Poincare meets Korn via Maxwell: Extending Korn's First Inequality to Incompatible Tensor Fields

TL;DR

This paper extends Korn's first inequality to incompatible tensor fields in three-dimensional domains, unifying it with Poincare's inequality and broadening its applicability in mathematical analysis.

Contribution

It introduces a novel inequality that generalizes Korn's first inequality to incompatible tensor fields, connecting it with Poincare's inequality through Maxwell's estimate.

Findings

Unified inequality for incompatible tensor fields and skew-symmetric tensors

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

Utilization of Helmholtz decompositions and Maxwell estimate

Abstract

For a bounded three-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.