Mappings of Finite Distortion: Decay of the Jacobian

arXiv: Complex Variables, 2005

Let $F\in W_{loc}^{1,n}(\Omega;\Bbb R^n)$ be a mapping with non-negative Jacobian $J_F(x)=\text{det} DF(x)\ge 0$ a.e. in a domain $\Omega\in \Bbb R^n$. The dilatation of the mapping $F$ is defined, almost everywhere in $\Omega$, by the formula $$K(x)={{|DF(x)|^n}\over {J_F(x)}}.$$ If $K(x)$ is bounded a.e., the mapping is said to be quasiregular. Quasiregular mappings are a generalization to higher dimensions of holomorphic mappings. The theory of higher dimensional quasiregular mappings began with Re\v{s}hetnyak's theorem, stating that non constant quasiregular mappings are continuous, discrete and open. In some problems appearing in the theory of non-linear elasticity, the boundedness condition on $K(x)$ is too restrictive. Tipically we only know that $F$ has finite dilatation, that is, $K(x)$ is finite a.e. and $K(x)^p$ is integrable for some value $p$. In two dimensions, Iwaniec and \v{S}verak [IS] have shown that $K(x)\in L^1_{loc}$ is sufficient to guarantee the conclusion...

Mathematical Research Letters, 2005

Bulletin of the American Mathematical Society, 1995

Let F ∈ W loc 1 , n ( Ω ; R n ) {F \in W_{{\text {loc}}}^{1,n}(\Omega ;{\mathbb {R}^n})} be a mapping with nonnegative Jacobian J F ( x ) = det D F ( x ) ≥ 0 {{J_F}(x) = \det DF(x) \geq 0} for a.e. x in a domain Ω ⊂ R n {\Omega \subset {\mathbb {R}^n}} . The dilatation of F is defined (almost everywhere in Ω {\Omega } ) by the formula \[ K ( x ) = | D F ( x ) | n J F ( x ) . K(x) = \frac {{|DF(x){|^n}}}{{{J_F}(x)}}. \] Iwaniec and Šverák [IS] have conjectured that if p ≥ n − 1 {p \geq n - 1} and K ∈ L l o c p ( Ω ) {K \in L_{loc}^p(\Omega )} then F must be continuous, discrete and open. Moreover, they have confirmed this conjecture in the two-dimensional case n = 2. In this article, we verify it in the higher-dimensional case n ≥ 2 {n \geq 2} whenever p > n − 1 {p > n - 1} .

The Michigan Mathematical Journal, 2001

Annales- Academiae Scientiarum Fennicae Mathematica

We study mappings f : Ω → R n whose distortion functions K l (x, f) , l = 1, 2, . . . , n − 1 , are in general unbounded but subexponentially integrable. The main result is the weak compactness principle. It asserts that a family of mappings with prescribed volume integral Ω J(x, f) dx , and with given subexponential norm l √ K l ExpA of a distortion function, is closed under weak convergence. The novelty of this result is twofold. Firstly, it requires integral bounds on the distortions K l (x, f) which are weaker than those for the usual outer distortion. Secondly, the category of subexponential bounds is optimal to fully describe the compactness principle for mappings of unbounded distortion, even when outer distortion is used.

Revista Matemática Iberoamericana, 2000

We establish continuity, openness and discreteness, and the condition (N ) for mappings of finite distortion under minimal integrability assumptions on the distortion. 2000 Mathematics Subject Classification: 30C65.

Proceedings of the London Mathematical Society, 2005

Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 2005

Let f ∈ W 1,n (Ω, R n ) be a continuous mapping so that the components of the preimage of each y ∈ R n are compact. We show that f is open and discrete if |Df (x)| n K(x)J f (x) a.e. where K(x) 1 and K n−1 /Φ(log(e + K)) ∈ L 1 (Ω) for a function Φ that satisfies ∞ 1 1/Φ(t) dt = ∞ and some technical conditions. This divergence condition on Φ is shown to be sharp.  2005 Elsevier SAS. All rights reserved.

Journal d'Analyse Mathématique, 2008

In all dimensions k = 1, ..., n − 1, we show that mappings f in R n with finite distortion of hyperarea satisfy certain modulus inequalities in terms of inner and outer dilatation of the mappings.

Siberian Mathematical Journal

Annales de l’institut Fourier, 2002

Journal d'Analyse Mathématique, 2004

General properties of mappings of finite metric distortion and of finite length distortion are studied. Uniqueness, equicontinuity, boundary behavior and removability of singularities are obtained under minimal additional assumptions.

Israel Journal of Mathematics, 2003

Reports of the National Academy of Sciences of Ukraine

The present paper is a natural continuation of our previous papers [1-3], where the reader can find the corresponding historic comments and a discussion of many definitions and relevant results. The given papers were devoted to the theory of the boundary behavior of mappings with finite distortion by Iwaniec. Here, we will develop the theory of the boundary behavior of the so-called mappings with finite length distortion first introduced in [4] for n  , 2 n , see also Chapter 8 in [5]. As was shown in [6], such mappings, generally speaking, are not mappings with finite distortion by Iwaniec, because their first partial derivatives can be not locally integrable. At the same time, this class is a natural generalization of the well-known classes of bi-Lipschitz mappings, as well as isometries and quasi-isometries.

Dynamical Systems, 2012

We give an example illustrating that two notions of bounded distortion for C 1 expanding maps in R are different.