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Symmetry and identification technique in shape optimization

Sufficient conditions of optimality in shape optimization problems are studied in the present paper. Shape optimization problems are the problems in which an objective function depends on the domain through the solution of the boundary-value problem defined on the domain. There are many such problems in various branches of science and high-technology industries whenever the best shape for a physical system is required. The method of deriving sufficient conditions suggested here is based on taking into account the symmetry that shape optimization problems have. We mean the objective function depends an the shape of the domain and does not depend on the place of the domain in the plane. That is why extremum in such problems is not proper and it makes obtaining of the effective sufficient conditions nontrivial. The main idea of our method is special projecting of the perturbated domain on the set of the domains having optimal shape. We prove here that this method leads to effective sufficient conditions for the problems having symmetry. In order to show the ideas of the method more clear we consider one of the classical problems as an illustrative example and establish the optimality in this problem proving all up to final point. Throughout the paper we are discussing how to use the general ideas of this method for other shape optimization problems.