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  -