Nonsmooth Analysis and Quasi-convexiication in Elastic Energy Minimization Problems

We consider an energy minimization problem for a two-component composite with xed volume fractions. We study two questions. The rst is the dependence of the minimum energy on the constraints and parameters. The second is the rigorous justi-cation of the method of Lagrange multipliers for this problem. We are able to treat only cases with periodic or aane boundary condition. We show that the constrained energy is a smooth and convex function of the constraints. We also nd that the La-grange 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.