On an Extension of Korn's First Inequality to Incompatible Tensor Fields on Domains of Arbitrary Dimensions

TL;DR

This paper generalizes Korn's first inequality to incompatible tensor fields on arbitrary-dimensional domains, unifying it with Poincare's inequality through advanced mathematical tools.

Contribution

It introduces a novel extension of Korn's first inequality applicable to incompatible tensor fields, linking it with Poincare's inequality and employing Helmholtz decompositions and Maxwell estimates.

Findings

Unified inequality for incompatible tensor fields and skew-symmetric cases

Reduction to Korn's and Poincare's inequalities in special cases

Application of Helmholtz decompositions and Maxwell estimates

Abstract

For a bounded N-dimensional domain with Lipschitz boundary we extend Korn's first inequality to incompatible tensor fields. For compatible tensor fields our estimate reduces to a non-standard variant of the well known Korn's first inequality. On the other hand, for skew-symmetric tensor fields our new estimate turns to Poincare's inequality. Therefore, our result may be viewed as a natural common generalization of Korn's first and Poincare's inequality. Decisive tools for this unexpected estimate are the classical Korn's first inequality, Helmholtz decompositions for mixed boundary conditions and the Maxwell estimate.