Elliptic Partial Differential Equations and Quasiconformal
Mappings in the Plane focuses
on the subject of planar quasiconformal mappings, with an emphasis on
building a link between analysis and geometry. The book is a organized
by iterating various subjects in PDEs: harmonic analysis, dynamical
systems and topology, hold together by the common theme of quasiconformal
mappings and their applications. It is remarkably interesting that all
the topics can be formulated in the form of second order elliptic
PDEs, which has been a well-investigated subject in the context of
calculus of variations. Among all the topics, I personally like the
chapter on inverse problems, which previously has not been studied
from a geometric perspective. The current treatment by this book will
definitely bring some new depth to a challenging field.
The book starts with an introduction providing the most important
applications of quasiconformal mappings, including holomorphic
dynamical systems, singular integral operators, etc. It also provides
a motivating example on the technical aspects of the book through a
two-dimensional hydrodynamical model. The introduction is followed by two
chapters on the background and fundamentals on quasiconformal
mappings.
For readers who need some additional background knowledge in
analysis, two chapters are devoted to complex analysis and the Beltrami
equation. The subject of elliptic PDEs is then discussed in greater detail in Chapters
6, 7, and 8. The existence and uniqueness of the nonlinear Riemann
mapping for different domain geometries are established in Chapter 9, and
the conformal equivalence of Riemann structures is proved in Chapter
10.
The authors then move to the topic of holomorphic dynamical
systems, and they investigate interesting topics such as Hausdorff
dimension. After some estimates on Beltrami operators using tools
from harmonic analysis, Pucci's conjecture for PDE's not of
divergence form is throughly studied and is further studied in Chapter
20. Applications such as inverse problems and calculus of variations are
also discussed in detail.
The chapters are clearly written and are very user friendly for readers who may not have a diversified background but are
interested in finding how analysis can be useful in revealing
geometry, and vice versa. The book could be used by experts who want
an updated account of all the recent progress on the subject as well
as by graduate students who need an encyclopedic reference. I found the book to be very enjoyable.
Editor's Note: More information, including a free preview of the first chapter, are available on the publisher's website.