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Mappings of finite distortion: The zero set of the Jacobian

Mappings of finite distortion: The zero set of the Jacobian J. Eur. Math. Soc. 5, 95–105 (2003) Digital Object Identifier (DOI) 10.1007/s10097-002-0046-9 Pekka Koskela · Jan Malý Mappings of finite distortion: The zero set of the Jacobian Received November 20, 2001 / final version received October 3, 2002 Published online November 19, 2002 –  c Springer-Verlag & EMS 2002 1. Introduction This paper is part of our program to establish the fundamentals of the theory of mappings of finite distortion [8], [1], [10], [15], [16], [9] which form a natural generalization of the class of mappings of bounded distortion, also called quasireg- ular mappings. Our research continues earlier developments on mappings of finite distortion, e.g. [5], [14], [7], [18]. Let us begin with the definition. We assume that  ⊂ R is a connected open set. We say that a mapping f :  → R has finite distortion if: 1,1 (FD-1) f ∈ W (,R ). loc (FD-2) The Jacobian determinant J(x, f ) of f is locally integrable. (FD-3) There is a measurable function K = K(x) ≥ 1, finite almost everywhere, such that f satisfies the distortion inequality |Df(x)| ≤ K(x) J(x, f ) a.e. We arrive at the usual definition of a mapping of http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of the European Mathematical Society Springer Journals

Mappings of finite distortion: The zero set of the Jacobian

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Publisher
Springer Journals
Copyright
Copyright © 2002 by Springer-Verlag Berlin Heidelberg & EMS
Subject
Mathematics; Mathematics, general
ISSN
1435-9855
DOI
10.1007/s10097-002-0046-9
Publisher site
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Abstract

J. Eur. Math. Soc. 5, 95–105 (2003) Digital Object Identifier (DOI) 10.1007/s10097-002-0046-9 Pekka Koskela · Jan Malý Mappings of finite distortion: The zero set of the Jacobian Received November 20, 2001 / final version received October 3, 2002 Published online November 19, 2002 –  c Springer-Verlag & EMS 2002 1. Introduction This paper is part of our program to establish the fundamentals of the theory of mappings of finite distortion [8], [1], [10], [15], [16], [9] which form a natural generalization of the class of mappings of bounded distortion, also called quasireg- ular mappings. Our research continues earlier developments on mappings of finite distortion, e.g. [5], [14], [7], [18]. Let us begin with the definition. We assume that  ⊂ R is a connected open set. We say that a mapping f :  → R has finite distortion if: 1,1 (FD-1) f ∈ W (,R ). loc (FD-2) The Jacobian determinant J(x, f ) of f is locally integrable. (FD-3) There is a measurable function K = K(x) ≥ 1, finite almost everywhere, such that f satisfies the distortion inequality |Df(x)| ≤ K(x) J(x, f ) a.e. We arrive at the usual definition of a mapping of

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Journal of the European Mathematical Society – Springer Journals

Published: Jun 1, 2003

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