# Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (PMS-48)

Kari Astala
Gaven Martin
Pages: 696
https://www.jstor.org/stable/j.ctt7sjhd

1. Front Matter
(pp. i-vi)
(pp. vii-xiv)
3. Preface
(pp. xv-xviii)
Kari Astala, Tadeusz Iwaniec and Gaven Martin
4. Chapter 1 Introduction
(pp. 1-11)

This book relates the most modern aspects and most recent developments in the theory of planar quasiconformal mappings and their application in conformal geometry, partial differential equations (PDEs) and nonlinear analysis. There are profound applications in such wide-ranging areas as holomorphic dynamical systems, singular integral operators, inverse problems, the geometry of mappings and, more generally, the calculus of variations—all of which are presented here. It is a simply amazing fact that the mathematics that underpins the geometry, structure and dimension of such concepts as Julia sets and limit sets of Kleinian groups, the spaces of moduli of Riemann surfaces,...

5. Chapter 2 A Background in Conformal Geometry
(pp. 12-47)

We have mentioned the strong connections between PDEs and geometry. In this chapter we give a gentle introduction to conformal geometry in the plane and describe some of the connections between conformal and Riemannian geometry with PDEs in two dimensions. We shall also try to give a clear account of how the PDEs we shall spend much of our time studying arise from a number of differing perspectives.

In this book the development of the theory of quasiconformal mappings really starts at the beginning of Chapter 3. The material in the present chapter will be quite familiar to those with...

6. Chapter 3 The Foundations of Quasiconformal Mappings
(pp. 48-91)

One of the more interesting and important recent developments in the theory of quasiconformal mappings has been the development of the study of these mappings on more general spaces than those that are locally Euclidean. In particular, the work of Heinonen and Koskela [161] develops the theory in certain metric spaces where some reasonable measure theoretic and geometric properties hold. Part of the motivation behind this work is that often while studying classical problems in geometry one is naturally led to singular spaces as quotients or, for instance, to the boundaries of geometric objects such as groups. In these situations...

7. Chapter 4 Complex Potentials
(pp. 92-160)

The basic tools for solving general linear elliptic systems are singular integral operators. Even for nonlinear systems, the methods from the linear theory often provide the estimates needed to establish existence and regularity. In the setting of quasilinear elliptic equations, the fact that each solution satisfies some linear elliptic equation makes the appproach via singular integrals particularly successful.

In our approach we shall meet three fundamental operators and many variations of them. It is the purpose of this chapter to develop the theory of these operators. This chapter may also be viewed as an introduction to harmonic analysis in the...

8. Chapter 5 The Measurable Riemann Mapping Theorem: The Existence Theory of Quasiconformal Mappings
(pp. 161-194)

It is the aim of this section to present a complete proof, in as elementary fashion as possible, of the classical existence theory for quasiconformal mappings, or what has come to be known as the measurable Riemann mapping theorem, originally due to Morrey in 1938 [271]. This theorem is a cornerstone in the various interactions among PDEs, complex analysis and geometry, the basic themes of this monograph. Roughly, it states that any bounded measurable conformal structure—or any measurable ellipse field—is equivalent to the usual (round) structure via a quasiconformal change of coordinates. The result underpins the relationship between...

9. Chapter 6 Parameterizing General Linear Elliptic Systems
(pp. 195-209)

In previous chapters we have established simple methods to identify all solutions to the$\mathbb{C}$-linear Beltrami equation${f_{\bar z}} = \mu {f_z}$. We found that it suffices to find one global homeomorphic solution$f:\;\mathbb{C}\; \to \;\mathbb{C}$to the equation and then every other solution$g:\;\Omega \; \to \;\mathbb{C}$can be represented as the composition

$g = \phi \; \circ \;f$, where$\phi :f(\Omega )\; \to \;$is holomorphic

This parameterization, the Stoilow factorization, is a powerful tool heavily used elsewhere in this book and in more general circumstances to develop a deeper understanding of the properties of solutions to the Beltrami equation. These include unique continuation, openess and discreteness, the argument principle, normal family properties, the isolation of branch...

