arXiv subjects

Zofia Grochulska

Publications and source records attributed to Zofia Grochulska.

5 recordsLinked to original sources

arXiv2025

$L^{1}_{loc}$-convergence of Jacobians of Sobolev homeomorphisms via area formula

We prove that given a sequence of homeomorphisms $f_k: Ω\to \mathbb{R}^n$ convergent in $W^{1,p}(Ω, \mathbb{R}^n)$, $p \geq 1$ for $n =2$ and $p > n-1$ for $n \geq 3$, to a homeomorphism $f$ which maps sets of measure zero onto sets of measure zero, Jacobians $Jf_k$ converge to $Jf$ in $L^1_{loc}(Ω)$. We prove it via Federer's area formula and investigation of when $|f_k(E)| \to |f(E)|$ as $k \to \infty$ for Borel subsets $E \Subset Ω$.

math.CA↗

arXiv2025

Gluing diffeomorphisms, bi-Lipschitz mappings and homeomorphisms

Cerf and Palais independently proved a remarkable result about extending diffeomorphisms defined on smooth balls in a manifold to global diffeomorphisms of the manifold onto itself. We explain Palais' argument and show how to extend it to the class of homeomorphisms and bi-Lipschitz homeomorphisms. While Palais' argument is surprising, it is elementary and short. However, its extension to bi-Lipschitz homeomorphisms and homeomorphisms requires deep results: the stable homeomorphism and the annulus theorems.

math.GT↗

arXiv2023

Constructing diffeomorphisms and homeomorphisms with prescribed derivative

We prove that for any measurable mapping $T$ into the space of matrices with positive determinant, there is a diffeomorphism whose derivative equals $T$ outside a set of measure less than $\varepsilon$. We use this fact to prove that for any measurable mapping $T$ into the space of matrices with non-zero determinant (with no sign restriction), there is an almost everywhere approximately differentiable homeomorphism whose derivative equals $T$ almost everywhere.

math.CA↗

arXiv2022

Local estimates for vectorial Rudin-Osher-Fatemi type problems in one dimension

We consider the Rudin-Osher-Fatemi variational denoising model with general regularizing term in one-dimensional, vector-valued setting. We obtain local estimates on the singular part of the variation measure of the minimizer in terms of the singular part of the variation measure of the datum. In the case of homogeneous regularizer, we prove local estimates on the whole variation measure of the minimizer and deduce an analogous result for the gradient flow of the regularizer.

math.AP↗

arXiv2022

Characterizations of generalized John domains in $\mathbb{R}^n$ via metric duality

In this paper, we extend the characterization of John disks obtained by Näkki and Väisälä [Exp. Math. 1991] to generalized John domains in higher dimensions under mild assumptions. The main ingredient in this characterization is to use the higher dimensional analogues of the local linear connectivity (LLC) and homological bounded turning properties introduced by Väisälä in his study of metric duality theory [Math. Scan. 1997]. Somewhat surprisingly, we constructed a uniform domain in $\R^3$, which is topologically simple, such that the complementary domain fails to be homotopically $1$-bounded turning. In particular, this shows that a similar characterization of generalized John domains in terms of higher dimensional homotopic bounded turning does not hold in dimension three.

math.GN↗