Mappings of finite distortion: Sharp Orlicz-conditions

Mathematical Research Letters, 2005

Journal of the European Mathematical Society, 2003

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

Rendiconti Lincei - Matematica e Applicazioni, 2000

We study regularity properties of mappings of finite distortion. We show that some sort of self-improvement phenomena hold also when only subexponential integrability is assumed for the distortion function. We extend to this setting results by Faraco, Koskela and Zhong [9] and Bildhauer, Fuchs and Zhong .

Journal of Geometric Analysis, 2012

We establish the sharp degree of integrability for the reciprocal of the Jacobian determinant of an open and discrete mapping with finite, p-integrable distortion.

arXiv (Cornell University), 2022

For an arbitrary convex function Ψ : [1, ∞) → [1, ∞), we consider uniqueness in the following two related extremal problems: Problem A (boundary value problem): Establish the existence of, and describe the mapping f , achieving inf f D Ψ(K(z, f)) dz : f :D →D a homeomorphism in W 1,1 0 (D) + f 0. Here the data f 0 :D →D is a homeomorphism of finite distortion with D Ψ(K(z, f 0)) dz < ∞-a barrier. Next, given two homeomorphic Riemann surfaces R and S and data f 0 : R → S a diffeomorphism.

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.

Annali di Matematica Pura ed Applicata (1923 -)

In the paper we investigate continuity of Orlicz-Sobolev mappings W 1,P (M, N) of finite distortion between smooth Riemannian n-manifolds, n ≥ 2, under the assumption that the Young function P satisfies the so-called divergence condition ∞ 1 P(t)/t n+1 dt = ∞. We prove that if the manifolds are oriented, N is compact, and the universal cover of N is not a rational homology sphere, then such mappings are continuous. That includes mappings with D f ∈ L n and, more generally, mappings with D f ∈ L n log −1 L. On the other hand, if the space W 1,P is larger than W 1,n (for example if D f ∈ L n log −1 L), and the universal cover of N is homeomorphic to S n , n = 4, or is diffeomorphic to S n , n = 4, then we construct an example of a mapping in W 1,P (M, N) that has finite distortion and is discontinuous. This demonstrates a new global-to-local phenomenon: Both finite distortion and continuity are local properties, but a seemingly local fact that finite distortion implies continuity is a consequence of a global topological property of the target manifold N .

Siberian Mathematical Journal

Proceedings of the London Mathematical Society, 2005