UM

### Ulrich Menne

• Differential Geometry
• Analysis of PDEs
• Classical Analysis and ODEs
• Functional Analysis
• ## ArticleDecay estimates for the quadratic tilt-excess of integral varifolds

Archive for Rational Mechanics and Analysis 204 (1), 1–83, 2012

• Integral varifold
• tilt-excess
• pointwise decay estimate
• multiple-valued function
• second order quasilinear elliptic PDE

Posted by: Ulrich Menne

This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space with their first variation given by either a Radon measure or a function in some Lebesgue space. Pointwise decay results for the quadratic tilt-excess are established for those varifolds. The results are optimal in terms of the dimension of the varifold and the exponent of the Lebesgue space in most cases, for example if the varifold is not two-dimensional.

• ## ArticleA Sobolev Poincaré type inequality for integral varifolds

Calculus of Variations and Partial Differential Equations 38 (3-4), 369–408, 2010

• Integral varifold
• Sobolev Poincaré inequality
• multiple-valued function

Posted by: Ulrich Menne

DOIarXivMSC 2010: 49Q15 26B05

In this work a local inequality is provided which bounds the distance of an integral varifold from a multivalued plane (height) by its tilt and mean curvature. The bounds obtained for the exponents of the Lebesgue spaces involved are shown to be sharp.

• ## ArticleSome applications of the isoperimetric inequality for integral varifolds

Advances in Calculus of Variations 2 (3), 247–269, 2009

• Integral varifold
• locally bounded first variation
• height-excess
• tilt-excess

Posted by: Ulrich Menne

DOIarXivMSC 2010: 49Q15 26B35

In this work the isoperimetric inequality for integral varifolds of locally bounded first variation is used to obtain sharp estimates for the size of the set where the density quotient is small and to generalise Calderón's and Zygmund's theory of first order differentiability for functions in Lebesgue spaces from Lebesgue measure to integral varifolds.