Variational aspects of biharmonic maps

For a map between Riemannian manifolds, the tension field is the L2-gradient of the Dirichlet functional. Consider the second order functional given by the L2-norm of the tension field. Its critical points are called (intrinsic) biharmonic maps.

When we want to apply methods from the calculus of variations, such as the direct method, to this problem, we face a number of difficulties. Most importantly, the functional is not coercive on the natural Sobolev spaces. We discuss an approach to the problem via a relaxation of the functional, using tools from geometric measure theory.