Generalized Korn's inequality and conformal Killing vectors

TL;DR

This paper generalizes Korn's inequality by replacing the linearized strain tensor with its trace-free part, leading to a stronger inequality with applications in General Relativity, and identifies conformal Killing vectors as its kernel.

Contribution

It introduces a generalized Korn's inequality involving the trace-free strain tensor, expanding its applicability and connecting it to conformal Killing vectors.

Findings

The kernel of the new operator consists of conformal Killing vectors.

The generalized inequality is stronger than the classical Korn's inequality.

Applications are found in the context of General Relativity.

Abstract

Korn's inequality plays an important role in linear elasticity theory. This inequality bounds the norm of the derivatives of the displacement vector by the norm of the linearized strain tensor. The kernel of the linearized strain tensor are the infinitesimal rigid-body translations and rotations (Killing vectors). We generalize this inequality by replacing the linearized strain tensor by its trace free part. That is, we obtain a stronger inequality in which the kernel of the relevant operator are the conformal Killing vectors. The new inequality has applications in General Relativity.