Work will be done in the application of variational methods to differential equations. It will focus on three areas: critical point theory and Hamiltonian systems, on elliptic differential equations arising in curvature problems of differential geometry and on multiplicity questions for superlinear second order elliptic boundary value problems. Although superlinear equations have been studied for some time, certain results are only partially understood. One of the important ones is the existence of infinitely many solutions of the Dirichlet problem with null boundary values. Efforts will be made to determine some global reason applicable to all superlinear equations. This will have to be achieved through a more abstract approach, possible using new complexification ideas. Work on Hamiltonian systems will follow two directions. The first is to determine the degree of regularity of generalized solutions to Hamiltonian systems with singular potentials; the second is to continue studies on the Weinstein conjecture. This latter concerns periodic orbits on odd-dimensional compact manifolds. It is believed that every contact vector-field has such an orbit if the first homology group of the manifold is zero. Previous work on this conjecture has led to ideas of critical points at infinity and pseudo-orbits of contact forms. These, in turn, have produced partial solutions to the conjecture. Work of a more geometric nature will center on the Kazdan - Warner problem on spheres of dimension five or greater. The fundamental question is one of deciding when a given function is the curvature of a metric conformal to the standard one. Positive results for the three-sphere do not carry over to higher dimensions. In addition to making fundamental contributions to the fields of differential geometry and partial differential equations, this work can be applied to dynamical systems and potential theory.
