Sets Of Finite Perimeter Research Articles

Measure-Theoretic Properties of Level Sets of Distance Functions

We consider the level sets of distance functions from the point of view of geometric measure theory. This lays the foundation for further research that can be applied, among other uses, to the derivation of a shape calculus based on the level-set method. Particular focus is put on the $$(n-1)$$ -dimensional Hausdorff measure of these level sets. We show that, starting from a bounded set, all sub-level sets of its distance function have finite perimeter. Furthermore, if a uniform-density condition is satisfied for the initial set, one can even show an upper bound for the perimeter that is uniform for all level sets. Our results are similar to existing results in the literature, with the important distinction that they hold for all level sets and not just almost all. We also present an example demonstrating that our results are sharp in the sense that no uniform upper bound can exist if our uniform-density condition is not satisfied. This is even true if the initial set is otherwise very regular (i.e., a bounded Caccioppoli set with smooth boundary).

The integral of the normal and fluxes over sets of finite perimeter

Given two intersecting sets of finite perimeter, E_{1} and E_{2} , with unit normals \nu_{1} and \nu_{2} respectively, we obtain a bound on the integral of \nu_{1} over the reduced boundary of E_{1} inside E_{2} . This bound depends only on the perimeter of E_{2} . For any vector field F\colon\mathbb R^{n}\to\mathbb R^{n} with the property that F \in L^{\infty} and div F is a (signed) Radon measure, we obtain bounds on the flux of F over the portion of the reduced boundary of E_{1} inside E_{2} . These results are then applied to study the limit of surfaces with perimeter growing to infinity.

A NOTE ON SOME TOPOLOGICAL PROPERTIES OF SETS WITH FINITE PERIMETER

AbstractSome well-known results about the 2-density topology on ${\mathcal R}$ (in particular in the context of the Lusin–Menchoff property) are extended to τbm, i.e. the m-density topology on ${\mathcal R}$n with m ∈ (n,+∞). Every set of finite perimeter in ${\mathcal R}$n is equivalent (in measure) to a set in τbm0, where m0=n+1+${1\over n-1}$. There exists a set of finite perimeter in ${\mathcal R}$n which is not equivalent (in measure) to any member in the a.e.-modification of τbm, whatever m ∈ [n,+∞).

Boundary measures, generalized Gauss–Green formulas, and mean value property in metric measure spaces

We study mean value properties of harmonic functions in metric measure spaces. The metric measure spaces we consider have a doubling measure and support a (1, 1)-Poincaré inequality. The notion of harmonicity is based on the Dirichlet form defined in terms of a Cheeger differentiable structure. By studying fine properties of the Green function on balls, we characterize harmonic functions in terms of a mean value property. As a consequence, we obtain a detailed description of Poisson kernels. We shall also obtain a Gauss–Green type formula for sets of finite perimeter which posses a Minkowski content characterization of the perimeter. For the Gauss–Green formula we introduce a suitable notion of the interior normal trace of a regular ball.

Book Review: Sets of finite perimeter and geometric variational problems. An introduction to geometric measure theory

Book Review: Sets of finite perimeter and geometric variational problems. An introduction to geometric measure theory

On Some Perturbations of the Total Variation Image Inpainting Method. Part III: Minimization Among Sets with Finite Perimeter

We propose a model for the restoration of images consisting only of completely black or completely white regions with the use of Caccioppoli sets.

Area formula for centered Hausdorff measures in metric spaces

Area formula for centered Hausdorff measures in metric spaces

The integral of the normal and fluxes over sets of finite perimeter

The integral of the normal and fluxes over sets of finite perimeter

Rigidity of equality cases in Steiner’s perimeter inequality

Characterization results for equality cases and for rigidity of equality cases in Steiner's perimeter inequality are presented. (By rigidity, we mean the situation when all equality cases are vertical translations of the Steiner's symmetral under consideration.) We achieve this through the introduction of a suitable measure-theoretic notion of connectedness and a fine analysis of barycenter functions for sets of finite perimeter having segments as orthogonal sections with respect to an hyperplane.

