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Sep 23, 2013
09/13

by
Sławomir Kolasiński; Marta Szumańska

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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...

Source: http://arxiv.org/abs/1111.1141v2

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3.0

Jun 29, 2018
06/18

by
Sławomir Kolasiński

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For an arbitrary Radon measure $\mu$ we estimate the integrated discrete curvature of $\mu$ in terms of its centred variant of Jones' beta numbers. We farther relate integrals of centred and non-centred beta numbers. As a corollary, employing the recent result of Tolsa [Calc. Var. PDE, 2015], we obtain a partial converse of the theorem of Meurer [arXiv:1510.04523].

Topics: Classical Analysis and ODEs, Mathematics

Source: http://arxiv.org/abs/1605.00939

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31

Sep 20, 2013
09/13

by
Sławomir Kolasiński

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We study a modified version of Lerman-Whitehouse Menger-like curvature defined for m+2 points in an n-dimensional Euclidean space. For 1

Source: http://arxiv.org/abs/1205.4112v2

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Jul 20, 2013
07/13

by
Sławomir Kolasiński; Paweł Strzelecki; Heiko von der Mosel

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We give sufficient and necessary geometric conditions, guaranteeing that an immersed compact closed manifold $\Sigma^m\subset \R^n$ of class $C^1$ and of arbitrary dimension and codimension (or, more generally, an Ahlfors-regular compact set $\Sigma$ satisfying a mild general condition relating the size of holes in $\Sigma$ to the flatness of $\Sigma$ measured in terms of beta numbers) is in fact an embedded manifold of class $C^{1,\tau}\cap W^{2,p}$, where $p>m$ and $\tau=1-m/p$. The...

Source: http://arxiv.org/abs/1203.4688v2

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Sep 21, 2013
09/13

by
Sławomir Kolasiński

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We propose a notion of integral Menger curvature for compact, $m$-dimensional sets in $n$-dimensional Euclidean space and prove that finiteness of this quantity implies that the set is $C^{1,\alpha}$ embedded manifold with the H{\"o}lder norm and the size of maps depending only on the curvature. We develop the ideas introduced by Strzelecki and von der Mosel [Adv. Math. 226(2011)] and use a similar strategy to prove our results.

Source: http://arxiv.org/abs/1011.2008v6

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45

Sep 23, 2013
09/13

by
Simon Blatt; Sławomir Kolasiński

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In this paper we show that embedded and compact $C^1$ manifolds have finite integral Menger curvature if and only if they are locally graphs of certain Sobolev-Slobodeckij spaces. Furthermore, we prove that for some intermediate energies of integral Menger type a similar characterization of objects with finite energy can be given.

Source: http://arxiv.org/abs/1110.4786v1

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Jun 30, 2018
06/18

by
Yangqin Fang; Sławomir Kolasiński

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We consider the problem of minimising an inhomogeneous anisotropic elliptic functional in a class of closed $m$~dimensional subsets of~$\mathbf{R}^n$ which is stable under taking smooth deformations homotopic to identity and under local Hausdorff limits. We~prove that the minimiser exists inside the class and is an $(\mathscr H^m,m)$ rectifiable set in the sense of Federer. The class of competitors encodes a notion of spanning a~boundary. We admit unrectifiable and non-compact competitors and...

Topics: Optimization and Control, Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1704.06576

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Jun 27, 2018
06/18

by
Sławomir Kolasiński

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We provide a sufficient geometric condition for $\mathbb{R}^n$ to be countably $(\mu,m)$ rectifiable of class $\mathscr{C}^{1,\alpha}$ (using the terminology of Federer), where $\mu$ is a Radon measure having positive lower density and finite upper density $\mu$ almost everywhere. Our condition involves integrals of certain many-point interaction functions (discrete curvatures) which measure flatness of simplices spanned by the parameters.

Topics: Mathematics, Classical Analysis and ODEs

Source: http://arxiv.org/abs/1506.00507

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Jun 27, 2018
06/18

by
Sławomir Kolasiński; Paweł Strzelecki; Heiko von der Mosel

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In this paper, we establish compactness for various geometric curvature energies including integral Menger curvature, and tangent-point repulsive potentials, defined a priori on the class of compact, embedded $m$-dimensional Lipschitz submanifolds in ${\mathbb{R}}^n$. It turns out that due to a smoothing effect any sequence of submanifolds with uniformly bounded energy contains a subsequence converging in $C^1$ to a limit submanifold. This result has two applications. The first one is an...

Topics: Differential Geometry, Analysis of PDEs, Mathematics, Metric Geometry

Source: http://arxiv.org/abs/1504.04538