A Korn-type inequality in SBD for functions with small jump sets

TL;DR

This paper establishes a Korn-type inequality for special functions of bounded deformation in the plane, demonstrating that functions with small jump sets are close to rigid motions outside a small exceptional set.

Contribution

It introduces a Korn-type inequality in SBD for functions with small jump sets, linking the distance to rigid motions with the elastic strain in a planar setting.

Findings

Functions in SBD$^2$ with small jump sets are close to rigid motions outside a small set.

The inequality controls the distance from a rigid motion using the elastic strain.

Functions in SBD$^2$ have bounded variation away from a small exceptional region.

Abstract

We present a Korn-type inequality in a planar setting for special functions of bounded deformation. We prove that for each function in SBD$^2$ with sufficiently small jump set the distance of the function and its derivative from an infinitesimal rigid motion can be controlled in terms of the linearized elastic strain outside of a small exceptional set of finite perimeter. Particularly the result shows that each function in SBD$^2$ has bounded variation away from an arbitrarily small part of the domain.