Задачи максимизации собственных значений линейных эллиптических операторов

Maximization problems for eigenvalues of elliptic operators are considered. The problems under investigation are optimal control problems in coefficients, admissible controls form a weak* compact set of essentially bounded measurable functions, and convexity hypotheses on the coefficients of operators are made. The purpose of this article is twofold: (i) to derive necessary optimality conditions, which form a basis for efficient numerical solution; (ii) to describe the structure of the set of solutions for such a problem, to prove uniqueness criteria, and to characterize the case of non-uniqueness. The main idea of the article is that, even in the case of multiple eigenvalues, one can derive necessary optimality conditions, which involve only one eigenfunction. The derived necessary optimality conditions also make it possible to replace the original non-smooth extremal problem by the problem of finding a saddle point of a certain concrete functional. Applications of the results to optimal design problems for non-homogeneous columns and three-layered plates are given.