Korn's inequality and Jones eigenpairs

TL;DR

This paper extends Korn's inequality to vector fields with boundary conditions on subsets of Lipschitz domain boundaries and explores the existence of Jones eigenpairs under these partial boundary constraints, broadening the understanding of elasticity eigenproblems.

Contribution

It demonstrates Korn's inequality for vector fields with boundary conditions on subsets and establishes the existence of Jones eigenpairs under partial boundary constraints.

Findings

Korn's inequality holds for vector fields with zero normal or tangential trace on boundary subsets.

Jones eigenpairs exist on various domains with partial boundary conditions.

Eigenpairs can be characterized with either normal or tangential boundary constraints.

Abstract

In this paper we show that Korn's inequality \cite{ref:korn1906} holds for vector fields with a zero normal or tangential trace on a subset (of positive measure) of the boundary of Lipschitz domains. We further show that the validity of this inequality depends on the geometry of this subset of the boundary. We then consider the {\it Jones eigenvalue problem} which consists of the usual traction eigenvalue problem for the Lam\'e operator for linear elasticity coupled with a zero normal trace of the displacement on a non-empty part of the boundary. Here we extend previous works in the literature to show the Jones eigenpairs exist on a broad variety of domains even when the normal trace of the displacement is constrained only on a subset of the boundary. We further show that one can have eigenpairs of a modified eigenproblem in which the constraint on the normal trace on a subset of the boundary is replaced by one on the tangential trace.