Uniqueness of Young measure in some variational problems with an infinite number of wells

Abstract

We study some variational problems involving energy densities (functions that have to be minimized) experiencing an infinite number of wells. Such densities are encountered in the study of microstructure of some materials as crystals. We consider the energy minimization problem with a fixed Dirichlet boundary data related by a convex relation to some number N of wells. We give a necessary and sufficient condition for nonexistence of minimizers. In the absence of minimizers, we prove that the minimizing sequences converge to the boundary data and choose their gradients around each of the N wells with a probability which tends to be constant. Moreover, they generate a unique Young measure that represents the microstructure. Our analysis shows that the deformation gradient of such materials is only governed by the N wells even if the energy density vanishes at an infinite number of wells. Our results agree with the assumption made in most of analytical and computational investigations that the deformation gradient can be modeled by a limited number of wells.