The Variational Approach to Biharmonic Maps

When studying a certain class of geometric objects, it is often helpful to give special attention to the minimizers or maximizers of the quantities naturally associated to them. For instance, among all surfaces with fixed boundary, the ones that minimize the area are of special interest.For this project, we consider a different minimization principle. It is rooted in the theory of manifolds, which is the higher-dimensional equivalent of surfaces. The geometrical objects in question are mappings of one manifold onto another. The quantity that is to by minimized, can be thought of as a measure for the behaviour of the curvature under such a mapping. The solutions of this minimization problem are called biharmonic maps, and they give rise to nonlinear partial differential equations.The calculus of variations is a theory from mathematical analysis which provides tools and methods to study minimization problems and the corresponding differential equations. The problem of biharmonic maps, however, does not quite fit into the usual framework of this theory. The main object of this project is to reconcile the methods of the calculus of variations with the structure of the problem at hand.To this end, the space of geometrical objects will have to be extended appropriately. The partial differential equations of the problem have to be studied on the extended space, and the analytic methods used for this purpose have to be combined with the underlying geometry.The problem of biharmonic maps is only one of several problems with similar structures. Some of them are derived from geometry, others from mathematical models in physics or other fields. Results obtained for biharmonic maps are likely to find applications in one of the other theories, and vice versa. Therefore the research will not be limited to biharmonic maps, although they are at its centre.Not much is currently known about variational aspects of problems of this type. A successful study of these questions could make the powerful tools of the calculus of variations available to geometers or mathematical physicists studying biharmonic maps and related theories. In addition, it will add new methods to the theories of the calculus of variations and partial differential equations.