Minimal H{ö}lder regularity implying finiteness of integral Menger curvature

Details Minimal H{ö}lder regularity implying finiteness of integral Menger curvature Minimal H{ö}lder regularity implying finiteness of integral Menger curvature

2011-11-04

arxiv.org/abs/1111.1141v2

Mathematics

Functional Analysis

Scientific paper

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.

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