An improved Poincaré inequality

Hurri-Syrjänen, Ritva

doi:10.1090/S0002-9939-1994-1169032-X

An improved Poincaré inequality
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Proc. Amer. Math. Soc. 120 (1994), 213-222 Request permission

Abstract:

We show that a large class of domains $D$ in ${\mathbb {R}^n}$ including John domains satisfies the improved Poincaré inequality \[ ||u(x) - {u_D}|{|_{{L^q}(D)}} \leqslant c||\nabla u(x)d{(x,\partial D)^\delta }|{|_{{L^p}(D)}}\] 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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