On the domain dependence of the inf–sup and related constants via conformal mapping

In this paper we investigate the domain dependence of the inf–sup stability constant in the family of two-dimensional simply connected domains using its connection to the optimal constant figuring in Friedrichsʼ inequality for conjugate harmonic functions and the conformal mapping of the domain. A lower estimation of the inf–sup constant is also given in terms of the conformal mapping provided the boundary of the domain is smooth enough. We illustrate the results with several examples.