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

Abstract

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 ${\cal H}$ on a–possibly infinite dimensional–symplectic space H ², the existence of a solution for the Hamiltonian system $-J\dot u (t)\in \partial {\cal H}(u(t))$ 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].