## Solutions of Navier Equations and Their Representation Structure

Cao, Bintao
##### Description
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 (field) $O(n,\mbb{R})$-invariant generalization of the classical Laplace equation, which physically describes the vibration of a string. In this paper, we decompose the space of polynomial solutions of Navier equations into a direct sum of irreducible $O(n,\mbb{R})$-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--Lam\'e equations by Fourier expansion and Xu's method of solving flag partial differential equations.
Comment: 44 pages
##### Keywords
Mathematical Physics, Astrophysics, Mathematics - Analysis of PDEs, Mathematics - Quantum Algebra, Mathematics - Representation Theory, 17B10, 17B20, 35C99