arXiv subjects

Stanislav Hencl

Publications and source records attributed to Stanislav Hencl.

At least 19 recordsLinked to original sources

arXiv2026

Bounded Continuous weak quasiregular mappings that fail to be quasiregular

We show that, in dimensions $n\geq 3$, continuity and boundedness do not restore the Sobolev regularity conjecture of Iwaniec and Martin for weakly quasiregular mappings below the critical exponent. For every bounded domain $\Omega\subset\mathbb R^n$ and every $1\leq p<nK/(K+1)$, we construct a bounded continuous weakly $K$-quasiregular mapping $$ f\in W^{1,\,p}(\Omega;\,\mathbb R^n)\cap C(\Omega;\,\mathbb R^n) \cap L^\infty(\Omega;\mathbb R^n) $$ which fails to be quasiregular. We further construct weakly quasiregular mappings whose singular sets have Hausdorff dimension arbitrarily close to the maximal size permitted by their Sobolev regularity. These examples show that, the almost-everywhere sign condition on the Jacobian is too weak to serve as an orientation-preserving hypothesis below $W^{1,n}$. In contrast, we show that, for $n-1<p<n$, quasiregularity follows once this condition is replaced by a one-sided condition on the distributional degree (together with boundedness).

math.CV↗

arXiv2026

Ciarlet Ne\v{c}as condition in fractional Sobolev spaces

Let $s\in(\frac{n}{n+1},1)$, $\Omega\subset\mathbb{R}^n$ be an open set and let $f\in W^{s,n/s}(\Omega,\mathbb{R}^n)$ be mapping with positive distributional Jacobian $\mathcal{J}_f>0$ which models some deformation in fractional Nonlinear Elasticity. We show change of variables formula in this class and as a consequence we show that the analogue of Ciarlet-Ne\v{c}as condition $\mathcal{J}_f(\Omega)=|f(\Omega)|$ implies that our mapping is one-to-one a.e.

math.FA↗

arXiv2025

Note on injectivity in second-gradient Nonlinear Elasticity

Let $q>1$, $(1-\frac{1}{q})a\geq 1$ and let $\Omega\subset \mathbb{R}^2$ be Lipschitz domain. We show that planar mappings in the second order Sobolev space $f\in W^{2,q}(\Omega,\mathbb{R}^2)$ with $|J_f|^{-a}\in L^1(\Omega)$ are homeomorphism if they agree with a homeomorphism on the boundary. The condition $(1-\frac{1}{q})a\geq 1$ is sharp. We also have a new sharp result about the $\mathcal{H}^{n-1}$ measure of the projection of the set $\{J_f=0\}$ in $\mathbb{R}^n$.

math.AP↗

arXiv2025

Mission $p<n-1$: Possible -- Nonlinear Elasticity Beyond Conventional Limits

In this paper we prove the lower semicontinuity of a Neohookean-type energy for a model of Nonlinear Elasticity that allows, for the first time, for $p<n-1$. Our class of admissible deformations consists of weak limits of Sobolev $W^{1,p}$ homeomorphisms. We also introduce a model that allows for cavitations by studying weak limits of homeomorphisms that can open cavities at some points. In this model we add the measure of the created surface to the energy functional and for this functional we again prove lower semicontinuity.

math.AP↗

arXiv2025

Ball-Evans approximation problem: recent progress and open problems

In this paper we give a short overview about the Ball-Evans approximation problem, i.e. about the approximation of Sobolev homeomorphism by a sequence of diffeomorphisms (or piecewise affine homeomorphisms) and we recall the motivation for this problem. We show some recent planar results and counterexamples in higher dimension and we give a number of open problems connected to this problem and related fields.

math.FA↗

arXiv2024

$(INV)$ condition and regularity of the inverse

Let $f \colon \Omega \to \Omega' $ be a Sobolev mapping of finite distortion between planar domains $\Omega $ and $\Omega'$, satisfying the $(INV)$ condition and coinciding with a homeomorphism near $\partial\Omega $. We show that $f$ admits a generalized inverse mapping $h \colon \Omega' \to \Omega$, which is also a Sobolev mapping of finite distortion and satisfies the $(INV)$ condition. We also establish a higher-dimensional analogue of this result: if a mapping $f \colon \Omega \to \Omega' $ of finite distortion is in the Sobolev class $W^{1,p}(\Omega, \mathbb{R}^n)$ with $p > n-1$ and satisfies the $(INV)$ condition, then $f$ has an inverse in $W^{1,1}(\Omega', \mathbb{R}^n)$ that is also of finite distortion. Furthermore, we characterize Sobolev mappings satisfying $(INV)$ whose generalized inverses have finite $n$-harmonic energy.

