A relaxation of the intrinsic biharmonic energy

The tension field τ(u) of a map u from a domain Ω ⊂ ℝ m into a manifold N is the negative L 2-gradient of the Dirichlet energy. In this paper we study the intrinsic biharmonic energy functional. In order to overcome the lack of coercivity of T, we extend it to a larger space. We construct minimizers of the extended functional via the direct method and we study the relation between these minimizers and critical points of T. Our results are restricted to dimensions m ≤ 4.