In the work eigenvalue problems are considered for the elliptic operators with variable domain. Eigenvalues of these operators are taken as functionals of the domain. Using the one to one correspondence between bounded convex domains and their support functions variation of the domain is expressed by the variation of its support function and calculate the first variation of this functional. Using the obtained formulas behavior of the eigenvalues is investigated when the domain varies. Then shape optimization problems are considered consider for the eigenvalues, the necessary conditions of optimality relatively domain are proved, an algorithm is offered for the numerical solution of the considered problems.

As some characteristics of certain mechanical systems are described by the eigenvalues of the corresponding operators, external problems for the eigenvalues with respect to domain (shape optimization problems) indeed are mathematical formulation of the problem of finding of the optimal form that defines the critical value of the corresponding physical characteristics. Solution of such problems allows avoid some risk situations in the applications.