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Free keywords:
Mathematics, Functional Analysis, math.FA,Mathematics, Classical Analysis and ODEs, math.CA,
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
$C^{1,\alpha}$ 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.
