We consider the Kirchhoff equation for a vibrating body, in any dimension, in the presence of a time-periodic external forcing with period and amplitude . We treat both Dirichlet and space-periodic boundary conditions, and both analytic and Sobolev regularity. We prove the existence, regularity and local uniqueness of time-periodic solutions, using a Nash-Moser iteration scheme. The results hold for parameters in a Cantor set with asymptotically full measure as .
@article{ASNSP_2009_5_8_1_117_0, author = {Baldi, Pietro}, title = {Periodic solutions of forced {Kirchhoff} equations}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {117--141}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 8}, number = {1}, year = {2009}, mrnumber = {2512203}, zbl = {1180.35040}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2009_5_8_1_117_0/} }
TY - JOUR AU - Baldi, Pietro TI - Periodic solutions of forced Kirchhoff equations JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2009 SP - 117 EP - 141 VL - 8 IS - 1 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2009_5_8_1_117_0/ LA - en ID - ASNSP_2009_5_8_1_117_0 ER -
%0 Journal Article %A Baldi, Pietro %T Periodic solutions of forced Kirchhoff equations %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2009 %P 117-141 %V 8 %N 1 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2009_5_8_1_117_0/ %G en %F ASNSP_2009_5_8_1_117_0
Baldi, Pietro. Periodic solutions of forced Kirchhoff equations. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 8 (2009) no. 1, pp. 117-141. http://archive.numdam.org/item/ASNSP_2009_5_8_1_117_0/
[1] Asymptotic behaviour as of the solutions of linear hyperbolic equations with coefficients discontinuous in time (on a bounded domain), J. Differential Equations 39 (1981), 291–309. | MR | Zbl
,[2] Averaged evolution equations. The Kirchhoff string and its treatment in scales of Banach spaces, In: “Functional Analytic Methods in Complex Analysis and Applications to Partial differential Equations (Trieste, 1993)”, World Sci. Publ., River Edge, NJ, 1995, 220–254. | Zbl
,[3] Global solutions of the Cauchy problem for a nonlinear hyperbolic equation, In: “Nonlinear PDE’s and their Applications”, H. Brezis and J. L. Lions (eds.), Collége de France Seminar, Vol. VI, 1–26, Research Notes Math. Vol. 109, Pitman, Boston, 1984. | Zbl
and ,[4] On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc. 348 (1996), 305–330. | MR | Zbl
and ,[5] Forced vibrations of a nonhomogeneous string, SIAM J. Math. Anal. 40 (2008), 382–412. | MR | Zbl
and ,[6] Sur une classe d’équations fonctionelles aux dérivées partielles, Izv. Akad. Nauk SSSR Ser. Mat. 4 (1940), 17–26. | MR | Zbl
,[7] “Nonlinear Oscillations of Hamiltonian PDEs”, Progr. Nonlinear Differential Equations Appl. 74, H. Brézis ed., Birkhäuser, Boston, 2008. | MR
,[8] Cantor families of periodic solutions for completely resonant nonlinear wave equations, Duke Math. J. 134 (2006), 359–419. | MR | Zbl
and ,[9] Cantor families of periodic solutions of wave equations with nonlinearities, NoDEA Nonlinear Differential Equations Appl. 15 (2008), 247–276. | MR | Zbl
and ,[10] Sobolev periodic solutions of nonlinear wave equations in higher spatial dimensions, Arch. Ration. Mech. Anal., to appear. | MR | Zbl
and ,[11] Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE, Internat. Math. Res. Notices 11 (1994), 475–497. | MR | Zbl
,[12] Construction of periodic solutions of nonlinear wave equations in higher dimension, Geom. Funct. Anal. 5 (1995), 629–639. | EuDML | MR | Zbl
,[13] Quasi-periodic solutions of Hamiltonian perturbation of 2D linear Schrödinger equations, Ann. of Math. (2) 148 (1998), 363–439. | MR | Zbl
