Cantor families of periodic solutions for completely resonant wave equations

Abstract

We present recent existence results of Cantor families of small amplitude periodic solutions for completely resonant nonlinear wave equations. The proofs rely on the Nash-Moser implicit function theory and variational methods.

Keywords

Nonlinear wave equation / infinite dimensional Hamiltonian system / periodic solution / variational method / Lyapunov-Schmidt reduction / small divisor / Nash-Moser Theorem

References

[1] Ambrosetti A., Rabinowitz P. Dual variational methods in critical point theory and applications. J Func Anal, 1973, 14: 349-381.

[2] Baldi P., Berti M. Periodic solutions of wave equations for asymptotically full measure sets of frequencies. Rend Mat Acc Naz Lincei, 2006, 17: 257-277.

[3] Bambusi D., Paleari S. Families of periodic solutions of resonant PDEs. J Nonlinear Sci, 2001, 11: 69-87.

[4] Berti M., Bolle P. Periodic solutions of nonlinear wave equations with general nonlinearities. Comm Math Phys, 2003, 243: 315-328.

[5] Berti M., Bolle P. Multiplicity of periodic solutions of nonlinear wave equations. Nonlinear Anal, 2004, 56: 1011-1046.

[6] Berti M., Bolle P. Cantor families of periodic solutions for completely resonant nonlinear wave equations. Duke Math J, 2006, 134: 359-419.

[7] Berti M., Bolle P. Cantor families of periodic solutions for wave equations via a variational principle. Advances in Mathematics, 2008, 217: 1671-1727.

[8] Berti M, Bolle P. Cantor families of periodic solutions of wave equations with C^k nonlinearities. Nonlinear Differential Equations and Applications (in press)

[9] Bourgain J. Periodic solutions of nonlinear wave equations. Harmonic Analysis and Partial Differential Equations. Chicago Lectures in Math, 1999, Chicago: Univ Chicago Press, 69-97.

[10] Craig W., Wayne E. Newton’s method and periodic solutions of nonlinear wave equation. Comm Pure Appl Math, 1993, 46: 1409-1498.

[11] Fadell E. R., Rabinowitz P. Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems. Inv Math, 1978, 45: 139-174.

[12] Gentile G., Mastropietro V., Procesi M. Periodic solutions for completely resonant nonlinear wave equations. Comm Math Phys, 2005, 256: 437-490.

[13] Lidskij B. V., Shulman E. Periodic solutions of the equation u_tt-u_xx + u³ = 0. Funct Anal Appl, 1988, 22: 332-333.

[14] Moser J. Periodic orbits near an equilibrium and a theorem by Alan Weinstein. Comm Pure Appl Math, 1976, 29: 724-747.

[15] Struwe M. The existence of surfaces of constant mean curvature with free boundaries. Acta Math, 1988, 160: 19-64.

[16] Weinstein A. Normal modes for nonlinear Hamiltonian systems. Inv Math, 1973, 20: 47-57.

[17] Yuan X. Quasi-periodic solutions of completely resonant nonlinear wave equations. J Diff Equations, 2006, 230: 213-274.