Iterations of anti-selfdual Lagrangians and applications to Hamiltonian systems and multiparameter gradient flows

摘要

Anti-selfdual Lagrangians on a state space lift to path space provided one adds a suitable selfdual boundary Lagrangian. This process can be iterated by considering the path space as a new state space for the newly obtained anti-selfdual Lagrangian. We give here two applications for these remarkable permanence properties. In the first, we establish for certain convex-concave Hamiltonians H on a-possibly infinite dimensional-symplectic space H-2, the existence of a solution for the Hamiltonian system -Ju(t) is an element of aM(u(t)) that connects in a given time T > 0, two Lagrangian submanifolds. Another application deals with the construction of multiparameter flows, including those generated by vector fields that represent superpositions of skew-adjoint operators with gradients of convex potentials. Our methods are based on the new variational calculus for anti-selfdual Lagrangians developed in [5-7].