Optimal design and relaxation of variational problems. I. (English) Zbl 0609.49008

Kohn, Robert V.; Strang, Gilbert

Commun. Pure Appl. Math. 39, 113-137 (1986).

This three-part paper provides a detailed account of interrelation which exists between structural design optimization, relaxation of variational problems, and homogenization. The theory of optimal design is ultimately connected with two other trends since many optimum problems, i.e. those of distribution of materials, are initially ill-posed. Physically speaking, the way out lies in a suitable extension of the initial variational problem, this extension allowing all possible mixtures of given constituents as admissible materials, too. Mathematically, relaxation involves a special pointwise transformation of the integrand of a lower semi-continuous functional to the form which is satisfactory from the viewpoint of existence of a minimizer. A suitable technique, the s.c. quasiconvexification procedure, is developed to obtain that kind of transformation, and this one yields the extension which does not influence the lower bound of a minimized functional; moreover, it makes this bound attainable.

Reviewer: K.Lurie

MSC:

49J45	Methods involving semicontinuity and convergence; relaxation
74E30	Composite and mixture properties
74P99	Optimization problems in solid mechanics
49J20	Existence theories for optimal control problems involving partial differential equations

Keywords:

relaxation; homogenization; optimal design; quasiconvexification

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