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Formation of cracks under deformations with finite energy

  • Piotr HajłaszEmail author
  • Pekka Koskela
Article

Abstract.

With a map \(f: \Omega\to {\bf R}^n\), \(\Omega\subset {\bf R}^n\), that belongs to the John Ball class \(A_{p,q}^{ + }(\Omega)\) where n-1 < p < n and \(q\geq p/(p-1)\) one can associate a set valued map F whose values \(F(x)\subset {\bf R}^n\) are subsets of \({\bf R}^n\) describing the topological character of the singularity of f at \(x\in\Omega\). Šverak conjectured that \({\cal H}^{n-1}(F(S)) = 0\), where S is the set of points at which f is not continuous and \({\cal H}^{n-1}\) is the Hausdorff measure. The purpose of our paper is to confirm this expectation.

Keywords

Hausdorff Measure Finite Energy Ball Class Topological Character 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin/Heidelberg 2004

Authors and Affiliations

  1. 1.Institute of MathematicsWarsaw UniversityWarszawaPoland

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