arXiv subjects

Ondřej Bouchala

Publications and source records attributed to Ondřej Bouchala.

3 recordsLinked to original sources

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 $Ω\subseteq \mathbb{R}^n$ 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↗

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↗

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↗