Some Questions in Nonlinear Differential Equations

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Tools and methods from the calculus of variations will be developed and applied to a variety of problems in dynamical systems, partial differential equations and geometry. For dynamical systems, the existence of various kinds of connecting orbits for Hamiltonian systems will be established. There are related applications to geometry where the existence of minimal heteroclinic, homoclinic, and chaotic geodesics for the n-torus and other manifolds will be studied. For partial differential equations, it will be shown how some equations that arise in phase transition models admit a large number of different kinds of equilibrium states.The goal of our project is to develop new minimization and minimax methods and apply them to areas such as dynamical systems, geometry, and partial differential equations. The applications to dynamical systems include establishing the existence of orbits asymptotic to simple basic states like equilibrium points and periodic orbits. Related problems occur in geometry where the basic periodic states are periodic geodesics and we seek connecting states that are also geodesics. For partial differential equations, a class of phase transition models will be studied where we seek to understand what kinds of equilibrium states are possible.