Decay rates for the quadratic and super-quadratic tilt-excess of integral varifolds

Sławomir Kolasiński*, Ulrich Menne

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space satisfying integrability conditions on their first variation. Firstly, the study of pointwise power decay rates almost everywhere of the quadratic tilt-excess is completed by establishing the precise decay rate for two-dimensional integral varifolds of locally bounded first variation. In order to obtain the exact decay rate, a coercive estimate involving a height-excess quantity measured in Orlicz spaces is established. Moreover, counter-examples to pointwise power decay rates almost everywhere of the super-quadratic tilt-excess are obtained. These examples are optimal in terms of the dimension of the varifold and the exponent of the integrability condition in most cases, for example if the varifold is not two-dimensional. These examples also demonstrate that within the scale of Lebesgue spaces no local higher integrability of the second fundamental form, of an at least two-dimensional curvature varifold, may be deduced from boundedness of its generalised mean curvature vector. Amongst the tools are Cartesian products of curvature varifolds.

Original languageEnglish
Article number17
JournalNonlinear Differential Equations and Applications
Issue number2
Publication statusPublished - 2017 Apr 1
Externally publishedYes


  • Cartesian product of varifolds
  • Curvature varifold
  • First variation
  • Generalised mean curvature vector
  • Integral varifold
  • Orlicz space height-excess
  • Quadratic tilt-excess
  • Second fundamental form
  • Super-quadratic tilt-excess

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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