A note on the uniformity of the constant in the Poincaré inequality
arXiv:1208.6045
Abstract
The classical Poincaré inequality establishes that for any bounded regular domain $Ω\subset \R^N$ there exists a constant $C=C(Ω)>0$ such that $$ \int_Ω |u|^2\, dx \leq C \int_Ω |\nabla u|^2\, dx \ \ \forall u \in H^1(Ω),\ \int_Ω u(x) \, dx=0.$$ In this note we show that $C$ can be taken independently of $Ω$ when $Ω$ is in a certain class of domains. Our result generalizes previous results in this direction.
12 pages, 1 figure