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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 asymptotic behaviour of surfaces with prescribed mean curvature near meeting points of fixed and free boundaries

Frank Müller (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We study the shape of stationary surfaces with prescribed mean curvature in the Euclidean 3-space near boundary points where Plateau boundaries meet free boundaries. In deriving asymptotic expansions at such points, we generalize known results about minimal surfaces due to G. Dziuk. The main difficulties arise from the fact that, contrary to minimal surfaces, surfaces with prescribed mean curvature do not meet the support manifold perpendicularly along their free boundary, in general.

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 double bubble conjecture.

Hass, Joel, Hutchings, Michael, Schlafly, Roger (1995)

Electronic Research Announcements of the American Mathematical Society [electronic only]

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