Solutions of Navier equations and their representation structure

Navier equations are used to describe the deformation of a homogeneous, isotropic and linear elastic medium in the absence of body forces. Mathematically, the system is a natural vector O(n,R)-invariant generalization of the classical Laplace equation. In this paper, we decompose the space of polynomial solutions of Navier equations into a direct sum of irreducible O(n,R)-submodules and construct an explicit basis for each irreducible summand. Moreover, we explicitly solve the initial value problems for Navier equations and their wave-type extension—Lamé equations by Fourier expansion and Xu's method of solving flag partial differential equations. Our work might be counted as a continuation of Olver's important work on the algebraic study of elasticity in a certain sense.