INFINITE-DIMENSIONAL SYMPLECTIC CAPACITIES AND A SQUEEZING THEOREM FOR HAMILTONIAN PDES

We study partial differential equations of hamiltonian form and treat them as infinite-dimensional hamiltonian systems in a functional phase-space of x-dependent functions. In this phase space we construct an invariant symplectic capacity and prove a version of Gromov's (non)squeezing theorem. We give an interpretation of the theorem in terms of the ''energy transition to high frequencies'' problem.