NONSMOOTH ANALYSIS AND QUASI-CONVEXIFICATION IN ELASTIC ENERGY MINIMIZATION PROBLEMS

An energy minimization problem for a two-component composite with fixe d volume fraction is considered. Two questions are studied. The first is the dependence of the minimum energy on the constraints and paramet ers. The second is the rigorous justification of the method of Lagrang e multipliers for this problem. It is possible to treat only cases wit h periodic or affine boundary condition. It is also found that the con strained energy is a smooth and convex function of the constraints. It is shown that the Lagrange multiplier problem is a convex dual of the problem with constraints. Moreover, it is shown that these two result s are closely linked with each other. The main tools are the Hashin-Sh trikman variational principle and some results from nonsmooth analysis .