Table of contents

This page was last modified on: 05 April 2021, 16:08, CST

Differentiability theory for distributions and subsets of Euclidean space

U. Menne
[14] Pointwise differentiability of higher-order for distributions, 32 pages.
Anal. PDE, 14(2):323–354, 2021.
DOI: 10.2140/apde.2021.14.323. ArXiv: 1803.10855v2 [math.FA].
Abstract: For distributions, we build a theory of higher order pointwise differentiability comprising, for order zero, Łojasiewicz's notion of point value. Results include Borel regularity of differentials, higher order rectifiability of the associated jets, a Rademacher-Stepanov type differentiability theorem, and a Lusin type approximation. A substantial part of this development is new also for zeroth order. Moreover, we establish a Poincaré inequality involving the natural norms of negative order of differentiability. As a corollary, we characterise pointwise differentiability in terms of point values of distributional partial derivatives.

U. Menne, M. Santilli
[11] A geometric second-order-rectifiable stratification for closed subsets of Euclidean space, 14 pages.
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 19(3):1185–1198, 2019.
DOI: 10.2422/2036-2145.201703_021. ArXiv: 1703.09561v2 [math.CA].
Abstract: Defining the m-th stratum of a closed subset of an n dimensional Euclidean space to consist of those points, where it can be touched by a ball from at least n−m linearly independent directions, we establish that the m-th stratum is second-order rectifiable of dimension m and a Borel set. This was known for convex sets, but is new even for sets of positive reach. The result is based on a sufficient condition of parametric type for second-order rectifiability.

U. Menne
[9] Pointwise differentiability of higher order for sets, 31 pages.
Ann. Global Anal. Geom., 55(3), 591–621, 2019.
Springer Nature SharedIt:
DOI: 10.1007/s10455-018-9642-0. ArXiv: 1603.08587v2 [math.DG].
Abstract: The present paper develops two concepts of pointwise differentiability of higher order for arbitrary subsets of Euclidean space defined by comparing their distance functions to those of smooth submanifolds. Results include that differentials are Borel functions, higher order rectifiability of the set of differentiability points, and a Rademacher result. One concept is characterised by a limit procedure involving inhomogeneously dilated sets. The original motivation to formulate the concepts stems from studying the support of stationary integral varifolds. In particular, strong pointwise differentiability of every positive integer order is shown at almost all points of the intersection of the support with a given plane.

Survey on varifolds suitable for beginning graduate students

U. Menne
[12] The concept of varifold, 5 pages.
Notices Amer. Math. Soc., 64(10):1148–1152, 2017.
DOI: 10.1090/noti1589. ArXiv: 1705.05253v4 [math.DG].
Abstract: We survey – by means of 20 examples – the concept of varifold, as generalised submanifold, with emphasis on regularity of integral varifolds with mean curvature, while keeping prerequisites to a minimum. Integral varifolds are the natural language for studying the variational theory of the area integrand if one considers, for instance, existence or regularity of stationary (or, stable) surfaces of dimension at least three, or the limiting behaviour of sequences of smooth submanifolds under area and mean curvature bounds.

Partial differential equations on singular surfaces with mean curvature – foundations

U. Menne
[8] Sobolev functions on varifolds, 50 pages.
Proc. Lond. Math. Soc. (3), 113(6):725–774, 2016.
Wiley Content Sharing:
DOI: 10.1112/plms/pdw023. ArXiv: 1509.01178v3 [math.CA].
Abstract: 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 Hölder 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.

U. Menne
[6] Weakly differentiable functions on varifolds 112 pages.
Indiana Univ. Math. J., 65(3):977–1088, 2016.
DOI: 10.1512/iumj.2016.65.5829. ArXiv: 1411.3287v1 [math.DG].
Abstract: 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 Poincaré-type embeddings, embeddings into spaces of continuous and sometimes Hölder-continuous functions, and pointwise 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.

Regularity of singular surfaces with mean curvature

S. Kolasiński, U. Menne
[7] Decay rates for the quadratic and super-quadratic tilt-excess of integral varifolds, 56 pages.
NoDEA Nonlinear Differential Equations Appl., 24:Art. 17, 56, 2017.
DOI: 10.1007/s00030-017-0436-z. ArXiv: 1501.07037v2 [math.DG].
Abstract: 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.

U. Menne
[4] Second order rectifiability of integral varifolds of locally bounded first variation, 55 pages.
J. Geom. Anal., 23(2):709–763, 2013.
DOI: 10.1007/s12220-011-9261-5. ArXiv: 0808.3665v3 [math.DG].
Abstract: It is shown that every integral varifold in an open subset of Euclidean space whose first variation with respect to area is representable by integration can be covered by a countable collection of submanifolds of the same dimension of class 2 and that their mean curvature agrees almost everywhere with the variationally defined generalized mean curvature of the varifold.

U. Menne
[3] Decay estimates for the quadratic tilt-excess of integral varifolds, 83 pages.
Arch. Ration. Mech. Anal., 204(1):1–83, 2012.
DOI: 10.1007/s00205-011-0468-1. ArXiv: 0909.3253v3 [math.DG].
Abstract: 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.

Basic geometric properties of singular surfaces with mean curvature

U. Menne, C. Scharrer
[13] A novel type of Sobolev-Poincaré inequality for submanifolds of Euclidean space, 35 pages.
In revision (to include one or more applications), 2017.
ArXiv: 1709.05504v1 [math.DG].
Abstract: For functions on generalised connected surfaces (of any dimensions) with boundary and mean curvature, we establish an oscillation estimate in which the mean curvature enters in a novel way. As application we prove an a priori estimate of the geodesic diameter of compact connected smooth immersions in terms of their boundary data and mean curvature. These results are developed in the framework of varifolds. For this purpose, we establish that the notion of indecomposability is the appropriate substitute for connectedness and that it has a strong regularising effect; we thus obtain a new natural class of varifolds to study. Finally, our development leads to a variety of questions that are of substance both in the smooth and the nonsmooth setting.

U. Menne, C. Scharrer
[10] An isoperimetric inequality for diffused surfaces, 16 pages.
Kodai Math. J., 41(1):70–85, 2018.
DOI: 10.2996/kmj/1521424824. ArXiv: 1612.03823v2 [math.DG].
Abstract: For general varifolds in Euclidean space, we prove an isoperimetric inequality, adapt the basic theory of generalised weakly differentiable functions, and obtain several Sobolev type inequalities. We thereby intend to facilitate the use of varifold theory in the study of diffused surfaces.

U. Menne
[2] A Sobolev Poincaré type inequality for integral varifolds, 40 pages.
Calc. Var. Partial Differential Equations, 38(3-4):369–408, 2010.
DOI: 10.1007/s00526-009-0291-9. ArXiv: 0808.3660v2 [math.DG].
Abstract: 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.

U. Menne
[1] Some applications of the isoperimetric inequality for integral varifolds, 23 pages.
Adv. Calc. Var., 2(3):247–269, 2009.
DOI: 10.1515/ACV.2009.010. ArXiv: 0808.3652v1 [math.DG].
Abstract: 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.

Conference proceedings

U. Menne
[5] A sharp lower bound on the mean curvature integral with critical power for integral varifolds, 3 pages.
In abstracts from the workshop held July 22-28, 2012, Organized by Camillo De Lellis, Gerhard Huisken and Robert Jerrard, Oberwolfach Reports. Vol. 9, no. 3, 2012.
DOI: 10.4171/OWR/2012/36.

List of minor corrections

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