@article{Menne2016,
  author    = {Menne, Ulrich},
  title     = {Weakly Differentiable Functions on Varifolds},
  series = {Indiana University mathematics journal},
  volume    = {65},
  journal   = {Indiana University mathematics journal},
  publisher = {Indiana University, Department of Mathematics},
  address   = {Bloomington},
  issn      = {0022-2518},
  doi       = {10.1512/iumj.2016.65.5829},
  pages     = {977 -- 1088},
  year      = {2016},
  abstract  = {The present paper is intended to provide the basis for the study of weakly differentiable functions on rectifiable varifolds with locally bounded first variation. The concept proposed here is defined by means of integration-by-parts identities for certain compositions with smooth functions. In this class, the idea of zero boundary values is realised using the relative perimeter of superlevel sets. Results include a variety of Sobolev Poincare-type embeddings, embeddings into spaces of continuous and sometimes Holder-continuous functions, and point wise differentiability results both of approximate and integral type as well as coarea formulae. As a prerequisite for this study, decomposition properties of such varifolds and a relative isoperimetric inequality are established. Both involve a concept of distributional boundary of a set introduced for this purpose. As applications, the finiteness of the geodesic distance associated with varifolds with suitable summability of the mean curvature and a characterisation of curvature varifolds are obtained.},
  language  = {en}
}
@article{Menne2016,
  author    = {Menne, Ulrich},
  title     = {Sobolev functions on varifolds},
  series = {Proceedings of the London Mathematical Society},
  volume    = {113},
  journal   = {Proceedings of the London Mathematical Society},
  publisher = {Oxford Univ. Press},
  address   = {Oxford},
  issn      = {0024-6115},
  doi       = {10.1112/plms/pdw023},
  pages     = {725 -- 774},
  year      = {2016},
  abstract  = {This paper introduces first-order Sobolev spaces on certain rectifiable varifolds. These complete locally convex spaces are contained in the generally non-linear class of generalised weakly differentiable functions and share key functional analytic properties with their Euclidean counterparts. Assuming the varifold to satisfy a uniform lower density bound and a dimensionally critical summability condition on its mean curvature, the following statements hold. Firstly, continuous and compact embeddings of Sobolev spaces into Lebesgue spaces and spaces of continuous functions are available. Secondly, the geodesic distance associated to the varifold is a continuous, not necessarily Holder continuous Sobolev function with bounded derivative. Thirdly, if the varifold additionally has bounded mean curvature and finite measure, then the present Sobolev spaces are isomorphic to those previously available for finite Radon measures yielding many new results for those classes as well. Suitable versions of the embedding results obtained for Sobolev functions hold in the larger class of generalised weakly differentiable functions.},
  language  = {en}
}
@article{KolasinskiMenne2017,
  author    = {Kolasinski, Slawomir and Menne, Ulrich},
  title     = {Decay rates for the quadratic and super-quadratic tilt-excess of integral varifolds},
  series = {Nonlinear Differential Equations and Applications NoDEA},
  volume    = {24},
  journal   = {Nonlinear Differential Equations and Applications NoDEA},
  publisher = {Springer},
  address   = {Basel},
  issn      = {1021-9722},
  doi       = {10.1007/s00030-017-0436-z},
  pages     = {56},
  year      = {2017},
  abstract  = {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.},
  language  = {en}
}