Nonsmooth analysis and quasi-convexification in elastic energy minimization problems

We consider an energy minimization problem for a two-component composite with fixed volume fractions. We study two questions. The first is the dependence of the minimum energy on the constraints and parameters. The second is the rigorous justification of the method of Lagrange multipliers for this problem. We are able to treat only cases with periodic or affine boundary condition. We show that the constrained energy is a smooth and convex function of the constraints. We also find that the Lagrange multiplier problem is a convex dual of the problem with constraints. And we show that these two results are closely linked with each other. Our main tools are the Hashin-Shtrikman variational principle and some results from nonsmooth analysis.