Line energies for gradient vector fields in the plane

In this paper we study the singular perturbation of integral(1 - \del u\(2) )(2) by epsilon(2)\del(2)u\(2). This problem, which could be thought as the natural second order version of the classical singular perturbation of the potential energy integral(1 - u(2))(2) by epsilon(2)\del(2)u\(2), leads, a s 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.