The Cantor Manifold Theorem With Symmetry And Applications To PDEs

In this master thesis we introduce a new Cantor manifold theorem and then apply it to one-dimensional (1d) quasi-linear beam equations with periodic boundary conditions. We show that the above equation admits small-amplitude linearly stable quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamical system. The proof is based on a partial Birkhoff normal form and an infinite dimensional KAM theorem for Hamiltonians with symmetry(cf.[19]).The master thesis is organized as follows. In chapter1we will give some prepara-tions. In chapter2we will give the introduction and the main results. In chapter3the Hamiltonian of the nonlinear beam equation is written in infinitely many coordinates and then transformed into its Birkhoff normal form of order four. In chapter4based on the Cantor Manifold Theorem with symmetry one gets the main theorem(Theorem2.2). In chapter5we recall an infinite dimensional KAM theorem with symmetry from [19] and also improve it. Then one can use it to prove the Cantor Manifold Theorem with symmetry. Some technical lemmas are deferred in the Appendix.