Nonpotentiality of Sobolev system and construction of semibounded functional

Abstract

Works by S. L. Sobolev on small-amplitude oscillations of a rotating fluid in 1940's stimulated a great interest to such problems. After the publications of his works, I.G. Petrovsky emphasized the importance of studying general differential equations and systems not resolved with respect to the higher-order time derivative. In this connection, it is natural to study the issue on the existence of their variational formulations. It can be considered as the inverse problem of the calculus of variations. The main goal of this work is to study this problem for the Sobolev system. A key object is the criterion of potentiality. On this base, we prove a nonpotentiality for the operator of a boundary value problem for the Sobolev system of partial differential equations with respect to the classical bilinear form. We show that this system does not admit a matrix variational multiplier of the given form. Thus, the equations of the Sobolev system cannot be deduced from a classical Hamilton principle. We pose the question that whether there exists a functional semibounded on solutions of the given boundary value problem. Then we propose an algorithm for a constructive determining such functional. The main advantage of the constructed functional action is applications of direct variational methods.