Uniqueness of equilibrium configurations in solid crystals

In this article, under suitable assumptions, it is proved that inf(u is an element ofU Lambda) E[u] is dual to sup((a,b)){integral (Omega) a(F(x))dx integral (Lambda)b(y)dy}, where, E[u] := integral Omega>(*) over bar * (h( det Du) - F.)dx. Here, the infimum is performed over U-Lambda, the set of a ll orientation-preserving deformations u is an element of C-1(Ohm)(d) that are homeomorphisms from onto <(<Omega>)over bar> onto <(<Lambda>)over bar>, and the supremum is performed over the set of all upper semicontinuous fun ctions a, b such that a(z) + alphab(y) less than or equal to h(alpha) - y.z . This duality result turns out to be important in the study of existence a nd uniqueness of smooth minimizers of E. Note that M --> h(det M) is not co ercive and thus direct methods of the calculus of variations don't apply he re.