Korn and Poincar\'e-Korn inequalities: A different perspective

TL;DR

This paper offers a simple, unified perspective on Korn's and Poincaré-Korn inequalities for various domains and exponents, using classical functional analysis tools like the Riesz representation theorem.

Contribution

It provides a conceptually straightforward proof approach for Korn inequalities applicable to general exponents and domain classes, including Lipschitz domains.

Findings

Unified proof for Korn's inequalities for all p in (1, ∞)

Elementary derivations of Poincaré-Korn inequalities in diverse domains

Clarification of the role of Riesz representation in elasticity inequalities

Abstract

We present a concise point of view on the first and the second Korn's inequality for general exponent $p$ and for a class of domains that includes Lipschitz domains. Our argument is conceptually very simple and, for $p = 2$, uses only the classical Riesz representation theorem in Hilbert spaces. Moreover, the argument for the general exponent $1<p<\infty$ remains the same, the only change being invoking now the $q$-Riesz representation theorem (with $q$ the harmonic conjugate of $p$). We also complement the analysis with elementary derivations of Poincar\'e-Korn inequalities in bounded and unbounded domains, which are essential tools in showing the coercivity of variational problems of elasticity but also propedeutic to the proof of the first Korn inequality.