10. Chapter 7 The Concept of Ellipticity
(pp. 210-234)

One of the most important and central themes of this monograph concerns the concept of ellipticity as it pertains to systems of partial differential equations. The types of equations we shall explore evolved from the first-order system of Cauchy-Riemann equations, the same equations that everyone meets in a first course in complex analysis [6] and that play such a central role,

$u_x = v_y$

${u_y} = - {v_x}$,

and of course there are the related second-order equations for$u$and$v$, namely, the Laplace equation

$\Delta u = {u_{xx}} + {u_{yy}} = 0$

A prototype of a nonlinear elliptic PDE is provided,...

11. Chapter 8 Solving General Nonlinear First-Order Elliptic Systems
(pp. 235-258)

The purpose of this chapter is to establish general results concerning the existence, uniqueness and regularity of the solutions to the nonlinear elliptic equations

$\mathcal{F}(z,\;u,\;\nabla u) = 0$(8.1)

in two dimensions. As we discussed in Section 7.7, under the assumption of ellipticity, (8.1) reduces to the system

$\frac{{\partial f}} {{\partial \bar z}} = H\left( {z,\;f,\;\frac{{\partial f}} {{\partial z}}} \right)$

The existence theory for such systems can be established under surprisingly general conditions even in the nonlinear setting. It will become clear that, together with ellipticity, it is the Lusin measurability that is the key notion here.

We are primarily interested in global solutions, that is, solutions defined in the entire plane$\mathbb{C}$. There...

12. Chapter 9 Nonlinear Riemann Mapping Theorems
(pp. 259-274)

Let$\Omega \; \subset \;\mathbb{C}$be a domain. We say that$\Omega$is aJordan domainif$\partial \,\Omega$is a topological circle in Ĉ. More generally, we will use the following terminology: We say that the domain$\Omega$is a Jordan$n{\rm{ - }}$connected domain if$\partial \,\Omega$has$n$disjoint connected components, each of which is a Jordan curve.

The most important theorem about Jordan domains is the celebrated Riemann mapping theorem, which asserts that all Jordan domains are conformally equivalent to the disk.

Theorem 9.0.2.A domain$\Omega \; \subset \;\mathbb{C}$is a Jordan domain if and only if there is a homeomorphism$\varphi :\bar \Omega \; \to \;\bar \mathbb{D}$that is conformal...

13. Chapter 10 Conformal Deformations and Beltrami Systems
(pp. 275-288)

It is now time to return to the study of Riemannian and conformal structures in planar domains, one of the starting themes of our discussion in Section 2.1.

Let$\Omega$and$\Omega '$be planar domains with measurable conformal structures$G:\Omega \to {\mathbf{S}}{\text{(2)}}$and$H:\Omega ' \to {\mathbf{S}}{\text{(2)}}$. We have already seen that a conformal mapping$f$between the surfaces$(\Omega ,\;G)$and$(\Omega ',\;H)$can be viewed as a solution to the first-order system of differential equations

${D^t}f(z)H(f(z))\,Df(z)\: = \:J(z,\:f)G(z)$, for almost every$z\: \in \:\Omega$(10.1)

Therefore a complex interpretation of the Beltrami system (10.1) is required. It will be useful to study this theme in some depth.

One of...

14. Chapter 11 A Quasilinear Cauchy Problem
(pp. 289-292)

In this chapter we shall study existence and uniqueness problems for a quasilinear Cauchy problem. We will not aim for maximal generality even though a large class of equations is covered with the results obtained here. Our goal is rather to develop tools for our subsequent discussion of the theory of holomorphic motions, although for that application it is, in fact, sufficient to work with the smooth case.

