Sławomir Kolasiński

activity

20102026

most citedUniqueness of critical points of the anisotropic isoperimetric problem for finite perimeter sets

20 citations · 147 across the 16 of their papers we have counts for

collaborators

18 papers

math.AP2026

Regularity for $H$ -systems of bounded distortion

Sławomir Kolasiński

We consider weak solutions $u \in W^{1, n} (U, R^{n + 1})$ of the $H$ -system $- div (∣ D u ∣^{n - 2} D u) = (H \circ u) J u$ , where $…

math.AP2026

Rectifiability of every finite order for stationary integral varifolds

Sławomir Kolasiński

We give an alternative proof of a theorem of Brena, De Lellis and Franceschini (arXiv:2503.00649): the support of a stationary integral $d$ dimensional varifold in an open subset o…

math.AP2026

Geometry of the Atomic Condition

Jacek Jakimiuk, Sławomir Kolasiński, Maciej Leśniak

The class AC of integrands satisfying the atomic condition was introduced by De Philippis, De Rosa, and Ghiraldin in 2018. These integrands give rise to Almgren elliptic geometric…

math.AP2026

Perpendicularity and Locality for Codimension-One Varifolds with Bounded Anisotropic Mean Curvature

Sławomir Kolasiński, Mario Santilli

Suppose $F$ is an integrand associated with a uniformly convex $C^{3}$ -norm, and $V$ is a $n$ -dimensional varifold in an open subset of $R^{n + 1}$ such…

math.AP2025

Quadratic flatness and Regularity for Codimension-One Varifolds with Bounded Anisotropic Mean Curvature

Sławomir Kolasiński, Mario Santilli

We prove that if $V$ is a $n$ -dimensional varifold in an open subset of $R^{n + 1}$ with bounded anisotropic mean curvature such that $spt ∥ V ∥$ has locally…

math.OC2021

Regularity of distance functions from arbitrary closed sets

Sławomir Kolasiński, Mario Santilli

We investigate the distance function $δ_{K}^{ϕ}$ from an arbitrary closed subset $K$ of a~finite-dimensional Banach space $(R^{n}, ϕ)$ , equipped with a unifor…

math.AP2019★ 20 cited

Uniqueness of critical points of the anisotropic isoperimetric problem for finite perimeter sets

Antonio De Rosa, Sławomir Kolasiński, Mario Santilli

Given an elliptic integrand of class $C^{3}$ , we prove that finite unions of disjoint open Wulff shapes with equal radii are the only volume-constrained critical points…

math.AP2018★ 14 cited

Equivalence of the ellipticity conditions for geometric variational problems

Antonio De Rosa, Sławomir Kolasiński

We exploit the so called atomic condition, recently defined by De Philippis, De Rosa, and Ghiraldin in [Comm. Pure Appl. Math.] and proved to be necessary and sufficient for the va…

math.AP2017★ 17 cited

Existence of solutions to a general geometric elliptic variational problem

Yangqin Fang, Sławomir Kolasiński

We consider the problem of minimising an inhomogeneous anisotropic elliptic functional in a class of closed $m$ dimensional subsets of $R^{n}$ which is stable under taking s…

math.CA2016★ 1 cited

Estimating discrete curvatures in terms of beta numbers

Sławomir Kolasiński

For an arbitrary Radon measure $μ$ we estimate the integrated discrete curvature of $μ$ in terms of its centred variant of Jones' beta numbers. We farther relate integrals of centr…

math.CA2015★ 4 cited

Higher order rectifiability of measures via averaged discrete curvatures

Sławomir Kolasiński

We provide a sufficient geometric condition for $R^{n}$ to be countably $(μ, m)$ rectifiable of class $C^{1, α}$ (using the terminology of Federer), where $μ$ is a R…

math.DG2015★ 6 cited

Compactness and isotopy finiteness for submanifolds with uniformly bounded geometric curvature energies

Sławomir Kolasiński, Paweł Strzelecki, Heiko von der Mosel

In this paper, we establish compactness for various geometric curvature energies including integral Menger curvature, and tangent-point repulsive potentials, defined a priori on th…

math.DG2015★ 13 cited

Decay rates for the quadratic and super-quadratic tilt-excess of integral varifolds

Sławomir Kolasiński, Ulrich Menne

This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space satisfying integrability conditions on their first variation. Firstly, the study…

math.FA2012★ 17 cited

Geometric Sobolev-like embedding using high-dimensional Menger-like curvature

Sławomir Kolasiński

We study a modified version of Lerman-Whitehouse Menger-like curvature defined for m+2 points in an n-dimensional Euclidean space. For 1 <= l <= m+2 and an m-dimensional subset S o…

math.CA2012★ 19 cited

Characterizing $W^{2, p}$ ~submanifolds by $p$ -integrability of global curvatures

Sławomir Kolasiński, Paweł Strzelecki, Heiko von der Mosel

We give sufficient and necessary geometric conditions, guaranteeing that an immersed compact closed manifold $Σ^{m} \subset R^{n}$ of class $C^{1}$ and of arbitrary dimension and codimen…

math.FA2011★ 7 cited

Minimal Hölder regularity implying finiteness of integral Menger curvature

Sławomir Kolasiński, Marta Szumańska

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 $m$ -dimens…

math.FA2011★ 17 cited

Sharp Boundedness and Regularizing effects of the integral Menger curvature for submanifolds

Simon Blatt, Sławomir Kolasiński

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.…

math.AP2010★ 12 cited

Integral Menger curvature for sets of arbitrary dimension and codimension

Sławomir Kolasiński

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 se…

Regularity for H-systems of bounded distortion

Rectifiability of every finite order for stationary integral varifolds

Geometry of the Atomic Condition

Perpendicularity and Locality for Codimension-One Varifolds with Bounded Anisotropic Mean Curvature

Quadratic flatness and Regularity for Codimension-One Varifolds with Bounded Anisotropic Mean Curvature

Regularity of distance functions from arbitrary closed sets

Uniqueness of critical points of the anisotropic isoperimetric problem for finite perimeter sets

Equivalence of the ellipticity conditions for geometric variational problems

Existence of solutions to a general geometric elliptic variational problem

Estimating discrete curvatures in terms of beta numbers

Higher order rectifiability of measures via averaged discrete curvatures

Compactness and isotopy finiteness for submanifolds with uniformly bounded geometric curvature energies

Decay rates for the quadratic and super-quadratic tilt-excess of integral varifolds

Geometric Sobolev-like embedding using high-dimensional Menger-like curvature

Characterizing W2,p~submanifolds by p-integrability of global curvatures

Minimal Hölder regularity implying finiteness of integral Menger curvature

Sharp Boundedness and Regularizing effects of the integral Menger curvature for submanifolds

Integral Menger curvature for sets of arbitrary dimension and codimension

Regularity for $H$ -systems of bounded distortion

Characterizing $W^{2, p}$ ~submanifolds by $p$ -integrability of global curvatures