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31

Sep 23, 2013
09/13

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Pawel Strzelecki; Heiko von der Mosel

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We study a two-point self-avoidance energy $E_q$ which is defined for all rectifiable curves in $R^n$ as the double integral along the curve of $1/r^q$. Here $r$ stands for the radius of the (smallest) circle that is tangent to the curve at one point and passes through another point on the curve, with obvious natural modifications of this definition in the exceptional, non-generic cases. It turns out that finiteness of $E_q(\gamma)$ for $q\ge 2$ guarantees that $\gamma$ has no...

Source: http://arxiv.org/abs/1006.4566v2

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3.0

Jun 30, 2018
06/18

by
Katarzyna Mazowiecka; Paweł Strzelecki

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We prove that for each positive integer $N$ the set of smooth, zero degree maps $\psi\colon\mathbb{S}^2\to \mathbb{S}^2$ which have the following three properties: (1) there is a unique minimizing harmonic map $u\colon \mathbb{B}^3\to \mathbb{S}^2$ which agrees with $\psi$ on the boundary of the unit ball; (2) this map $u$ has at least $N$ singular points in $\mathbb{B}^3$; (3) the Lavrentiev gap phenomenon holds for $\psi$, i.e., the infimum of the Dirichlet energies $E(w)$ of all smooth...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1406.0601

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46

Sep 18, 2013
09/13

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Paweł Strzelecki; Marta Szumańska; Heiko von der Mosel

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We investigate knot-theoretic properties of geometrically defined curvature energies such as integral Menger curvature. Elementary radii-functions, such as the circumradius of three points, generate a family of knot energies guaranteeing self-avoidance and a varying degree of higher regularity of finite energy curves. All of these energies turn out to be charge, minimizable in given isotopy classes, tight and strong. Almost all distinguish between knots and unknots, and some of them can be...

Source: http://arxiv.org/abs/1209.1527v2

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26

Sep 21, 2013
09/13

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Paweł Goldstein; Paweł Strzelecki; Anna Zatorska-Goldstein

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We consider a class of fourth order elliptic systems which include the Euler-Lagrange equations of biharmonic mappings in dimension 4 and we prove that weak limit of weak solutions to such systems is again a weak solution to a limit system.

Source: http://arxiv.org/abs/1301.0927v1

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30

Jun 29, 2018
06/18

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Armin Schikorra; Paweł Strzelecki

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In this paper, we discuss two well-known open problems in the regularity theory for nonlinear, conformally invariant elliptic systems in dimensions $n\ge 3$, with a critical nonlinearity: $H$-systems (equations of hypersurfaces of prescribed mean curvature) and $n$-harmonic maps into compact Riemannian manifolds. For $n=2$ several solutions of these problems are known but they all break down in higher dimensions (unless one considers special cases, e.g. hypersurfaces of constant mean curvature...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1606.07891

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70

Sep 18, 2013
09/13

by
Pawel Strzelecki; Heiko von der Mosel

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We consider repulsive potential energies $\E_q(\Sigma)$, whose integrand measures tangent-point interactions, on a large class of non-smooth $m$-dimensional sets $\Sigma$ in $\R^n.$ Finiteness of the energy $\E_q(\Sigma)$ has three sorts of effects for the set $\Sigma$: topological effects excluding all kinds of (a priori admissible) self-intersections, geometric and measure-theoretic effects, providing large projections of $\Sigma$ onto suitable $m$-planes and therefore large $m$-dimensional...

Source: http://arxiv.org/abs/1102.3642v1

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15

Jun 27, 2018
06/18

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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

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55

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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5.0

Jun 30, 2018
06/18

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Krystian Kazaniecki; Michał Łasica; Katarzyna Ewa Mazowiecka; Paweł Strzelecki

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We prove an $\varepsilon$-regularity result for a wide class of parabolic systems $$ u_t-\text{div}\big(|\nabla u|^{p-2}\nabla u) = B(u, \nabla u) $$ with the right hand side $B$ growing like $|\nabla u|^p$. It is assumed that the solution $u(t,\cdot)$ is uniformly small in the space of functions of bounded mean oscillation. The crucial tool is provided by a sharp nonlinear version of the Gagliardo-Nirenberg inequality which has been used earlier in an elliptic context by T. Rivi\`ere and the...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1406.1978