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An improved Poincaré inequality


Author: Ritva Hurri-Syrjänen
Journal: Proc. Amer. Math. Soc. 120 (1994), 213-222
MSC: Primary 46E35; Secondary 26D20
DOI: https://doi.org/10.1090/S0002-9939-1994-1169032-X
MathSciNet review: 1169032
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Abstract: We show that a large class of domains $ D$ in $ {\mathbb{R}^n}$ including John domains satisfies the improved Poincaré inequality

$\displaystyle \vert\vert u(x) - {u_D}\vert{\vert _{{L^q}(D)}} \leqslant c\vert\vert\nabla u(x)d{(x,\partial D)^\delta }\vert{\vert _{{L^p}(D)}}$

where $ p \leqslant q \leqslant \tfrac{{np}} {{n - p(1 - \delta )}},\;p(1 - \delta ) < n,\;\delta \in [0,1],\;c = c(p,q,\delta ,D) < \infty $, and $ u$ is in an appropriate Sobolev class.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1169032-X
Keywords: Poincaré inequality, Poincaré domains, John domains, domains satisfying a quasihyperbolic boundary condition
Article copyright: © Copyright 1994 American Mathematical Society

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