Optimal Design and Relaxation of Variational Problems. III,
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This paper has attempted a synthesis of three subjects that have developed separately - optimal design, homogenization, and relaxation in the calculus of variations. The underlying theme is that of oscillations. They occur with certain definite patterns in minimizing sequences for the variational problem. The same patterns recur in nearly optimal designs and in the coefficients of the equation being homogenized. Each problem is resolved by determining the structure of those oscillations, and above all their macroscopic averages. It is those averages that solve the relaxed problem. They also determine the optimal design and the solution of the homogenized equation. We emphasize also the role of optimization. The homogenized equations and composite materials of interest here come not from arbitrary mixtures but from those that are extremal, since they correspond to minimizing sequences for the variational problem. The effective properties of these composites attain the optimal bounds for given materials in fixed proportions.
- Theoretical Mathematics