Investigation scheme for extremals of multidimensional variational problems (Q1907454)

The straightforward extension of finite-dimensional optimality conditions to the investigation of extremal problems in calculus of variations encounters some difficulties. This makes it necessary to resort to some special constructions, one of which is based on the simultaneous use of two spaces. The first space is a Banach space, where differentiability properties of involved functionals are studied, and the other is a Hilbert space, where the spectral properties of the corresponding differential are investigated. The scheme suggested in the paper is intended for optimization problems where the spectrum of the second differential is nonnegative and contains zero. In these cases testing an extremal for minimum can be reduced to the investigation for the minimum of a function of a finite number of variables at its critical points. The suggested scheme uses the idea of the well-known Lyapunov-Schmidt method for constructing bifurcation equations.