A Note on Linear Elliptic Systems on $\R^d$

TL;DR

This paper investigates the well-posedness of linear elliptic systems on ^d, introducing a variant of homogeneous Sobolev spaces to handle translation-invariant problems, with applications to anisotropic finite elasticity.

Contribution

It develops a new framework using modified Sobolev spaces to analyze linear elliptic systems on ^d, applicable to elasticity and similar translation-invariant problems.

Findings

Established well-posedness criteria for elliptic systems on ^d

Introduced a variant of homogeneous Sobolev spaces

Applicable to linearized models in anisotropic elasticity

Abstract

We are concerned with the well-posedness of linear elliptic systems posed on $\mathbb{R}^d$. The concrete problem of interest, for which we require this theory, arises from the linearization of the equations of anisotropic finite elasticity, however, our results are more generally applicable to translation-invariant problem posed on $\mathbb{R}^d$. We describe a variant of homogeneous Sobolev spaces, which are convenient for treating problems of this kind.