Related papers: Obstacles for Sobolev-homeomorphisms with low rank…

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 given a sequence of homeomorphisms $f_k: \Omega \to \mathbb{R}^n$ convergent in $W^{1,p}(\Omega, \mathbb{R}^n)$, $p \geq 1$ for $n =2$ and $p > n-1$ for $n \geq 3$, to a homeomorphism $f$ which maps sets of measure zero onto…

The Heisenberg groups are examples of sub-Riemannian manifolds homeomorphic, but not diffeomorphic to the Euclidean space. Their metric is derived from curves which are only allowed to move in so-called horizontal directions. We report on…

Let $K$ be a closed polydisc or ball in $\C^n$, and let $Y$ be a quasi projective algebraic manifold which is Zariski locally equivalent to $\C^p$, or a complement of an algebraic subvariety of codimension $\ge 2$ in such manifold. If $r$…

The Lefschetz fixed point theorem provides a powerful obstruction to the existence of minimal homeomorphisms on well-behaved spaces such as finite CW-complexes. We show that these obstructions do not hold for more general spaces. More…

This paper has the characteristics of a review paper in which results of shift-invariant subspaces of Sobolev type are summarized without proofs. The structure of shift-invariant spaces $V_s$, $s\in\mathbb{R}$, generated by at most…

In this paper, we obtain the sharp $k$-th order Sobolev inequalities in the hyperbolic space ${\H}^n$ for all $k=1,2,3,\cdots$. This gives an answer to an open question raised by Aubin in [5, p.$\;$176-177] for $W^{k,2}({\H}^n)$ with $k>1$.…

We classify all links whose Khovanov homology have ranks no greater than 8, and all three-component links whose Khovanov homology have ranks no greater than 12, where the coefficient ring is Z/2. The classification is based on the previous…

The multipullback quantization of complex projective spaces lacks the naive quantum CW-complex structure because the quantization of an embedding of the $n$-skeleton into the $(n+1)$-skeleton does not exist. To overcome this difficulty, we…

We introduce and study weak-2-local symmetric maps between C$^*$-algebras $A$ and $B$ as non necessarily linear nor continuous maps $\Delta: A\to B$ such that for each $a,b\in A$ and $\phi\in B^{*}$, there exists a symmetric linear map…

We establish a "low rank property" for Sobolev mappings that pointwise solve a first order nonlinear system of PDEs, whose smooth solutions have the so-called "contact property". As a consequence, Sobolev mappings from an open set of the…

In this paper, we establish a new improved Sobolev inequality based on a weighted Morrey space. To be precise, there exists $C=C(n,m,s,\alpha)>0$ such that for any $u,v \in {\dot{H}}^s(\mathbb{R}^{n})$ and for any $\theta \in…

We discuss $C^1$ regularity and developability of isometric immersions of flat domains into $\mathbb R^3$ enjoying a local fractional Sobolev $W^{1+s, \frac2s}$ regularity for $2/3 \le s< 1 $, generalizing the known results on Sobolev and…

From the Lytchak's result for polar foliations on an irreducible simply connected symmetric space $G/K$ of compact type and rank greater than one, we can derive that there exists no equifocal submanifold with non-flat section whose…

First of all, we prove that open mappings in Orlicz-Sobolev classes $W^{1,\phi}_{\rm loc}$ under the Calderon type condition on $\phi$ have the total differential a.e. that is a generalization of the well-known theorems of…

Let X and Y be bounded multiply connected Lipschitz domains in \R^2. We consider the class H_p (X, Y) of homeomorphisms h : X -> Y in the Sobolev space W^{1,p} (X, \R^2). We prove that the weak and strong closures of H_p (X, Y), 2 \le p<…

The classical Sobolev and Escobar inequalities are embedded into the same one-parameter family of sharp trace-Sobolev inequalities on half-spaces. Equality cases are characterized for each inequality in this family by tweaking a well-known…

The well-known Bohr--P\'al theorem asserts that for every continuous real-valued function $f$ on the circle $\mathbb T$ there exists a change of variable, i.e., a homeomorphism $h$ of $\mathbb T$ onto itself, such that the Fourier series of…

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…

Backward stochastic partial differential equations of parabolic type with variable coefficients are considered in the whole Euclidean space. Improved existence and uniqueness results are given in the Sobolev space $H^n$ ($=W^n_2$) under…