Poincaré inequalities in punctured domains

Autor(en)

E H Lieb, Robert Seiringer, Jakob Yngvason

Abstrakt

The classic PoincareŽ inequality bounds the Lq-norm of a function f in a bounded domain O ? Rn in terms of some Lp-norm of its gradient in O. We generalize this in two ways: In the first generalization we remove a set ? from O and concentrate our attention on ? = O \ ?. This new domain might not even be connected and hence no PoincareŽ 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 ? is arbitrarily small. A PoincareŽ inequality does hold, however, if one makes the additional assumption that f has a finite Lp gradient norm on the whole of O, not just on ?. The important point is that the PoincareŽ inequality thus obtained bounds the Lq-norm of f in terms of the Lp gradient norm on ? (not O) plus an additional term that goes to zero as the volume of ? goes to zero. This error term depends on ? only through its volume. Apart from this additive error term, the constant in the inequality remains that of the 'nice' domain O. 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.

Organisation(en)

Mathematische Physik

Externe Organisation(en)

Princeton University

Journal

Annals of Mathematics

Band

158

Seiten

1067-1080

Anzahl der Seiten

14

ISSN

0003-486X

DOI

https://doi.org/10.4007/annals.2003.158.1067

Publikationsdatum

2003

Peer-reviewed

Ja

ÖFOS 2012

103036 Theoretische Physik

Link zum Portal

https://ucrisportal.univie.ac.at/de/publications/99431f32-66f6-4b8f-8d8f-58288647efbe