Regular Article: Optimal Design Problems for Two-Phase Conducting Composites with Weakly Discontinuous Objective Functionals

Authors:
Yury Grabovsky
Affiliations:
Department of Mathematics, Temple University, Philadelphia, Pennsylvania, 19122
Venue:
Advances in Applied Mathematics
Year:
2001

Citing 0
Cited 3

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Relaxation of Optimal Control Problem Governed by Semilinear Elliptic Equation with Leading Term Containing Controls

Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications

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Abstract

We consider the relatively simple but non-trivial example of an optimal design problem with a weakly discontinuous objective functional. The objective functional is quadratic and was suggested by Tartar in his ''Remarks on Optimal Design'' paper. We analyze a problem of finding a layout of a conducting composite such that the fields in both phases provide least squares fit to a given field. The main result of the paper is an explicit formula for the relaxed optimal design problem, suitable for numerical solution. Further analysis of our explicit formula shows that the optimal layout is a rank one laminate locally, lending some support for Tartar's conjecture that the minimizing sequences always converge strongly.