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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Products of Poincaré domains


Author: Alexander Stanoyevitch
Journal: Proc. Amer. Math. Soc. 117 (1993), 79-87
MSC: Primary 46E35
DOI: https://doi.org/10.1090/S0002-9939-1993-1104403-8
MathSciNet review: 1104403
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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 $ \vert\vert u\vert{\vert _{{L^p}(\Omega )}} \leqslant K\vert\vert\nabla u\vert{\vert _{{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})\} $.


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DOI: https://doi.org/10.1090/S0002-9939-1993-1104403-8
Keywords: Poincaré domain, Poincaré inequality, Sobolev space, Neumann Laplacian
Article copyright: © Copyright 1993 American Mathematical Society

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