{"_id":"2357","abstract":[{"text":"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.","lang":"eng"}],"main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/math/0205088"}],"date_published":"2003-11-01T00:00:00Z","extern":1,"publication_status":"published","citation":{"ama":"Lieb É, Seiringer R, Yngvason J. Poincaré inequalities in punctured domains. <i>Annals of Mathematics</i>. 2003;158(3):1067-1080. doi:<a href=\"https://doi.org/10.4007/annals.2003.158.1067 \">10.4007/annals.2003.158.1067 </a>","chicago":"Lieb, Élliott, Robert Seiringer, and Jakob Yngvason. “Poincaré Inequalities in Punctured Domains.” <i>Annals of Mathematics</i>. Princeton University Press, 2003. <a href=\"https://doi.org/10.4007/annals.2003.158.1067 \">https://doi.org/10.4007/annals.2003.158.1067 </a>.","apa":"Lieb, É., Seiringer, R., &#38; Yngvason, J. (2003). Poincaré inequalities in punctured domains. <i>Annals of Mathematics</i>. Princeton University Press. <a href=\"https://doi.org/10.4007/annals.2003.158.1067 \">https://doi.org/10.4007/annals.2003.158.1067 </a>","ista":"Lieb É, Seiringer R, Yngvason J. 2003. Poincaré inequalities in punctured domains. Annals of Mathematics. 158(3), 1067–1080.","short":"É. Lieb, R. Seiringer, J. Yngvason, Annals of Mathematics 158 (2003) 1067–1080.","mla":"Lieb, Élliott, et al. “Poincaré Inequalities in Punctured Domains.” <i>Annals of Mathematics</i>, vol. 158, no. 3, Princeton University Press, 2003, pp. 1067–80, doi:<a href=\"https://doi.org/10.4007/annals.2003.158.1067 \">10.4007/annals.2003.158.1067 </a>.","ieee":"É. Lieb, R. Seiringer, and J. Yngvason, “Poincaré inequalities in punctured domains,” <i>Annals of Mathematics</i>, vol. 158, no. 3. Princeton University Press, pp. 1067–1080, 2003."},"issue":"3","month":"11","date_updated":"2021-01-12T06:57:00Z","type":"journal_article","doi":"10.4007/annals.2003.158.1067 ","page":"1067 - 1080","day":"01","author":[{"first_name":"Élliott","last_name":"Lieb","full_name":"Lieb, Élliott H"},{"orcid":"0000-0002-6781-0521","first_name":"Robert","last_name":"Seiringer","id":"4AFD0470-F248-11E8-B48F-1D18A9856A87","full_name":"Robert Seiringer"},{"full_name":"Yngvason, Jakob","last_name":"Yngvason","first_name":"Jakob"}],"publication":"Annals of Mathematics","quality_controlled":0,"title":"Poincaré inequalities in punctured domains","status":"public","publisher":"Princeton University Press","year":"2003","oa":1,"intvolume":"       158","volume":158,"date_created":"2018-12-11T11:57:11Z","publist_id":"4570"}