Goncharov∗ MATI — Russian State Technological University Orshanskay, 3, Moscow, 121552 Russia Received 14.07.2015, received in revised form 28.10.2015, accepted 11.12.2015 Some existence criteria for a certrain class of extremum problems involving eigenvalues of linear elliptic boundary-value problems (including ones in the form of variational inequalities) are proved. <...> The approach applied admits an extension to the case of extremum problems associated with eigenvalues of nonlinear boundary-value problems. <...> Some applications to optimal structural design and comparisons with results in the literature are given. <...> Keywords: eigenvalue optimization problem, elliptic boundary-value problem, variational inequality, existence theorem, optimal structural design. <...> A large number of such problems often arise in optimal structural design (see [1, 2] for more details). <...> For example, in order to widen a resonance-free frequency interval of some structure it is sufficient to maximize either its first natural frequency or the difference between the corresponding adjacent frequencies. <...> One of the most important characteristics of a structure is also the critical load under which the structure loses stability. <...> The frequencies of the natural oscillations of a structure and the critical load that causes buckling of the structure correspond to eigenvalues of appropriate boundary-value problems. <...> Thus, there exists a class of extremal problems for eigenvalue functionals in optimal structural design. <...> Optimization problems for eigenvalues of elliptic operators have been considered by many authors (see [1–9]). <...> For surveys on such problems we refer the reader to [1–3]. <...> Such problems, under the assumption that admissible controls form a weakly compact set of a Sobolev space, were considered in [2, 4]. <...> Let us advance some arguments in favour of consideration of broader sets of admissible controls for such problems. <...> Firstly, the condition of uniform boundness of the first-order weak derivatives of functions that belong to Sobolev spaces leads to using additional techniques, such as those utilizing penalty methods, to implement numerical procedures to derive optimal solutions to such problems. <...> Finally, classes of admissible controls arising in many applications whose elements are essentially bounded measurable functions are weak* compact without any artificial supplementary constraints. <...> Let us illustrate the essence of the second argument by means of an example. <...> For a thin <...>