TY  - JOUR
A1  - Kolasinski, Slawomir
A1  - Menne, Ulrich
T1  - Decay rates for the quadratic and super-quadratic tilt-excess of integral varifolds
JF  - Nonlinear Differential Equations and Applications NoDEA
N2  - 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.
KW  - Integral varifold
KW  - First variation
KW  - Generalised mean curvature vector
KW  - Quadratic tilt-excess
KW  - Super-quadratic tilt-excess
KW  - Orlicz space height-excess
KW  - Curvature varifold
KW  - Second fundamental form
KW  - Cartesian product of varifolds
Y1  - 2017
U6  - https://doi.org/10.1007/s00030-017-0436-z
SN  - 1021-9722
SN  - 1420-9004
VL  - 24
PB  - Springer
CY  - Basel
ER  - 
TY  - JOUR
A1  - Menne, Ulrich
T1  - Weakly Differentiable Functions on Varifolds
JF  - Indiana University mathematics journal
N2  - 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.
KW  - Rectifiable varifold
KW  - (generalised) wealdy differentiable function
KW  - distributional boundary
KW  - decomposition
KW  - relative isoperimetric inequality
KW  - Sobolev Poincare inequality
KW  - approximate differentiability
KW  - coarea formula
KW  - geodesic distance
KW  - curvature varifold
Y1  - 2016
U6  - https://doi.org/10.1512/iumj.2016.65.5829
SN  - 0022-2518
SN  - 1943-5258
VL  - 65
SP  - 977
EP  - 1088
PB  - Indiana University, Department of Mathematics
CY  - Bloomington
ER  - 
TY  - JOUR
A1  - Menne, Ulrich
T1  - Sobolev functions on varifolds
JF  - Proceedings of the London Mathematical Society
N2  - 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.
Y1  - 2016
U6  - https://doi.org/10.1112/plms/pdw023
SN  - 0024-6115
SN  - 1460-244X
VL  - 113
SP  - 725
EP  - 774
PB  - Oxford Univ. Press
CY  - Oxford
ER  -