**Authors**:
Vieri Mastropietro, Michela Procesi

**Title** *
Lindstedt series for periodic solutions
of beam equations with quadratic and nonlinear
dependent nonlinearities
*

**Abstract:**
We prove the existence of small amplitude periodic solutions, for a
large Lebesgue measure set, in the nonlinear beam equation with a weak
quadratic and velocity dependent nonlinearity and with Direchelet
boundary conditions. Such nonlinear PDE can be regarded as a simple
model describing oscillations of flexible structures like suspension
bridges in presence of an uniform wind flow. The periodic solutions
are explicitely constructed by a convergent perturbative expansion
which can be considered the analogue of the lindstedt series for the
invariant tori in classical mechanics. The periodic solutions are
defined only in a Cantor set, and resummation techniques of divergent
power series are used in order to control the small divisor problem.

**Key words: ** Small divisors, PDE

Vieri Mastropietro

Matematica, Universita' di Roma 2

V.le
della Ricerca Scientifica

00133 Roma, Italia

tel +39-06-7259-4209

em: mastropi@mat.uniroma2.it

http://ipparco.roma1.infn.it