scholarly journals A note on mappings of finite distortion: Examples for the sharp modulus of continuity

2011 ◽  
Vol 36 ◽  
pp. 531-536 ◽  
Author(s):  
Daniel Campbell ◽  
Stanislav Hencl
Keyword(s):  
Modulus Of Continuity ◽  
Finite Distortion ◽  
Mappings Of Finite Distortion

On nonhomeomorphic mappings with the inverse Poletsky inequality

2020 ◽  
Vol 17 (3) ◽  
pp. 414-436
Author(s):  
Evgeny Sevost'yanov ◽  
Serhii Skvortsov ◽  
Oleksandr Dovhopiatyi
Keyword(s):  
Boundary Behavior ◽  
Hölder Continuity ◽  
Continuous Extension ◽  
Quasiconformal Mappings ◽  
Holder Continuity ◽  
Hölder Continuous ◽  
Finite Distortion ◽  
Mappings Of Finite Distortion ◽  
Lipschitz Continuous ◽  
Inverse Inequality

As known, the modulus method is one of the most powerful research tools in the theory of mappings. Distortion of modulus has an important role in the study of conformal and quasiconformal mappings, mappings with bounded and finite distortion, mappings with finite length distortion, etc. In particular, an important fact is the lower distortion of the modulus under mappings. Such relations are called inverse Poletsky inequalities and are one of the main objects of our study. The use of these inequalities is fully justified by the fact that the inverse inequality of Poletsky is a direct (upper) inequality for the inverse mappings, if there exist. If the mapping has a bounded distortion, then the corresponding majorant in inverse Poletsky inequality is equal to the product of the maximum multiplicity of the mapping on its dilatation. For more general classes of mappings, a similar majorant is equal to the sum of the values of outer dilatations over all preimages of the fixed point. It the class of quasiconformal mappings there is no significance between the inverse and direct inequalities of Poletsky, since the upper distortion of the modulus implies the corresponding below distortion and vice versa. The situation significantly changes for mappings with unbounded characteristics, for which the corresponding fact does not hold. The most important case investigated in this paper refers to the situation when the mappings have an unbounded dilatation. The article investigates the local and boundary behavior of mappings with branching that satisfy the inverse inequality of Poletsky with some integrable majorant. It is proved that mappings of this type are logarithmically Holder continuous at each inner point of the domain. Note that the Holder continuity is slightly weaker than the classical Holder continuity, which holds for quasiconformal mappings. Simple examples show that mappings of finite distortion are not Lipschitz continuous even under bounded dilatation. Another subject of research of the article is boundary behavior of mappings. In particular, a continuous extension of the mappings with the inverse Poletsky inequality is obtained. In addition, we obtained the conditions under which the families of these mappings are equicontinuous inside and at the boundary of the domain. Several cases are considered: when the preimage of a fixed continuum under mappings is separated from the boundary, and when the mappings satisfy normalization conditions. The text contains a significant number of examples that demonstrate the novelty and content of the results. In particular, examples of mappings with branching that satisfy the inverse Poletsky inequality, have unbounded characteristics, and for which the statements of the basic theorems are satisfied, are given.

scholarly journals A Note on Extremal Mappings of Finite Distortion

2005 ◽  
Vol 12 (2) ◽  
pp. 231-237 ◽  
Author(s):  
Stanislav Hencl ◽  
Pekka Koskela ◽  
Jani Onninen
Keyword(s):  
Finite Distortion ◽  
Mappings Of Finite Distortion

scholarly journals Mappings of finite distortion: Sharp Orlicz-conditions

2003 ◽  
pp. 857-872 ◽  
Author(s):  
Janne Kauhanen ◽  
Pekka Koskela ◽  
Jan Malý ◽  
Jani Onninen ◽  
Xiao Zhong
Keyword(s):  
Finite Distortion ◽  
Mappings Of Finite Distortion

scholarly journals Optimal regularity for planar mappings of finite distortion

2010 ◽  
Vol 27 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Kari Astala ◽  
James T. Gill ◽  
Steffen Rohde ◽  
Eero Saksman
Keyword(s):  
Optimal Regularity ◽  
Finite Distortion ◽  
Mappings Of Finite Distortion
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