Lectures on mappings of finite distortion
type: Item
instance of: scholarly article
title: Lectures on mappings of finite distortion
zbMATH Open document ID: 1293.30051
DOI: 10.1007/978-3-319-03173-6
author: Stanislav Hencl
author: Pekka Koskela
published in: Lecture Notes in Mathematics
publication date: 2013-11-28T00:00:00Z
review text: The title of the book resembles the well-known monograph by \textit{J. Väisälä} [Lectures on \(n\)-dimensional quasiconformal mappings. Berlin-Heidelberg-New York: Springer-Verlag (1971; Zbl 0221.30031)], which was also published in the Lecture Notes in Mathematics series. Both books deeply investigate the geometry and analysis of maps that are not necessarily smooth, but preserve some of the conformal structure of the Euclidean space \(\mathbb R^n\). The Jacobian matrix \(Df\) of a quasiconformal mapping satisfies the ``distortion inequality'' \(\|Df\|^n \leq K\det Df\) almost everywhere; the map is also required to be in the Sobolev class \(W_{\mathrm{loc}}^{1,n}\) and to be a homeomorphism. The requirements for \(f\) to be a mapping of finite distortion are weaker: in the distortion inequality, the constant \(K\) is replaced by a function \(K(x)\) that must be finite almost everywhere, but may be unbounded. Additionally, a mapping of finite distortion need not be a homeomorphism, and its Sobolev regularity can be as low as \(W_{\mathrm{loc}}^{1,1}\). For most results, some integrability assumption are imposed on \(K\); these vary widely. The case of bounded distortion function, \(K\in L^\infty\), is covered in detail in the book by \textit{Yu. G. Reshetnyak} [Space mappings with bounded distortion. Translations of Mathematical Monographs, 73. Providence, RI: American Mathematical Society (AMS) (1989; Zbl 0667.30018)]. The potential unboundedness of \(K\) brings along many additional challenges. On the other hand, this greater generality allows the methods of geometric mapping theory to penetrate some geometric variational problems (such as those arising in nonlinear elasticity) in which the restriction to \(K\in L^\infty\) would not be natural. The relatively recent book by \textit{T. Iwaniec} and \textit{G. Martin} [Geometric function theory and nonlinear analysis. Oxford University Press (2001; Zbl 1045.30011)] made a convincing case that the theory of mappings…
Mathematics Subject Classification ID: 30C65
Mathematics Subject Classification ID: 49Q15
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 6234401
zbMATH Keywords: mappings of finite distortion
zbMATH Keywords: Sobolev mappings
zbMATH Keywords: Jacobian determinant
zbMATH Keywords: null sets
reviewed by: Leonid V. Kovalev
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-03173-6
OpenAlex ID: W654833068
description: scientific article; zbMATH DE number 6234401