# Poincaré inequalities in punctured domains

Lieb É, Seiringer R, Yngvason J. 2003. Poincaré inequalities in punctured domains. Annals of Mathematics. 158(3), 1067–1080.

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Lieb, Élliott H;
Seiringer, Robert

^{ISTA}^{}; Yngvason, JakobAbstract

The classic Poincaré inequality bounds the L q-norm of a function f in a bounded domain Ω ⊂ ℝ n in terms of some L p-norm of its gradient in Ω. We generalize this in two ways: In the first generalization we remove a set Τ from Ω and concentrate our attention on Λ = Ω \ Τ. This new domain might not even be connected and hence no Poincaré inequality can generally hold for it, or if it does hold it might have a very bad constant. This is so even if the volume of Τ is arbitrarily small. A Poincaré inequality does hold, however, if one makes the additional assumption that f has a finite L p gradient norm on the whole of Ω, not just on Λ. The important point is that the Poincaré inequality thus obtained bounds the L q-norm of f in terms of the L p gradient norm on Λ (not Ω) plus an additional term that goes to zero as the volume of Τ goes to zero. This error term depends on Τ only through its volume. Apart from this additive error term, the constant in the inequality remains that of the 'nice' domain Ω. In the second generalization we are given a vector field A and replace ∇ by ∇ + iA(x) (geometrically, a connection on a U(1) bundle). Unlike the A = 0 case, the infimum of ∥(∇ + iA)f∥ p over all f with a given ∥f∥ q is in general not zero. This permits an improvement of the inequality by the addition of a term whose sharp value we derive. We describe some open problems that arise from these generalizations.

Publishing Year

Date Published

2003-11-01

Journal Title

Annals of Mathematics

Volume

158

Issue

3

Page

1067 - 1080

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### Cite this

Lieb É, Seiringer R, Yngvason J. Poincaré inequalities in punctured domains.

*Annals of Mathematics*. 2003;158(3):1067-1080. doi:10.4007/annals.2003.158.1067Lieb, É., Seiringer, R., & Yngvason, J. (2003). Poincaré inequalities in punctured domains.

*Annals of Mathematics*. Princeton University Press. https://doi.org/10.4007/annals.2003.158.1067Lieb, Élliott, Robert Seiringer, and Jakob Yngvason. “Poincaré Inequalities in Punctured Domains.”

*Annals of Mathematics*. Princeton University Press, 2003. https://doi.org/10.4007/annals.2003.158.1067 .É. Lieb, R. Seiringer, and J. Yngvason, “Poincaré inequalities in punctured domains,”

*Annals of Mathematics*, vol. 158, no. 3. Princeton University Press, pp. 1067–1080, 2003.Lieb, Élliott, et al. “Poincaré Inequalities in Punctured Domains.”

*Annals of Mathematics*, vol. 158, no. 3, Princeton University Press, 2003, pp. 1067–80, doi:10.4007/annals.2003.158.1067 .**All files available under the following license(s):**

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