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

TL;DR

This paper establishes explicit Nečas-Lions inequalities with symmetric gradients on star-shaped domains using Babuška-Aziz inequalities and Fourier techniques, providing new estimates for divergence operators.

Contribution

It introduces a Nečas-Lions inequality with symmetric gradients on star-shaped domains, leveraging a first order Babuška-Aziz inequality and Fourier transform methods.

Findings

Explicit constants in the inequality depending on domain parameters

Derivation of higher order estimates for divergence operators

Extension of inequalities to 2D and 3D star-shaped domains

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.