Line energies for gradient vector fields in the plane

Abstract.

In this paper we study the singular perturbation of $\int (1-|\nabla u|^2)^2$ by $\varepsilon^2|\nabla^2u|^2$. This problem, which could be thought as the natural second order version of the classical singular perturbation of the potential energy $\int (1-u^2)^2$ by $\varepsilon^2|\nabla u|^2$, 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.