math.FA↗

arXiv2024

Weak limits of Sobolev homeomorphisms are one to one

We prove that the key property in models of Nonlinear Elasticity which corresponds to the non-interpenetration of matter, i.e. injectivity a.e., can be achieved in the class of weak limits of homeomorphisms under very minimal assumptions. Let $\Omega\subseteq \mathbb{R}^n$ be a domain and let $p>\left\lfloor\frac{n}{2}\right\rfloor$ for $n\geq 4$ or $p\geq 1$ for $n=2,3$. Assume that $f_k\in W^{1,p}$ is a sequence of homeomorphisms such that $f_k\rightharpoonup f$ weakly in $W^{1,p}$ and assume that $J_f>0$ a.e. Then we show that $f$ is injective a.e.

math.FA↗

arXiv2024

Non-interpenetration of rods derived by $\Gamma$-limits

Ensuring non-interpenetration of matter is a fundamental prerequisite when modeling the deformation response of solid materials. In this contribution, we thoroughly examine how this requirement, equivalent to the injectivity of deformations within bulk structures, manifests itself in dimensional-reduction problems. Specifically, we focus on the case of rods embedded in a two-dimensional plane. Our results focus on $\Gamma$-limits of energy functionals that enforce an admissible deformation to be a homeomorphism. These $\Gamma$-limits are evaluated along a passage from the bulk configuration to the rod arrangement. The proofs rely on the equivalence between the weak and strong closures of the set of homeomorphisms from $\mathbb{R}$ to $\mathbb{R}^2$, a result that is of independent interest and that we establish in this paper, too.

math.AP↗

arXiv2023

Weak Limit of $W^{1,2}$ Homeomorphisms in $\mathbb{R}^3$ Can Have Any Degree

In this paper for every $k\in\mathbb{Z}$ we construct a sequence of weakly converging homeomorphisms $h_m\colon B(0,10)\to\mathbb{R}^3$, $h_m\rightharpoonup h$ in $W^{1,2}(B(0,10))$, such that $h_m(x)=x$ on $\partial B(0,10)$ and for every $r\in \left(\tfrac5{16},\tfrac{7}{16}\right)$ the degree of $h$ with respect to the ball $B(0,r)$ is equal to $k$ on a set of positive measure.

math.FA↗

arXiv2023

Extension of planar H\"older homeomorphisms

Let $\alpha$ \in (0; 1). We show that any $\alpha$-H\"older homeomorphism from the unit circle in the plane to the plane can be extended to an $\alpha$-H\"{o}lder homeomorphism from the whole unit disc.

math.CV↗

arXiv2022

Weak limit of homeomorphisms in $W^{1,n-1}$: invertibility and lower semicontinuity of energy

Let $\Omega$, $\Omega'\subset\mathbb{R}^n$ be bounded domains and let $f_m\colon\Omega\to\Omega'$ be a sequence of homeomorphisms with positive Jacobians $J_{f_m} >0$ a.e. and prescribed Dirichlet boundary data. Let all $f_m$ satisfy the Lusin (N) condition and $\sup_m \int_{\Omega}(|Df_m|^{n-1}+A(|\text{cof} Df_m|)+\phi(J_f))<\infty$, where $A$ and $\varphi$ are positive convex functions. Let $f$ be a weak limit of $f_m$ in $W^{1,n-1}$. Provided certain growth behaviour of $A$ and $\varphi$, we show that $f$ satisfies the (INV) condition of Conti and De Lellis, the Lusin (N) condition, and polyconvex energies are lower semicontinuous.

