Conservation Laws and Solutions of the First Boundary Value Problem for the Equations of Two- and Three-Dimensional Elasticity

Abstract

If a system of differential equations admits a continuous transformation group, then, in some cases, the system can be represented as a combination of two systems of differential equations. These systems, as a rule, are of smaller order than the original one. This information pertains to the linear equations of elasticity theory. The first system is automorphic and is characterized by the fact that all of its solutions are obtained from a single solution using transformations in this group. The second system is resolving, with its solutions passing into themselves under the group action. The resolving system carries basic information about the original system. The present paper studies the automorphic and resolving systems of two- and three-dimensional time-invariant elasticity equations, which are systems of first-order differential equations. We have constructed infinite series of conservation laws for the resolving systems and automorphic systems. There exist infinitely many such laws, since the systems of elasticity equations under consideration are linear. Infinite series of linear conservation laws with respect to the first derivatives are constructed in this article. It is these laws that permit solving the first boundary value problem for the equations of elasticity theory in the two- and three-dimensional cases. The solutions are constructed by quadratures, which are calculated along the boundary of the studied domains.