Abstract
An energy minimization problem for a twocomponent composite with fixed volume fraction is considered. Two questions are studied. 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. It is possible to treat only cases with periodic or affine boundary condition. It is also found that the constrained 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 results are closely linked with each other. The main tools are the Hashin-Shtrikman variational principle and some results from nonsmooth analysis.
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