Completeness of representation of solutions for stationary homogeneous isotropic elastic/viscoelastic systems

Abstract

The usual Helmholtz decomposition gives a decomposition of any vector valued function into a sum of a gradient of a scalar function and a rotation of a vector valued function under some mild condition. In this paper we show that the vector valued function of the second term i.e., the divergence free part of this decomposition can be further decomposed into a sum of a vector valued function polarized in one component and the rotation of a vector valued function also polarized in the same component. Hence the divergence free part only depends on two scalar functions. We refer to this as a special Helmholtz decomposition. Further we show the so-called completeness of representation associated to this decomposition for the stationary wave field of a homogeneous, isotropic viscoelastic medium. That is, this wave field can be expressed as a special Helmholtz decomposition and each of its scalar functions satisfies a Helmholtz equation. Our completeness of representation is useful for solving boundary value problems in a cylindrical domain for several partial differential equations of systems in mathematical physics such as stationary isotropic homogeneous elastic/viscoelastic equations of a system.

Levenberg-Marquardt method for solving inverse problem of MRE based on the modified stationary Stokes system

“…By applying the curl operator to the first equation of ( 15), we have ∇ × u = 0 in Ω. According to the Boggio type lemma [5] (this reference is very old, for an easy access see lemma 3.2 in [8]), there exist two vector functions v and ṽ satisfying…”