Symmetry method and sufficient condition of optimality in a class of domain optimization problems

Sufficient conditions of optimality in shape optimization problems are studied in present paper. The method of deriving sufficient conditions suggested here is based on taking into account the symmetry that shape optimization problems have. The main idea of our method is special projecting of the perturbed 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.