Mappings of finite distortion: condition N

2005

This paper investigates the self-improving integrability properties of the so-called mappings of finite distortion. Let KðxÞX1 be a measurable function defined on a domain OCR n ; nX2; and such that expðbKðxÞÞAL 1 loc ðOÞ; b40: We show that there exist two universal constants c 1 ðnÞ; c 2 ðnÞ with the following property: Let f be a mapping in W 1;1 loc ðO; R n Þ with jDf ðxÞj n pKðxÞJðx; f Þ for a.e. xAO and such that the Jacobian determinant Jðx; f Þ is locally in L 1 log Àc1ðnÞb L: Then automatically Jðx; f Þ is locally in L 1 log c2ðnÞb LðOÞ: This result constitutes the appropriate analog for the self-improving regularity of quasiregular mappings and clarifies many other interesting properties of mappings of finite distortion. Namely, we obtain novel results on the size of removable singularities for bounded mappings of finite distortion, and on the area distortion under this class of mappings. r

Mathematika, 1979

Siberian Mathematical Journal

Inventiones Mathematicae, 2001

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} .

Journal of the European Mathematical Society, 2003

Transactions of the American Mathematical Society, 2003

We establish an essentially sharp modulus of continuity for mappings of subexponentially integrable distortion.

Information Sciences, 2013

Let l = (l n) be a universal fuzzy measure and let MðlÞ be the set of all l-measurable sets, i.e. sets A & N for which the limit l ⁄ (A) = lim n?1 l n (A \ {1, 2,. . ., n}) exists. We are studying properties of measurability preserving injective mappings, i.e. injective mappings p : N ! N such that A 2 MðlÞ implies pðAÞ 2 MðlÞ. Under some assumptions on l we prove l ⁄ (p(A)) = kl ⁄ (A) for all A 2 MðlÞ, where k ¼ l à ðpðNÞÞ.

Forum Mathematicum, 2018

In this work, we are concerned with the study of the N-Lusin property in metric measure spaces. A map satisfies that property if sets of measure zero are mapped to sets of measure zero. We prove a new sufficient condition for the N-Lusin property using a weak and pointwise Lipschitz-type estimate. Relations with approximate differentiability in metric measure spaces and other sufficient conditions for the N-Lusin property will be provided.

Annales de l’institut Fourier, 2002