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 ρ; we exhibit a bound for the constant appearing in that inequality, which is explicit with respect to R and ρ. Crucial tools in the derivation of such a bound 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 Bogovskiĭ's operator, with upper bounds on the corresponding constants that are explicit with respect to R and ρ.