# Relaxation of Optimization Problems with Equations Containing the Operator ∇ · ⅅ · ∇: An Application to the Problem of Elastic Torsion

## Abstract

In the preceding chapter, we investigated the problem of optimal distribution of the specific resistance of the working fluid in a channel of an MHD power generator. This problem represents a typical example of optimization problems which arise when a control function ⅅ (either scalar or tensor) enters the main part of the elliptic operator ∇ · ⅅ · ∇: of the second order. As was pointed out in the Introduction, the first step in such problems consists of building the *G*-closure of the initially given set *U* of controls ⅅ; i.e., an extension of this set to some wider set *GU* including, in addition to *U*, all possible composites assembled of the initially given components—elements of the set *U*. The problem of obtaining an invariant description of this wider set is also essential for the theory of composite materials. For a given microstructure of a composite, its effective characteristics may be obtained by the procedure of *homogenization* (see Refs. 403, 405, 406, 421, 427, 441, 525, and 526) which required the solution of some fairly complicated auxiliary boundary-value problems. On the other hand, for many applications, particularly for optimal control and optimal design problems, a description of the *whole set* of materials which may be obtained from some initially given compounds with the aid of the process of mixing is of central interest.^{476–479,499,514}.

## Keywords

Layered Composite Straight Segment Torsional Rigidity Switching Line Layered Microstructure## Preview

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