Irregular 1-sets on the graphs of continuous functions

One can define in a natural way irregular 1-sets on the graphs of several fractal functions, like Takagi’s function, Weierstrass-Cellerier type functions and the typical continuous function. These irregular 1-sets can be useful during the investigation of level-sets and occupation measures of these functions. For example, we see that for Takagi’s function and for certain Weierstrass-Cellerier functions the occupation measure is singular with respect to the Lebesgue measure and for almost every level the level set is finite.     

Growth Estimates for Cauchy Integrals of Measures and Rectifiability

We show that if μ is a finite Borel measure on the complex plane such that

Structure of entropy solutions to the eikonal equation

In this paper, we establish rectifiability of the jump set of an S ¹–valued conservation law in two space–dimensions. This conservation law is a reformulation of the eikonal equation and is motivated by the singular limit of a class of variational problems. The only assumption on the weak solutions is that the entropy productions are (signed) Radon measures, an assumption which is justified by the variational origin. The methods are a combination of Geometric Measure Theory and elementary geometric arguments used to classify blow–ups.?The merit of our approach is that we obtain the structure as if the solutions were in BV, without using the BV–control, which is not available in these variationally motivated problems. Received June 24, 2002 / final version received November 12, 2002?Published online February 7, 2003     

Rectifiability of Measures with Locally Uniform Cube Density

The conjecture that Radon measures in Euclidean space with positivefinite density are rectifiable was a central problem in GeometricMeasure Theory for fifty years. This conjecture was positivelyresolved by Preiss in 1986, using methods entirely dependenton the symmetry of the Euclidean unit ball. Since then, dueto reasons of isometric immersion of metric spaces into l andthe uncommon nature of the sup norm even in finite dimensions,a popular model problem for generalising this result to non-Euclideanspaces has been the study of 2-uniform measures in . The rectifiability or otherwise of these measureshas been a well-known question. In this paper the stronger result that locally 2-uniform measuresin are rectifiable is proved. This is the first result that proves rectifiability, from aninitial condition about densities, for general Radon measuresof dimension greater than 1 outside Euclidean space. 2000 MathematicalSubject Classification: 28A75.     

关于极限limx→0sinx/x的注记

讨论了极限limx→0sinx/x=1,认为该极限本质上与(单位)圆周可求长是等价的.     

Principal values for the Cauchy integral and rectifiability

We give a geometric characterization of those positive finite measures on with the upper density finite at -almost every , such that the principal value of the Cauchy integral of ,

{\varepsilon}} \frac{1}{\xi-z}\, d\mu(\xi),\end{displaymath}">

exists for -almost all . This characterization is given in terms of the curvature of the measure . In particular, we get that for , -measurable (where is the Hausdorff -dimensional measure) with , if the principal value of the Cauchy integral of exists -almost everywhere in , then is rectifiable.

     

Quasiminimal crystals with a volume constraint and uniform rectifiability

We establish here, in a quite general context, uniform rectifiability properties for quasiminimal crystals with a volume constraint. Namely we prove that to any quasiminimal crystal with a volume constraint corresponds a unique equivalent open set whose boundary is Ahlfors-regular and which satisfies the so-called condition B. Moreover implicit bounds in these properties, which imply the uniform rectifiability of the boundary, can be chosen universal. As a consequence we give a universal upper bound for the number of connected components of reduced quasiminimizers and we also prove that quasiminimal crystals with a volume constraint actually satisfy, in some universal way, an apparently stronger quasiminimality condition where admissible perturbations are not required to be volume-preserving anymore.     

The co-area formula for Sobolev mappings

We extend Federer's co-area formula to mappings belonging to the Sobolev class , , m$">, and more generally, to mappings with gradient in the Lorentz space . This is accomplished by showing that the graph of in is a Hausdorff -rectifiable set.

     

Functions with prescribed singularities

The distributional k-dimensional Jacobian of a map u in the Sobolev space W^1,k-1 which takes values in the the sphere S^k-1 can be viewed as the boundary of a rectifiable current of codimension k carried by (part of) the singularity of u which is topologically relevant. The main purpose of this paper is to investigate the range of the Jacobian operator; in particular, we show that any boundary M of codimension k can be realized as Jacobian of a Sobolev map valued in S^k-1. In case M is polyhedral, the map we construct is smooth outside M plus an additional polyhedral set of lower dimension, and can be used in the constructive part of the proof of a -convergence result for functionals of Ginzburg-Landau type, as described in [2]. Mathematics Subject Classification (2000) 46E35 (53C65, 49Q15, 26B10, 58A25)     

A nicely behaved singular integral on a purely unrectifiable set

We construct an example of a purely 1-unrectifiable AD-regular set in the plane such that the limit

exists and is finite for almost every for some class of antisymmetric Calderón-Zygmund kernels. Moreover, the singular integral operators associated with these kernels are bounded in , where has a positive measure.