J. Eur. Math. Soc. 9, 1–43 c  European Mathematical Society 2007 Arkady Poliakovsky Upper bounds for singular perturbation problems involving gradient fields Received May 29, 2005 Abstract. We prove an upper bound for the Aviles–Giga problem, which involves the minimization of the energy E ε (v) = ε     ∇ 2 v   2 dx + ε −1   (1 − |∇v| 2 ) 2 dx over v ∈ H 2 (), where ε> 0 is a small parameter. Given v ∈ W 1,∞ () such that ∇v ∈ BV and |∇v|= 1 a.e., we construct a family {v ε } satisfying: v ε → v in W 1,p () and E ε (v ε ) → 1 3  J ∇v |∇ + v −∇ − v| 3 d H N −1 as ε goes to 0. 1. Introduction Consider the energy functional E ε (v) = ε   |∇ 2 v| 2 + 1 ε   (1 − |∇v| 2 ) 2 (1.1) where  is a C 2 bounded domain in R N , v is a scalar function and ε is a small parameter. Energies similar to (1.1) appear in different physical situations: smectic liquid crystals, blisters in thin films, micromagnetics (see [13] and the references therein). Clearly, one expects that any limit of the minimizers to (1.1) should satisfy the eikonal equation |∇v|= 1 a.e. in . (1.2) Aviles and Giga [3] made a conjecture, based on a certain ansatz for the minimizers, that the limiting energy should take the form E(v) = 1 3  J ∇v |∇ + v −∇ − v| 3 d H N −1 , where J ∇v is the jump set of ∇v and ∇ ± v are the traces of ∇v on the two sides of the jump set (see Section 2 below for the exact definitions of the notions needed from the theory of functions of bounded variation). Most of the results on this problem treat the two-dimensional case N = 2 (an example due to De Lellis [6] shows that the Aviles– Giga ansatz does not hold for N ≥ 3), so we assume N = 2 in the review of the known results below. Support for the Aviles–Giga conjecture was given in the work of Jin and A. Poliakovsky: Department of Mathematics, Technion-I.I.T, 32000 Haifa, Israel; e-mail: maarkady@techunix.technion.ac.il