The Stokes paradox in inhomogeneous elastostatics

TL;DR

This paper investigates the uniqueness and decay properties of solutions to the inhomogeneous elastostatic problem in 2D exterior domains, establishing conditions under which solutions exist, are unique, and decay at infinity.

Contribution

It proves the existence, uniqueness, and decay rates of solutions to inhomogeneous elastostatics in 2D exterior Lipschitz domains, extending classical results to inhomogeneous materials.

Findings

Unique solutions exist under a compatibility condition.

Solutions decay at a rate depending on elastic properties.

If elasticities tend to a homogeneous state, solutions decay faster.

Abstract

We prove that the displacement problem of inhomogeneous elastostatics in a two--dimensional exterior Lipschitz domain has a unique solution with finite Dirichlet integral $\u$, vanishing uniformly at infinity if and only if the boundary datum satisfies a suitable compatibility condition (Stokes' paradox). Moreover, we prove that it is unique under the sharp condition $\u=o(\log r)$ and decays uniformly at infinity with a rate depending on the elasticities. In particular, if these last ones tend to a homogeneous state at large distance, then $\u=O(r^{-\alpha})$, for every $\alpha<1$.