabilité du mouvement,
Ann. Sc. Fac. Toulouse,
2 (
1907), 203-474. |
MR 21186[25]
B. V. LIDSKIJ-
E. I. SHULMAN,
Periodic solutions of the equation $u_{tt} - u_{xx} + u^3 = 0$,
Funct. Anal. Appl.,
22 (
1980), 332-333. |
Zbl 0837.35012[26]
S. B. KUKSIN,
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]
J. MOSER,
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]
J. MOSER,
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]
H. POINCARÉ,
Les Méthodes nouvelles de la Mécanique Céleste,
Gauthier Villars, Paris,
1892. |
Jbk 30.0834.08[31]
J. PÖSCHEL,
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]
P. RABINOWITZ,
Minimax methods in critical point theory with applications to differential equations,
CBMS Regional Conference Series in Mathematics,
65. |
MR 845785 |
Zbl 0609.58002[34]
A. WEINSTEIN,
Normal modes for Nonlinear Hamiltonian Systems,
Inv. Math,
20 (
1973), 47-57. |
MR 328222 |
Zbl 0264.70020
Con il contributo del Ministero per i Beni e le Attività Culturali