Optimal regularity for planar mappings of finite distortion

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

Mathematical Research Letters, 2005

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.

Inventiones Mathematicae, 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.

Journal of the European Mathematical Society, 2003

Siberian Mathematical Journal

The Michigan Mathematical Journal, 2001

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 of Geometric Analysis, 2010

ABSTRACT We study how planar Sobolev homeomorphisms distort sets of Hausdorff dimension strictly less than two. We measure the image size by means of a generalized Hausdorff measure. As an application, we obtain a sharp generalized dimension distortion estimate for mappings of exponentially integrable distortion. KeywordsMapping of finite distortion-Generalized Hausdorff measure-Sobolev homeomorphism Mathematics Subject Classification (2000)30C65