Mappings of finite distortion: The sharp modulus of continuity

Inventiones Mathematicae, 2001

Illinois Journal of Mathematics

We study mappings of finite distortion whose distortion functions are locally subexponentially integrable. We establish a local modulus of continuity estimate for the inverse of such a map. As applications, we describe the possible expansion and compression of certain Hausdorff measures and Minkowski contents under such mappings. We also exhibit examples that describe the extent to which our results are sharp.

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.

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

Journal of the European Mathematical Society, 2003

Mathematical Research Letters, 2005

The Michigan Mathematical Journal, 2001

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.

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.

Siberian Mathematical Journal