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Tangency properties of sets with finite geometric curvature energies

Sebastian Scholtes (2012)

Fundamenta Mathematicae

We investigate tangential regularity properties of sets of fractal dimension, whose inverse thickness or integral Menger curvature energies are bounded. For the most prominent of these energies, the integral Menger curvature ℳ p α ( X ) : = ∫ X ∫ X ∫ X κ p ( x , y , z ) d X α ( x ) d X α ( y ) d X α ( z ) , where κ(x,y,z) is the inverse circumradius of the triangle defined by x,y and z, we find that ℳ p α ( X ) < ∞ for p ≥ 3α implies the existence of a weak approximate α-tangent at every point of the set, if some mild density properties hold. This includes the scale invariant case p = 3 for...

The BV-energy of maps into a manifold : relaxation and density results

Mariano Giaquinta, Domenico Mucci (2006)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Let  𝒴   be a smooth compact oriented riemannian manifoldwithout boundary, and assume that its 1 -homology group has notorsion. Weak limits of graphs of smooth maps  u k : B n → 𝒴   with equibounded total variation give riseto equivalence classes of cartesian currents in  cart 1 , 1 ( B n 𝒴 )   for which we introduce a natural B V -energy.Assume moreover that the first homotopy group of   𝒴   iscommutative. In any dimension   n   we prove that every element  T   in   cart 1 , 1 ( B n 𝒴 )   can be approximatedweakly in the sense of currents by a sequence of graphs...

The Curvature of a Set with Finite Area

Elisabetta Barozzi (1994)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In a paper, by myself, E. Gonzalez and I. Tamanini (see [2]), it was proven that all sets of finite perimeter do have a non trivial variational property, connected with the mean curvature of their boundaries. In the present article, that variational property is made more precise.

The mean curvature of a Lipschitz continuous manifold

Elisabetta Barozzi, Eduardo Gonzalez, Umberto Massari (2003)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

The paper is devoted to the description of some connections between the mean curvature in a distributional sense and the mean curvature in a variational sense for several classes of non-smooth sets. We prove the existence of the mean curvature measure of ∂ ⁡ E by using a technique introduced in [4] and based on the concept of variational mean curvature. More precisely we prove that, under suitable assumptions, the mean curvature measure of ∂ ⁡ E is the weak limit (in the sense of distributions) of the mean...

Transport problems and disintegration maps

Luca Granieri, Francesco Maddalena (2013)

ESAIM: Control, Optimisation and Calculus of Variations

By disintegration of transport plans it is introduced the notion of transport class. This allows to consider the Monge problem as a particular case of the Kantorovich transport problem, once a transport class is fixed. The transport problem constrained to a fixed transport class is equivalent to an abstract Monge problem over a Wasserstein space of probability measures. Concerning solvability of this kind of constrained problems, it turns out that in some sense the Monge problem corresponds to a...

Type-II singularities of two-convex immersed mean curvature flow

Theodora Bourni, Mat Langford (2016)

Geometric Flows

We show that any strictly mean convex translator of dimension n ≥ 3 which admits a cylindrical estimate and a corresponding gradient estimate is rotationally symmetric. As a consequence, we deduce that any translating solution of the mean curvature flow which arises as a blow-up limit of a two-convex mean curvature flow of compact immersed hypersurfaces of dimension n ≥ 3 is rotationally symmetric. The proof is rather robust, and applies to a more general class of translator equations. As a particular...

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