Korn's inequality in anisotropic Sobolev spaces

TL;DR

This paper extends Korn's inequality to anisotropic Sobolev spaces, demonstrating its validity and related inequalities like Poincare's in these spaces, with implications for continuum mechanics discretizations.

Contribution

It proves Korn's inequality and its nonlinear extension in anisotropic Sobolev spaces, broadening their applicability in mathematical analysis.

Findings

Korn's inequality holds in anisotropic Sobolev spaces.

An extension involving nonlinear maps is valid in these spaces.

Poincare's inequality also applies in anisotropic Sobolev spaces.

Abstract

Korn's inequality has been at the heart of much exciting research since its first appearance in the beginning of the 20th century. Many are the applications of this inequality to the analysis and construction of discretizations of a large variety of problems in continuum mechanics. In this paper, we prove that the classical Korn inequality holds true in {\it anisotropic Sobolev spaces}. We also prove that an extension of Korn's inequality, involving non-linear continuous maps, is valid in such spaces. Finally, we point out that another classical inequality, Poincare's inequality, also holds in anisotropic Sobolev spaces.