A note on the uniformity of the constant in the Poincaré inequality

Abstract

The classical Poincaré inequality establishes that for any bounded regular domain $\Omega \subset {\mathbb{R}}^N$ there exists a constant $C=C(\Omega )>0$ such that $${\int }_{\Omega }|u{|}^{2}dx\le C{\int }_{\Omega }|\nabla u{|}^{2}dx\forall u\in H^1(\Omega ),{\int }_{\Omega }u(x)dx=0.$$ In this note we show that $C$ can be taken independently of $\Omega$ when $\Omega$ is in a certain class of domains. Our result generalizes previous results in this direction.

12 pages, 1 figure

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