A Ne\v{c}as-Lions inequality with symmetric gradients on star-shaped domains based on a first order Babu\v{s}ka-Aziz inequality

Abstract

We prove a Ne\v{c}as-Lions inequality with symmetric gradients on two and three dimensional domains of diameter $R$ that are star-shaped with respect to a ball of radius $\rho$; the constants in the inequality are explicit with respect to $R$ and $\rho$. Crucial tools in deriving this inequality are a first order Babu\v{s}ka-Aziz inequality based on Bogovski\u{i}'s construction of a right-inverse of the divergence and Fourier transform techniques proposed by Dur\'an. As a byproduct, we derive arbitrary order estimates in arbitrary dimension for that operator.