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Topological obstructions to continuity of Orlicz–Sobolev mappings of finite distortion

  • Paweł Goldstein
  • Piotr Hajłasz
Article
  • 27 Downloads

Abstract

In the paper we investigate continuity of Orlicz–Sobolev mappings \(W^{1,P}(M,N)\) of finite distortion between smooth Riemannian n-manifolds, \(n\ge 2\), under the assumption that the Young function P satisfies the so-called divergence condition \(\int _1^\infty P(t)/t^{n+1}\, \hbox {d}t=\infty \). We prove that if the manifolds are oriented, N is compact, and the universal cover of N is not a rational homology sphere, then such mappings are continuous. That includes mappings with \(Df\in L^n\) and, more generally, mappings with \(Df\in L^n\log ^{-1}L\). On the other hand, if the space \(W^{1,P}\) is larger than \(W^{1,n}\) (for example if \(Df\in L^n\log ^{-1}L\)), and the universal cover of N is homeomorphic to \(\mathbb {S}^n\), \(n\ne 4\), or is diffeomorphic to \(\mathbb {S}^n\), \(n=4\), then we construct an example of a mapping in \(W^{1,P}(M,N)\) that has finite distortion and is discontinuous. This demonstrates a new global-to-local phenomenon: Both finite distortion and continuity are local properties, but a seemingly local fact that finite distortion implies continuity is a consequence of a global topological property of the target manifold N.

Keywords

Orlicz–Sobolev mappings Rational homology spheres Mappings of finite distortion 

Mathematics Subject Classification

Primary 30C65 Secondary 46E35 58C07 

Notes

Acknowledgements

The authors would like to thank Armin Schikorra for a helpful discussion about fractional Sobolev spaces. Funding was provided by Narodowe Centrum Nauki (Grant No. 2012/05/E/ST1/03232) and National Science Foundation (Grant No. DMS-1800457).

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© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Mathematics, Faculty of Mathematics, Informatics and MechanicsUniversity of WarsawWarsawPoland
  2. 2.Department of MathematicsUniversity of PittsburghPittsburghUSA

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