Line energies for gradient vector fields in the plane

In this paper we study the singular perturbation of ∫ (1−|∇u|2)2 by ε2|∇2u|2. This problem, which could be thought as the natural second order version of the classical singular perturbation of the potential energy ∫ (1 − u2)2 by ε2|∇u|2, leads, as in the first order case, to energy concentration effects on hypersurfaces. In the two dimensional case we study the natural domain for the limiting energy and prove a compactness theorem in this class.