Tangency properties of sets with finite geometric curvature energies

• Autorzy: Sebastian Scholtes
• Języki publikacji: EN

Abstrakty

EN

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
$ℳ_{p}^{α}(X): = ∫_{X}∫_{X}∫_{X} κ^{p}(x,y,z) d𝓗 ^{α}_{X}(x)d𝓗 ^{α}_{X}(y)d𝓗 ^{α}_{X}(z)$,
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 $ℳ ¹_{p}$, for which, to the best of our knowledge, no regularity properties have been established before. Furthermore we prove that for α = 1 these exponents are sharp, i.e., if p lies below the threshold value of scale invariance, then there exists a set containing points with no weak approximate 1-tangent, but such that the energy is still finite. Moreover we demonstrate that weak approximate tangents are the most we can expect. For the other curvature energies analogous results are shown.

Kategorie tematyczne

• 28A75: Length, area, volume, other geometric measure theory
• 49Q15: Geometric measure and integration theory, integral and normal currents

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2012
• Tom: 218
• Numer: 2
• Strony: 165-191

Daty

wydano

2012

Twórcy

autor

Sebastian Scholtes

• Institut für Mathematik, RWTH Aachen University, Templergraben 55, D-52062 Aachen, Germany

Identyfikatory

DOI

10.4064/fm218-2-4