Lectures on mappings of finite distortion
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 of finite distortion is deep, rich in open problems, and has natural connections with the theory of elasticity. There were numerous new developments in the theory since 2001, many of which appear in this book. Chapter 1 is introductory; it motivates the concept of a mapping of finite distortion by considering the invertibility of maps in the Sobolev setting. Chapter 2 concerns the continuity of maps of finite distortion, as well as their differentiability. Images and preimages of null sets are considered in Chapter 4. Chapter 5 contains a large number of strong results concerning the regularity of inverses of Sobolev and BV maps. Most of these results are very recent and are due to the authors of the book and their collaborators. Chapter 6 investigates the integrability of the Jacobian determinant \(\det Df\) and of its reciprocal. The concluding Chapter 7 is a collection of recent results that did not fit in the preceding chapters. In order to make the book more self-contained, the authors included 10 appendices on relevant topics of geometric measure theory, real analysis and partial differential equations. Possible directions of future research are outlined by 26 open problems stated throughout the book. Much of the material, especially in Chapters 5--7, appears in book form for the first time. The book is a welcome addition to the growing body of literature on mappings of finite distortion.
scientific article; zbMATH DE number 6234401