The energy in a square membrane Ω subject to constant viscous damping on a subset $\omega\subset \Omega$ decays exponentially in time as soon as ω satisfies a geometrical condition known as the “Bardos-Lebeau-Rauch” condition. The rate $\tau(\omega)$ of this decay satisfies $\tau(\omega)= 2 \min( -\mu(\omega), g(\omega))$ (see Lebeau [Math. Phys. Stud.19 (1996) 73–109]). Here $\mu(\omega)$ denotes the spectral abscissa of the damped wave equation operator and $g(\omega)$ is a number called the geometrical quantity of ω and defined as follows. A ray in Ω is the trajectory generated by the free motion of a mass-point in Ω subject to elastic reflections on the boundary. These reflections obey the law of geometrical optics. The geometrical quantity $g(\omega)$ is then defined as the upper limit (large time asymptotics) of the average trajectory length. We give here an algorithm to compute explicitly $g(\omega)$ when ω is a finite union of squares.

We prove that, contrary to the L1-Nash inequality, there exists a second best constant for the L2-Nash inequality on any smooth compact Riemannian manifold.

We prove that, contrary to the L1-Nash inequality, there exists a second best constant for the L2-Nash inequality on any smooth compact Riemannian manif