Related papers: On the bi-Sobolev planar homeomorphisms and their …

Let $\Omega\subseteq\mathbb R^2$ be a domain and let $f\in W^{1,1}(\Omega,\mathbb R^2)$ be a homeomorphism (between $\Omega$ and $f(\Omega)$). Then there exists a sequence of smooth diffeomorphisms $f_k$ converging to $f$ in…

Let $\Omega\subseteq\mathcal{R}^2$ be a domain, let $X$ be a rearrangement invariant space and let $f\in W^{1}X(\Omega,\mathcal{R}^2)$ be a homeomorphism between $\Omega$ and $f(\Omega)$. Then there exists a sequence of diffeomorphisms…

Let $\Omega\subset \mathbb{R}^{n}$ be a bounded open set. Given $1\leq m_1,m_2\leq n-2$, we construct a homeomorphism $f :\Omega\to \Omega$ that is H\"older continuous, $f$ is the identity on $\partial \Omega$, the derivative $D f$ has rank…

The approximation of Sobolev homeomorphisms by smooth diffeomorphisms is well understood in first-order spaces $W^{1,p}$, but remains largely open in the second-order space $W^{2,1}$ due to a fundamental tension between curvature control…

Every homeomorphism h : X -> Y between planar open sets that belongs to the Sobolev class W^{1,p}(X,Y), 1<p<\infty, can be approximated in the Sobolev norm by 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 $\epsilon$ measure. We show that this piecewise quadratic map can be approximated by diffeomorphisms…

We prove that planar homeomorphisms can be approximated by diffeomorphisms in the Sobolev space $W^{1,2}$ and in the Royden algebra. As an application, we show that every discrete and open planar mapping with a holomorphic Hopf differential…

Let X and Y be planar Jordan domains of the same finite connectivity, Y being inner chordarc regular (such are Lipschitz domains). Every homeomorphism h:X->Y in the Sobolev space $W^{1,2}$ extends to a continuous map between closed domains.…

We present three novel classifications of the weak sequential (and strong) limits in $W^{1,p}$ of planar diffeomorphisms. We introduce a concept called the QM condition which is a kind of separation property for pre-images of closed…

We study global regularity properties of Sobolev homeomorphisms on $n$-dimensional Riemannian manifolds under the assumption of $p$-integrability of its first weak derivatives in degree $p\geq n-1$. We prove that inverse homeomorphisms have…

We show that a planar bi-Lipschitz orientation-preserving homeomorphism can be approximated in the $W^{1,p}$ norm, together with its inverse, with an orientation-preserving homeomorphism which is piecewise affine or smooth.

We show that Sobolev space $W^1_1(\Omega)$ of any planar one-connected domain $\Omega$ has the Bounded Approximation property. The result holds independently from the properties of the boundary of $\Omega$. The prove is based on a new…

Given any $f$ a locally finitely piecewise affine homeomorphism of $\Omega \subset \mathbb{R}^d$ onto $\Delta \subset \mathbb{R}^d$ (for $d=3, 4$) such that $f\in W^{1,p}(\Omega, \mathbb{R}^d)$ and $f^{-1}\in W^{1,q}(\Delta, \mathbb{R}^d)$,…

In this paper, we first prove some propositions of Sobolev spaces defined on a locally finite graph $G=(V,E)$, which are fundamental when dealing with equations on graphs under the variational framework. Then we consider a nonlinear…

First of all, we establish compactness of continuous mappings of the Orlicz--Sobolev classes $W^{1,\varphi}_{\rm loc}$ with the Calderon type condition on $\varphi$ and, in particular, of the Sobolev classes $W^{1,p}_{\rm loc}$ for $p>n-1$…

In the article we study mappings of Carnot groups satisfy moduli inequalities. We prove that homeomorphisms satisfy the moduli inequalities ($Q$-homeomor\-phisms) with a locally integrable function $Q$ are Sobolev mappings. On this base in…

Let $\Omega\subset \mathbb{R}^{n}$ be a bounded open set. Given $2\leq m\leq n$, we construct a convex function $\phi :\Omega\to \mathbb{R}$ whose gradient $f= \nabla \phi$ is a H\"older continuous homeomorphism, $f$ is the identity on…

This paper investigates the differentiability of weak limits of bi-Sobolev homeomorphisms. Given $p>n-1$, consider a sequence of homeomorphisms $f_k$ with positive Jacobians $J_{f_k} >0$ almost everywhere and $\sup_k(\|f_{k}\|_{W^{1,n-1}} +…

Let $(M,g)$ be a closed Riemannian manifold of dimension $n$, and $k\geq 1$ an integer such that $n>2k$. We show that there exists $B_0>0$ such that for all $u \in H^{k}(M)$, \[\|u\|_{L^{2^\sharp}(M)}^2 \leq K_0^2 \int_M |\Delta_g^{k/2}…

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.