Periodic solutions of nonlinear vibrating strings and duality principles
Haïm Brézis · Bulletin of the American Mathematical Society · 1983
Introduction.Our lecture deals with the study of T-periodic solutions for the nonlinear vibrating string equation:Here g denotes a continuous function on R such that g(0) = 0 and f(x, t) is a given T-periodic function of t.Problem (1) may be viewed as an infinite-dimensional Hamiltonian system (let us recall that H. Poincaré has abundantly investigated the question of periodic solutions for finite-dimensional Hamiltonian systems; see [50]).Indeed if we set p = u and q -u n then (1) becomes è(;H-";Hï)where the Hamiltonian H is defined on the space #0(0, w) X L 2 (0, IT) byand G denotes a primitive of g.We shall be concerned with two distinct questions.Question 1. Existence of forced vibrations; that is, given ƒ(JC, /) find at least one solution of (1).Question 2. Existence of free vibrations (or "breathers"); that is, assume ƒ = 0 and find at least one nonzero solution of (1).