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Minimal Hölder regularity implying finiteness of integral Menger curvature

MPG-Autoren
http://pubman.mpdl.mpg.de/cone/persons/resource/persons96543

Kolasinski,  Slawomir
Geometric Measure Theory, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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1111.1141.pdf
(Preprint), 561KB

s00229-012-0565-y.pdf
(beliebiger Volltext), 266KB

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Zitation

Kolasinski, S., & Szumańska, M. (2013). Minimal Hölder regularity implying finiteness of integral Menger curvature. Manuscripta Mathematica, 141(1-2), 125-147. doi:10.1007/s00229-012-0565-y.


Zitierlink: http://hdl.handle.net/11858/00-001M-0000-0013-B36A-4
Zusammenfassung
We study two families of integral functionals indexed by a real number $p > 0$. One family is defined for 1-dimensional curves in $\R^3$ and the other one is defined for $m$-dimensional manifolds in $\R^n$. These functionals are described as integrals of appropriate integrands (strongly related to the Menger curvature) raised to power $p$. Given $p > m(m+1)$ we prove that $C^{1,\alpha}$ regularity of the set (a curve or a manifold), with $\alpha > \alpha_0 = 1 - \frac{m(m+1)}p$ implies finiteness of both curvature functionals ($m=1$ in the case of curves). We also show that $\alpha_0$ is optimal by constructing examples of $C^{1,\alpha_0}$ functions with graphs of infinite integral curvature.