### abstract

- We prove an upper bound for the Aviles - Giga problem, which involves the minimization of the energy E-epsilon(v) = epsilon integral(Omega) \del(2) v\(2) dx + epsilon(-1) integral(Omega)(1 - |del v|(2))(2) dx over v is an element of H-2(Omega), where epsilon > 0 is a small parameter. Given v is an element of W-1,W-infinity(Omega) such that del v is an element of BV and |del v| = 1 a. e., we construct a family {v(epsilon)} satisfying: v(epsilon) --> v in W-1,W-p(Omega) and E-epsilon(v(epsilon)) --> 1/3 integral (J del v)|del(+) v - del(-)v|(3) dH(N-1) as epsilon goes to 0.