- 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 Γ -convergence result. The second problem involves the Aviles–Giga functional, ɛ ∫ Ω | ∇ 2 v | 2 d x + 1 ɛ ∫ Ω ( 1 − | ∇ v | 2 ) 2 d x , for which we construct upper bounds via a sequence of functions whose limit has gradient in BV. To cite this article: A. Poliakovsky, C. R. Acad. Sci. Paris, Ser. I 341 (2005).