The differential equation we are interested in here is

$\frac{{\partial g}} {{\partial \bar z}}\quad = \quad \Psi (z,\;g)$(11.1)

$g(z) \to {z_0}$as$z\: \to \:\infty$(11.2)

This equation lies slightly outside our theme of ellipticity, yet the reader will see that it plays...

15. Chapter 12 Holomorphic Motions
(pp. 293-315)

The notion of holomorphic motions, introduced by Mañé, Sad and Sullivan [237], explains in a striking manner the many connections quasiconformal mappings have to holomorphic dynamics, Teichmüller theory and many other areas of complex analysis. It shows that a holomorphic deformation of the identity mapping, in the space of injections of a given set$A$into$\mathbb{C}$, yields only quasisymmetric mappings. An important fact concerning holomorphic motions is Słodkowski’s generalized$\lambda {\text{ - lemma}}$[329], which shows that any holomorphic motion of any set$A \subset$Ĉ admits an extension to a holomorphic motion of the entire Riemann sphere Ĉ. Thus the quasisymmetric mappings obtained...

16. Chapter 13 Higher Integrability
(pp. 316-361)

In this chapter we will meet some of the most important recent advances in the theory of planar quasiconformal mappings. The results will be used in wide and diverse settings and will provide a bridge between complex dynamics and the regularity theory and singularity structure of second-order elliptic equations.

Finding the exact degree of smoothness of a$K$-quasiconformal mapping is the primary question we shall address in this chapter. It should not be surprising that knowing such estimates provides bounds on how a mapping might distort the measure of a given set. Bojarski’s Theorem 5.4.2 shows that the differential of...

17. Chapter 14 ${L^p}$-Theory of Beltrami Operators
(pp. 362-388)

Classically the homeomorphic solutions to the Beltrami equation

$\frac{{\partial f}} {{\partial \,\bar z}} = \mu (z)\,\frac{{\partial f}} {{\partial z}}$

are constructed via the Neumann iteration procedure such as that by Bojarski [64], although there do exist other, more function theoretic approaches, for example, in Lehto-Virtanen [229].

The${L^p}$-boundedness of the Beurling transform was a crucial ingredient in the success of the Neumann iteration method and provided the first results toward the self-improving regularity of quasiconformal mappings.

Now that we have been able to establish, through the theory of holomorphic motions, the precise higher-integrability properties of quasiconformal mappings, we wish to return to the Neumann procedure and see what improvements the...

18. Chapter 15 Schauder Estimates for Beltrami Operators
(pp. 389-402)

The fundamental ideas concerning the Hölder regularity of solutions to PDEs started with J. Schauder’s work [321, 322] in the early 1930s. He was largely concerned with linear, quasilinear and nonlinear elliptic equations of second order. Suppose we have a linear elliptic equation

$\sum\limits_{j,\,l = 1}^n {{a_{jl}}(x)\,{u_{{x_j}{x_l}}}} = h(x),\quad \quad x\; \in \;\Omega \; \subset \;{\mathbb{R}^n}$,

where the coefficients and the right hand side are Hölder-continuous of exponent$0\; < \;\alpha \; < \;1$; 1; that is,${a_{jl}},\;h\; \in \;C_{loc}^\alpha (\Omega )$. Schauder proved that the solutions$u$lie in the space${C^{2,\,\alpha }}(\Omega )$using singular integrals acting in the space${C^\alpha }({\mathbb{R}^n})$; the${L^p}$-bounds for singular integrals came later in 1952 in the pioneering work of Calderón and Zygmund [86]. Schauder’s basic...

19. Chapter 16 Applications to Partial Differential Equations
(pp. 403-471)

As we pointed out in the introduction to this book, in physics and mathematics conservation laws and equations of motion or state are typically described by divergence-type second-order differential equations. In that introduction we carefully worked through an example from the calculus of variations—essentially a Lagrangian approach to deformations. There are many other classical examples with some interesting modern overtones. For instance, Noether’s theorem [288] is a basic result from theoretical physics, and it implies that a conservation law, and therefore a second-order equation of divergence-type, can be derived from any continuous symmetry of a physical system. For example,...

