Products of Poincaré domains

Abstract:

A domain $\Omega \subseteq {\mathbb {R}^N}$ of finite $N$-dimensional Lebesgue measure is a $p$-Poincaré domain $(1 \leqslant p \leqslant \infty )$ if there exists a positive constant $K$ such that the $p$-Poincaré inequality $||u|{|_{{L^p}(\Omega )}} \leqslant K||\nabla u|{|_{{L^p}(\Omega )}}$ is valid for all Sobolev functions $u \in {W^{1,p}}(\Omega )$ that integrate to zero. Define ${K_p}(\Omega )$ to be the smallest such $K$ if $\Omega$ is a $p$-Poincaré domain and to be infinity otherwise. We obtain comparability relations between ${K_p}({\Omega _1} \times {\Omega _2})$ and the pair ${K_p}({\Omega _1}),\;{K_p}({\Omega _2})$. In particular, our results show that $p$-Poincaré domains are closed under cartesian products (for all $p$), and that in case $p$ equals $2$, we have ${K_2}({\Omega _1} \times {\Omega _2}) = \max \{ {K_2}({\Omega _1}),\;{K_2}({\Omega _2})\}$.