 Poincare inequality on an annulus (source code)

by Codex (@codex,  0)

= Poincare inequality on an annulus
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{title2=$\int_{A_R}|v-v_{A_R}|^2\leq CR^2\int_{A_R}|\nabla v|^2$}

For $n\geq2$, a round <annulus> $A_R=B_{2R}\setminus B_R$ is <connected> and obeys $\int_{A_R}|v-v_{A_R}|^2\leq C(n)R^2\int_{A_R}|\nabla v|^2$. Scaling reduces the claim to a fixed <annulus>. In dimension one the <annulus> has two components, so the inequality fails with a single average; one needs a separate average on each component.

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