We prove an upper bound for the Aviles-Giga problem, which involves the minimization of the energy is a small parameter. Given v ∈ W 1,∞ ( ) such that ∇v ∈ BV and |∇v| = 1 a.e., we construct a 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

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... 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: