Application of Scans and Fractional Power Integrands

  • Thierry De Pauw
  • Robert Hardt
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 59)


In this note we describe the notion of a rectifiable scan and consider some applications [DH1], [DH2] to Plateau-type minimization problems. “Scans” were first introduced in the work [HR1] of Tristan Rivière and the second author to adequately describe certain bubbling phenomena. There, the behaviour of certain W 1,3 weakly convergent sequences of smooth maps from four-dimensional domains into S 2 led to the consideration of a necessarily infinite mass generalization of a rectifiable current. The definition of a scan is motivated by the fact that a rectifiable current can be expressed in terms of its lower-dimensional slices by oriented affine subspaces. By integral geometry, the slicing function for the rectifiable current is a mass integrable function of the subspaces. With a scan one considers more general such functions that are not necessarily mass integrable.


Soap Film Geometric Measure Theory Rectifiable Current Interior Regularity Plateau Problem 
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Copyright information

© Springer Basel AG 2004

Authors and Affiliations

  • Thierry De Pauw
    • 1
  • Robert Hardt
    • 2
  1. 1.Université Catholique de LouvainBelgium
  2. 2.Rice UniversityUSA

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