arXiv subjects

Marta Szumańska

Publications and source records attributed to Marta Szumańska.

4 recordsLinked to original sources

arXiv2018

Kinetic energy represented in terms of moments of vorticity and applications

We study 2d vortex sheets with unbounded support. First we show a version of the Biot- Savart law related to a class of objects including such vortex sheets. Next, we give a formula associating the kinetic energy of a very general class of ows with certain moments of their vorticities. It allows us to identify a class of vortex sheets of unbounded support being only ?-finite measures (in particluar including measures ωsuch that ω(R2) = \infty), but with locally finite kinetic energy. One of such examples are celebrated Kaden approximations. We study them in details. In particular our estimates allow us to show that the kinetic energy of Kaden approximations in the neighbourhood of an origin is dissipated, actually we show that the energy is pushed out of any ball centered in the origin of the Kaden spiral. The latter result can be interpreted as an artificial viscosity in the center of a spiral.

math-ph↗

arXiv2012

On some knot energies involving Menger curvature

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 shown to be uniquely minimized by round circles. Bounds on the stick number and the average crossing number, some non-trivial global lower bounds, and unique minimization by circles upon compaction complete the picture.

math.CA↗

arXiv2012

On Whitney-type characterization of approximate differentiability on metric measure spaces

We study approximately differentiable functions on metric measure spaces admitting a Cheeger differentiable structure. The main result is a Whitney-type characterization of approximately differentiable functions in this setting. As an application, we prove a Stepanov-type theorem and consider approximate differentiability of Sobolev, BV and maximal functions.

math.CA↗

arXiv2011

Minimal Hölder regularity implying finiteness of integral Menger curvature

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.

math.FA↗