A Nečas-Lions inequality with symmetric gradients on star-shaped domains based on a first order Babuška-Aziz inequality

We prove a Nečas-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 Babuška-Aziz inequality based on Bogovskiĭ’s construction of a right-inverse of the divergence and Fourier transform techniques proposed by Durán. As a byproduct, we derive arbitrary order estimates in arbitrary dimension for that operator.