Poincare inequalities in punctured domains
成果类型：
Article
署名作者：
Lieb, EH; Seiringer, R; Yngvason, J
刊物名称：
ANNALS OF MATHEMATICS
ISSN/ISSBN：
0003-486X
DOI：
10.4007/annals.2003.158.1067
发表日期：
2003
页码：
1067-1080
关键词：
摘要：
The classic Poincare inequality bounds the L-q-norm of a function f in a bounded domain Omega subset of R-n in terms of some L-P-norm of its gradient in Omega. We generalize this in two ways: In the first generalization we remove a set Gamma from Omega and concentrate our attention on Lambda = Omega \ Gamma. This new domain might not even be connected and hence no Poincare 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 Gamma is arbitrarily small. A Poincare inequality does hold, however, if one makes the additional assumption that f has a finite L-p gradient norm on the whole of Omega, not just on Lambda. The important point is that the Poincare inequality thus obtained bounds the L-q-norm of f in terms of the L-p gradient norm on Lambda (not Omega) plus an additional term that goes to zero as the volume of IF goes to zero. This error term depends on Gamma only through its volume. Apart from this additive error term, the constant in the inequality remains that of the 'nice' domain Q. In the second generalization we are given a vector field A and replace del by del + iA(x) (geometrically, a connection on a U(1) bundle). Unlike the A = 0 case, the infimum of parallel to (del + iA) f parallel to(p) over all f with a given parallel tofparallel to(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.