Solutions of Navier Equations and Their Representation Structure

TL;DR

This paper analyzes the structure of polynomial solutions to Navier equations, decomposes them into irreducible modules, and explicitly solves initial value problems using Fourier and flag PDE methods.

Contribution

It provides a decomposition of polynomial solutions into irreducible modules and explicit solutions to initial value problems for Navier and Lamé equations.

Findings

Decomposition of polynomial solutions into irreducible $O(n,\mathbb{R})$-submodules.

Explicit basis construction for each irreducible component.

Solutions to initial value problems via Fourier expansion and flag PDE methods.

Abstract

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.