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

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