manuscripta mathematica

, Volume 127, Issue 2, pp 249–273

Weighted Hardy inequalities and the size of the boundary

Article

DOI: 10.1007/s00229-008-0208-5

Cite this article as:
Lehrbäck, J. manuscripta math. (2008) 127: 249. doi:10.1007/s00229-008-0208-5
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Abstract

We establish necessary and sufficient conditions for a domain \({\Omega \subset \mathbb{R}^n}\) to admit the (p, β)-Hardy inequality \({\int_{\Omega} |u|^p d_{\Omega}^{\beta-p} \leq C \int_{\Omega} |\nabla u|^p d_{\Omega}^\beta}\) , where d(x) = dist(x, ∂Ω) and \({u \in C_0^\infty(\Omega)}\) . Our necessary conditions show that a certain dichotomy holds, even locally, for the dimension of the complement Ωc when Ω admits a Hardy inequality, whereas our sufficient conditions can be applied in numerous situations where at least a part of the boundary ∂Ω is “thin”, contrary to previously known conditions where ∂Ω or Ωc was always assumed to be “thick” in a uniform way. There is also a nice interplay between these different conditions that we try to point out by giving various examples.

Mathematics Subject Classification (2000)

Primary 46E35 26D15 Secondary 28A78 

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of JyväskyläJyväskyläFinland

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