Formation of cracks under deformations with finite energy

  • Piotr HajłaszEmail author
  • Pekka Koskela


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.


Hausdorff Measure Finite Energy Ball Class Topological Character 
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© Springer-Verlag Berlin/Heidelberg 2004

Authors and Affiliations

  1. 1.Institute of MathematicsWarsaw UniversityWarszawaPoland

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