Decay rates for the quadratic and super-quadratic tilt-excess of integral varifolds
- 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 boundednessThis 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.…
Author details: | Slawomir KolasinskiORCiD, Ulrich MenneORCiDGND |
---|---|
DOI: | https://doi.org/10.1007/s00030-017-0436-z |
ISSN: | 1021-9722 |
ISSN: | 1420-9004 |
Title of parent work (English): | Nonlinear Differential Equations and Applications NoDEA |
Publisher: | Springer |
Place of publishing: | Basel |
Publication type: | Article |
Language: | English |
Date of first publication: | 2017/03/20 |
Publication year: | 2017 |
Release date: | 2022/06/13 |
Tag: | 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 |
Volume: | 24 |
Number of pages: | 56 |
Funding institution: | Foundation for Polish Science; NCN [2013/10/M/ST1/00416] |
Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik |
DDC classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
Peer review: | Referiert |