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arXiv · 1111.1141

Minimal Hölder regularity implying finiteness of integral Menger curvature

Sławomir Kolasiński·Marta Szumańska
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Abstract

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 $C^{1,α}$ regularity of the set (a curve or a manifold), with $α> α_0 = 1 - \frac{m(m+1)}p$ implies finiteness of both curvature functionals ($m=1$ in the case of curves). We also show that $α_0$ is optimal by constructing examples of $C^{1,α_0}$ functions with graphs of infinite integral curvature.

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Sławomir Kolasiński, Marta Szumańska. 2011-11-17. Minimal Hölder regularity implying finiteness of integral Menger curvature. https://doi.org/10.1007/s00229-012-0565-y

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