Uniqueness of Equilibrium Configurations in Solid Crystals

In this article, under suitable assumptions, it is proved that $ \inf_{\u \in \calUlambda} E[\u]$ is dual to $\sup_{(a,b)} \{ \int_\Omega a(\F(\x)) d\x + \int_\Lambda b(\y) d\y\},$ where, $E[\u]:= \int_\Omega (h(\det D\u) -\F \cdot \u) d\x.$ Here, the infimum is performed over $\calUlambda$, the set of all orientation-preserving deformations $\u \in C^1(\Omega)^d$ that are homeomorphisms from $\bar \Omega$ onto $\bar \Lambda,$ and the supremum is performed over the set of all upper semicontinuous functions $a,b$ such that $a(\z) +\alpha b(\y) \leq h(\alpha) -\y \cdot \z.$ This duality result turns out to be important in the study of existence and uniqueness of smooth minimizers of E. Note that $M \rightarrow h(\det M)$ is not coercive and thus direct methods of the calculus of variations don't apply here.