Calculus of Variations and Geometric Measure Theory

G. Di Fratta - F. Solombrino

Korn and Poincaré-Korn inequalities: A different perspective

created by difratta on 30 Jul 2023


Submitted Paper

Inserted: 30 jul 2023
Last Updated: 30 jul 2023

Year: 2023


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 Poincaré-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.