TY - JOUR

T1 - Decay rates for the quadratic and super-quadratic tilt-excess of integral varifolds

AU - Kolasiński, Sławomir

AU - Menne, Ulrich

N1 - Publisher Copyright:
© 2017, Springer International Publishing.

PY - 2017/4/1

Y1 - 2017/4/1

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.

AB - 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 - Cartesian product of varifolds

KW - Curvature varifold

KW - First variation

KW - Generalised mean curvature vector

KW - Integral varifold

KW - Orlicz space height-excess

KW - Quadratic tilt-excess

KW - Second fundamental form

KW - Super-quadratic tilt-excess

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U2 - 10.1007/s00030-017-0436-z

DO - 10.1007/s00030-017-0436-z

M3 - Article

AN - SCOPUS:85015813162

SN - 1021-9722

VL - 24

JO - Nonlinear Differential Equations and Applications

JF - Nonlinear Differential Equations and Applications

IS - 2

M1 - 17

ER -