A method for establishing upper bounds for singular perturbation problems

We present two applications of a new method for proving upper bounds for singular perturbation problems involving maps of bounded variation. The two problems are of first and second order, respectively. The first is a minimization problem, related to the question of optimal lifting for BV-maps with values in S-1, for which we prove a Gamma-convergence result. The second problem involves the Aviles-Giga functional, epsilon integral(Omega)vertical bar del(2)v vertical bar(2)dx + 1/epsilon integral(Omega)(1-vertical bar del v vertical bar(2))(2) dx, for which we construct upper bounds via a sequence of functions whose limit has gradient in BV.