## On the inf-sup constant of the divergence alias LBB constant

Martin Costabel and Monique Dauge

Slides and extended abstract of a talk given at the Conference on Computational Electromagnetism and Acoustics, Oberwolfach, January 20-26, 2013.

The inf-sup constant $\beta(\Omega)$ of a domain $\Omega\in\mathbb{R}^d$ is $$\beta(\Omega) = \ {\inf_{q\in L^2_\circ(\Omega)}} \ \ {\sup_{v\in H^1_0(\Omega)^d}} \ \ \frac{\big\langle \mathrm{div}\, v, q \big\rangle_\Omega} {\Vert{v}\Vert_{1,\Omega}\,\vert{q}\vert_{0,\Omega}}$$ We address its relations with the spectrum of the Schur complement of the Stokes operator, the Cosserat spectrum, the Friedrichs constant and the Horgan-Payne angle. Domains with narrow pass play an important role.

Jan 2013

 Slides    Extended abstract Oberwolfach Reports, 13, 1, 69-72 (2013)