Random sets of finite perimeter

An approach to modelling random sets with locally finite perimeter as random elements in the corresponding subspace of L1 functions is suggested. A Crofton formula for flat sections of the perimeter is shown. Finally, random processes of particles with finite perimeter are introduced and it is shown that their union sets are random sets with locally finite perimeter.

Strict interior approximation of sets of finite perimeter and functions of bounded variation

It is well known that sets of finite perimeter can be strictly approximated by smooth sets, while, in general, one cannot hope to approximate anopensetΩ\Omegaof finite perimeter inRn\mathbb {R}^nstrictlyfrom within. In this note we show that, nevertheless, the latter type of approximation is possible under the mild hypothesis that the(n−1)(n{-}1)-dimensional Hausdorff measure of the topological boundary∂Ω\partial \Omegaequals the perimeter ofΩ\Omega. We also discuss an optimality property of this hypothesis, and we establish a corresponding result on strict approximation ofBVBV-functions from a prescribed Dirichlet class.

Regularity of Free Boundaries in Anisotropic Capillarity Problems and the Validity of Young’s Law

Local volume-constrained minimizers in anisotropic capillarity problems develop free boundaries on the walls of their containers. We prove the regularity of the free boundary outside a closed negligible set, showing in particular the validity of Young’s law at almost every point of the free boundary. Our regularity results are not specific to capillarity problems, and actually apply to sets of finite perimeter (and thus to codimension one integer rectifiable currents) arising as minimizers in other variational problems with free boundaries.

Extensions and traces of functions of bounded variation on metric spaces

Extensions and traces of functions of bounded variation on metric spaces

BV functions and sets of finite perimeter in sub-Riemannian manifolds

BV functions and sets of finite perimeter in sub-Riemannian manifolds

The configuration space and principle of virtual power for rough bodies

In the setting of an n-dimensional Euclidean space, the duality between velocity fields on the class of admissible bodies and Cauchy fluxes is studied using tools from geometric measure theory. A generalized Cauchy flux theory is obtained for sets whose measure theoretic boundaries may be as irregular as flat (n − 1)-chains. Initially, bodies are modeled as normal n -currents induced by sets of finite perimeter. A configuration space comprising Lipschitz embeddings induces virtual velocities given by locally Lipschitz mappings. A Cauchy flux is defined as a real valued function on the Cartesian product of (n − 1)-currents and locally Lipschitz mappings. A version of Cauchy’s postulates implies that a Cauchy flux may be uniquely extended to an n-tuple of flat (n − 1)-cochains. Thus, the class of admissible bodies is extended to include flat n-chains and a generalized form of the principle of virtual power is presented. Wolfe’s representation theorem for flat cochains enables the identification of stress as an n-tuple of flat (n − 1)-forms representing the flat (n − 1)-cochains associated with the Cauchy flux.

Perimeter under Multiple Steiner Symmetrizations

Steiner symmetrization in n linearly independent directions transforms every compact subset of $\mathbb {R}^{n}$ into a set of finite perimeter.

How to recognize convexity of a set from its marginals

How to recognize convexity of a set from its marginals

On sets of finite perimeter in Wiener spaces: reduced boundary and convergence to halfspaces

The theory of sets of finite perimeter and BV functions in Wiener spaces, i.e., Banach spaces endowed with a Gaussian Borel probability measure γ, was initiated by Fukushima and Hino in [9, 10, 11], and has been further investigated in [12, 1, 2, 3]. The basic question one would like to consider is the research of infinite-dimensional analogues of the classical fine properties of BV functions and sets of finite perimeter in finite-dimensional spaces. The class of sets of finite Gaussian perimeter E in a Gaussian Banach space (X, γ) is defined by the integration by parts formula ˆ

Capillary surfaces and floating bodies

We investigate capillary surfaces with boundary components on a floating body. The unknowns of this problem, the free surface and the position and orientation of the floating body are determined by minimizing the corresponding energy; besides gravitation and cohesion forces, we consider also adhesion between the fluid and both the wall of the container and the floating body. Existence of a solution is shown in the class of Caccioppoli sets under suitable restrictions on the data, as well as for surfaces that are graphs of real functions.

Existence and regularity of minimizers of a functional for unsupervised multiphase segmentation

Existence and regularity of minimizers of a functional for unsupervised multiphase segmentation