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 Γ-convergence result. The second problem involves the Aviles–Giga functional, $?{\int }_{\Omega }{|{\mathrm{?}}^{2}v|}^2\mathrm{d}x+\frac{1}{?}{\int }_{\Omega }{(1?{|\mathrm{?}v|}^2)}^2\mathrm{d}x$, 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).