,[14] Periodic solutions of nonlinear wave equations, In: “Harmonic Analysis and Partial Differential Equations”, Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL, 1999, 69–97. | MR | Zbl
,[15] On the nonlinear vibration problem of the elastic string, Quart. Appl. Math. 3 (1945), 157–165. | MR | Zbl
,[16] A note on the vibrating string, Quart. Appl. Math. 7 (1949), 97–101. | MR | Zbl
,[17] “Problèmes de Petits Diviseurs dans les Équations aux Dérivées Partielles”, Panoramas et Synthèses, 9, Société Mathématique de France, Paris, 2000. | MR
,[18] Newton’s method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math. 46 (1993), 1409–1501. | MR | Zbl
and ,[19] A class of nonlinear hyperbolic problems with global solutions, Arch. Ration. Mech. Anal. 124 (1993), 201–219. | MR | Zbl
and ,[20] Infinite systems of nonlinear oscillation equations related to the string, Proc. Amer. Math. Soc. 23 (1969), 459–468. | MR | Zbl
,[21] Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order, Proc. Japan Acad. 43 (1967), 82–86. | MR | Zbl
,[22] The initial value problem for a stretched string, Quart. Appl. Math. 38 (1980/81), 289–311. | MR | Zbl
and ,[23] “Vorlesungen über Mathematische Physik: Mechanik”, Teubner, Leipzig, 1876, Ch. 29. | JFM
,[24] “Nearly Integrable Infinite-Dimensional Hamiltonian Systems”, Lecture Notes in Math. 1556, Springer, Berlin, 1993. | MR | Zbl
,[25] On some questions in boundary value problems of mathematical physics, In: “Contemporary Developments in Continuum Mechanics and PDE’s”, G. M. de la Penha and L. A. Medeiros (eds.), North-Holland, Amsterdam, 1978. | Zbl
,[26] Espaces de fonctions et distributions du type de Gevrey et problemes aux limites paraboliques, Ann. Mat. Pura Appl. (4) 68 (1965), 341–417. | MR | Zbl
and ,[27] “Problemes aux Limites non Homogenes et Applications”, Dunod, Paris, 1968. | Zbl
and ,[28] On the global solvability of Kirchhoff equation for non-analytic initial data, J. Differential Equations 211 (2005), 38–60. | MR | Zbl
,[29] A new technique for the construction of solutions of nonlinear differential equations, Proc. Natl. Acad. Sci. USA 47 (1961), 1824–1831. | MR | Zbl
,[30] A rapidly convergent iteration method and non-linear differential equations. I. II., Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 20 (1966), 265–315 & 499–535. | EuDML | Numdam | MR | Zbl
,[31] Nonlinear vibration of an elastic string, J. Sound Vibration 8 (1968), 134–146. | Zbl
,[32] Nash-Moser theory for standing water waves, Arch. Ration. Mech. Anal. 159 (2001), 1–83. | MR | Zbl
and ,[33] A certain class of quasilinear hyperbolic equations, (Russian), Mat. Sb. (N.S.) 96 (138) (1975), 152–166, 168. (English transl.: Math. USSR-Sb. 25 (1975), 145–158). | MR
,[34] A KAM theorem for some nonlinear PDEs, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15 (1996), 119–148. | EuDML | Numdam | MR | Zbl
,[35] Periodic solutions of nonlinear hyperbolic partial differential equations. II, Comm. Pure Appl. Math. 22 (1969), 15–39. | MR | Zbl
,[36] “Methods of Modern Mathematical Physics”, Academic Press, Inc., New York, 1978. | MR
and ,[37] The Cauchy problem for Kirchhoff equations, Rend. Sem. Mat. Fis. Milano 62 (1994), 17–51. | MR | Zbl
,[38] Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys. 127 (1990), 479–528. | MR | Zbl
,[39] Generalized Implicit Function Theorems, In: L. Nirenberg, “Topics in Nonlinear Functional Analysis”, Courant Inst. of Math. Sciences, New York Univ., New York, 1974, Chapter VI. | MR | Zbl
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