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.

We consider the Kirchhoff equation for a vibrating body, in any dimension, in the presence of a time-periodic external forcing with period 2π/ω and amplitude ε. 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 (ω,ε) in a Cantor set with asymptotically full measure as ε0.

Classification: 35L70, 45K05, 35B10, 37K55
Baldi, Pietro 1

1 Dipartimento di Matematica e Applicazioni, “R. Caccioppoli”, Università degli Studi di Napoli, “Federico II”, Via Cintia, Monte S. Angelo, 80126 Napoli, Italia
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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] A. Arosio, Asymptotic behaviour as t+ 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] A. Arosio, 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] A. Arosio and S. Spagnolo, 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

[4] A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc. 348 (1996), 305–330. | MR | Zbl

[5] P. Baldi and M. Berti, Forced vibrations of a nonhomogeneous string, SIAM J. Math. Anal. 40 (2008), 382–412. | MR | Zbl

[6] S. N. Bernstein, Sur une classe d’équations fonctionelles aux dérivées partielles, Izv. Akad. Nauk SSSR Ser. Mat. 4 (1940), 17–26. | MR | Zbl

[7] M. Berti, “Nonlinear Oscillations of Hamiltonian PDEs”, Progr. Nonlinear Differential Equations Appl. 74, H. Brézis ed., Birkhäuser, Boston, 2008. | MR

[8] M. Berti and P. Bolle, Cantor families of periodic solutions for completely resonant nonlinear wave equations, Duke Math. J. 134 (2006), 359–419. | MR | Zbl

[9] M. Berti and P. Bolle, Cantor families of periodic solutions of wave equations with C k nonlinearities, NoDEA Nonlinear Differential Equations Appl. 15 (2008), 247–276. | MR | Zbl

[10] M. Berti and P. Bolle, Sobolev periodic solutions of nonlinear wave equations in higher spatial dimensions, Arch. Ration. Mech. Anal., to appear. | MR | Zbl

[11] J. Bourgain, 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] J. Bourgain, Construction of periodic solutions of nonlinear wave equations in higher dimension, Geom. Funct. Anal. 5 (1995), 629–639. | EuDML | MR | Zbl

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

[14] J. Bourgain, 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] G. F. Carrier, On the nonlinear vibration problem of the elastic string, Quart. Appl. Math. 3 (1945), 157–165. | MR | Zbl

[16] G. F. Carrier, A note on the vibrating string, Quart. Appl. Math. 7 (1949), 97–101. | MR | Zbl

[17] W. Craig, “Problèmes de Petits Diviseurs dans les Équations aux Dérivées Partielles”, Panoramas et Synthèses, 9, Société Mathématique de France, Paris, 2000. | MR

[18] W. Craig and E. Wayne, Newton’s method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math. 46 (1993), 1409–1501. | MR | Zbl

[19] P. D’Ancona and S. Spagnolo, A class of nonlinear hyperbolic problems with global solutions, Arch. Ration. Mech. Anal. 124 (1993), 201–219. | MR | Zbl

[20] R. W. Dickey, Infinite systems of nonlinear oscillation equations related to the string, Proc. Amer. Math. Soc. 23 (1969), 459–468. | MR | Zbl

[21] D. Fujiwara, 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] J. M. Greenberg and S. C. Hu, The initial value problem for a stretched string, Quart. Appl. Math. 38 (1980/81), 289–311. | MR | Zbl

[23] G. Kirchhoff, “Vorlesungen über Mathematische Physik: Mechanik”, Teubner, Leipzig, 1876, Ch. 29. | JFM

[24] S. Kuksin, “Nearly Integrable Infinite-Dimensional Hamiltonian Systems”, Lecture Notes in Math. 1556, Springer, Berlin, 1993. | MR | Zbl

[25] J. L. Lions, 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] J. L. Lions and E. Magenes, 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

[27] J. L. Lions and E. Magenes, “Problemes aux Limites non Homogenes et Applications”, Dunod, Paris, 1968. | Zbl

[28] R. Manfrin, On the global solvability of Kirchhoff equation for non-analytic initial data, J. Differential Equations 211 (2005), 38–60. | MR | Zbl

[29] J. Moser, A new technique for the construction of solutions of nonlinear differential equations, Proc. Natl. Acad. Sci. USA 47 (1961), 1824–1831. | MR | Zbl

[30] J. Moser, 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] R. Narasimha, Nonlinear vibration of an elastic string, J. Sound Vibration 8 (1968), 134–146. | Zbl

[32] P. Plotnikov and J. Toland, Nash-Moser theory for standing water waves, Arch. Ration. Mech. Anal. 159 (2001), 1–83. | MR | Zbl

[33] S. I. Pohozaev, 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] J. Pöschel, A KAM theorem for some nonlinear PDEs, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15 (1996), 119–148. | EuDML | Numdam | MR | Zbl

[35] P. Rabinowitz, Periodic solutions of nonlinear hyperbolic partial differential equations. II, Comm. Pure Appl. Math. 22 (1969), 15–39. | MR | Zbl

[36] M. Reed and B. Simon, “Methods of Modern Mathematical Physics”, Academic Press, Inc., New York, 1978. | MR

[37] S. Spagnolo, The Cauchy problem for Kirchhoff equations, Rend. Sem. Mat. Fis. Milano 62 (1994), 17–51. | MR | Zbl

[38] C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys. 127 (1990), 479–528. | MR | Zbl

[39] E. Zehnder, 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