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Poincaré inequalities in punctured domains

2003; Princeton University; Volume: 158; Issue: 3 Linguagem: Inglês

10.4007/annals.2003.158.1067

1939-8980

Élliott H. Lieb, Robert Seiringer, Jakob Yngvason,

Elasticity and Material Modeling

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 Γ from Ω and concentrate our attention on Λ = Ω \ Γ.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 Γ 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 Γ goes to zero.This error term depends on Γ 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.