math.FA↗

arXiv2021

Sobolev homeomorphic extensions from two to three dimensions

We study the basic question of characterizing which boundary homeomorphisms of the unit sphere can be extended to a Sobolev homeomorphism of the interior in 3D space. While the planar variants of this problem are well-understood, completely new and direct ways of constructing an extension are required in 3D. We prove, among other things, that a Sobolev homeomorphism $φ\colon \mathbb R^2 \to \mathbb R^2$ in $W_{loc}^{1,p} (\mathbb R^2 , \mathbb R^2)$ for some $p\in [1,\infty )$ admits a homeomorphic extension $h \colon \mathbb R^3 \to \mathbb R^3$ in $W_{loc}^{1,q} (\mathbb R^3, \mathbb R^3) $ for $1\le q < \frac{3}{2}p$. Such an extension result is nearly sharp, as the bound $q=\frac{3}{2}p$ cannot be improved due to the Hölder embedding. The case $q=3$ gains an additional interest as it also provides an $L^1$-variant of the celebrated Beurling-Ahlfors extension result.

math.CA↗

arXiv2021

Weak limit of homeomorphisms in $W^{1,n-1}$ and (INV) condition

Let $\Omega,\Omega'\subset\mathbb{R}^3$ be Lipschitz domains, let $f_m:\Omega\to\Omega'$ be a sequence of homeomorphisms with prescribed Dirichlet boundary condition and $\sup_m \int_{\Omega}(|Df_m|^2+1/J^2_{f_m})<\infty$. Let $f$ be a weak limit of $f_m$ in $W^{1,2}$. We show that $f$ is invertible a.e., more precisely it satisfies the (INV) condition of Conti and De Lellis and thus it has all the nice properties of mappings in this class. Generalization to higher dimensions and an example showing sharpness of the condition $1/J^2_f\in L^1$ are also given. Using this example we also show that unlike the planar case the class of weak limits and the class of strong limits of $W^{1,2}$ Sobolev homeomorphisms in $\mathbb{R}^3$ are not the same.

math.FA↗

arXiv2020

Approximation of planar Sobolev $W^{2,1}$ homeomorphisms by Piecewise Quadratic Homeomorphisms and Diffeomorphisms

Given a Sobolev homeomorphism $f\in W^{2,1}$ in the plane we find a piecewise quadratic homeomorphism that approximates it up to a set of $ε$ measure. We show that this piecewise quadratic map can be approximated by diffeomorphisms in the $W^{2,1}$ norm on this set.

math.FA↗

arXiv2020

A sense preserving Sobolev homeomorphism with negative Jacobian almost everywhere

For every $1\leq p<\frac{3}{2}$ we construct a Sobolev homeomorphism $f\in W^{1,p}([-1,1]^4,[-1,1]^4)$ such that $f(x)=x$ for every $x\in \partial[-1,1]^4$ but $J_f<0$ a.e.

math.AP↗

arXiv2019

Injectivity almost everywhere for weak limits of Sobolev homeomorphisms

Let $Ω\subset\mathbb{R}^n$ be an open set and let $f\in W^{1,p}(Ω,\mathbb{R}^n)$ be a weak (sequential) limit of Sobolev homeomorphisms. Then $f$ is injective almost everywhere for $p>n-1$ both in the image and in the domain. For $p\leq n-1$ we construct a strong limit of homeomorphisms such that the preimage of a point is a continuum for every point in a set of positive measure in the image and a topological image of a point is a continuum for every point in a set of positive measure in the domain.

math.CA↗

arXiv2019

On distributional adjugate and derivative of the inverse

Let $Ω\subset\er^3$ be a domain and let $f\colonΩ\to\er^3$ be a bi-$BV$ homeomorphism. Very recently in \cite{HKL} it was shown that the distributional adjugate of $Df$ (and thus also of $Df^{-1}$) is a matrix-valued measure. In the present paper we show that the components of $\Adj Df$ are equal to components of $Df^{-1}(f(U))$ as measures and that the absolutely continuous part of the distributional adjugate $\Adj Df$ equals to the pointwise adjugate $\adj Df(x)$ a.e. We also show the equivalence of several approaches to the definition of the distributional adjugate.

math.FA↗

arXiv2019

Weak regularity of the inverse under minimal assumptions

Let $Ω\subset\mathbb{R}^3$ be a domain and let $f\in BV_{\operatorname{loc}}(Ω,\mathbb{R}^3)$ be a homeomorphism such that its distributional adjugate is a finite Radon measure. We show that its inverse has bounded variation $f^{-1}\in BV_{\operatorname{loc}}$. The condition that the distributional adjugate is finite measure is not only sufficient but also necessary for the weak regularity of the inverse.

math.CA↗