20. Chapter 17 PDEs Not of Divergence Type: Pucci’s Conjecture
(pp. 472-489)

In the previous chapter we made a fairly thorough study of second-order elliptic PDEs of divergence-type. However, in two dimensions the methods and techniques developed there actually apply as well to equations that are not of divergence-type. Such equations arise naturally in many different contexts, for instance in stochastics, as Monge Ampere– type equations (for transport and related problems) and also in the linearization of nonlinear elliptic PDEs.

The inequality of Alexandrov, Bakel’man and Pucci is a basic tool in the theory of linear elliptic partial differential equations that are not in divergence form, as well as in the more...

21. Chapter 18 Quasiconformal Methods in Impedance Tomography: Calderón’s Problem
(pp. 490-513)

In impedance tomography one aims to determine the internal structure of a body from electrical measurements on its surface. Such methods have a variety of different applications for instance in engineering and medical diagnostics. For a general expository presentation for medical applications, see [106].

In 1980 A.P. Calderón showed that the impedance tomography problem admits a clear and precise mathematical formulation. Indeed, suppose that$\Omega \; \subset \;{\mathbb{R}^n}$is a bounded domain with connected complement and let$\sigma :\Omega \to (0,\;\infty )$be a measurable function that is bounded away from zero and infinity. According to Theorem 16.1.1, the Dirichlet problem

$\nabla \; \cdot \;\sigma \,\nabla u\;\; = \;\;0\;\;\;{\rm{in}}\;\Omega {\rm{,}}$(18.1)

${\left. u \right|_{\partial \,\Omega }}\;\; = \;\;\phi \; \in \;{W^{1,\,1/2}}(\partial \,\Omega )$(18.2)

22. Chapter 19 Integral Estimates for the Jacobian
(pp. 514-526)

In this chapter, besides providing some fundamental estimates for the Jacobian of a quasiconformal mapping, we aim to connect the theory of quasiconformal mappings with central problems in the calculus of variations and in particular the notions of quasiconvexity and rank-one convexity due to Morrey [273]. We show how conjectures relating these ideas would provide the answers to important questions in the theory of quasiconformal mappings such as for instance the precise form of the$p{\rm{ - }}$norms of the Beurling transform. Actually, the natural place for this discussion is in${\mathbb{R}^n}$, and so for this chapter we will frame some of...

23. Chapter 20 Solving the Beltrami Equation: Degenerate Elliptic Case
(pp. 527-585)

This chapter is focused on the study of the degenerate elliptic equation

$\frac{{\partial f}} {{\partial \bar z}} = \mu (z)\frac{{\partial f}} {{\partial z}}$, (20.1)

where$|\,\mu (z)\,|\; < \;1$almost everywhere, but one might have

${\left\| \mu \right\|_\infty } = 1$(20.2)

These equations arise naturally in hydrodynamics, nonlinear elasticity, holomorphic dynamics and several other related areas. Let us briefly consider two examples:

In two-dimensional hydrodynamics we have seen that the fluid velocity (the complex gradient of the potential function (see (16.51) ) satisfies a Beltrami equation that degenerates as the flow approaches a critical value, the local speed of sound (see (16.54)). What happens to these equations and their solutions as we approach or break the speed...

24. Chapter 21 Aspects of the Calculus of Variations
(pp. 586-623)

The theory of mappings of finite distortion arose out of a need to extend the ideas and applications of the classical theory of quasiconformal mappings to the degenerate elliptic setting. There one finds concrete applications in materials science, particularly nonlinear elasticity and critical phase phenomena, and in the calculus of variations. In this chapter we consider applications of mappings of finite distortion to some interesting problems in the calculus of variations.

Here we discuss recent advances in the study of existence and uniqueness properties for mappings between planar domains whose boundary values are prescribed and have the smallest mean distortion....

25. Appendix: Elements of Sobolev Theory and Function Spaces
(pp. 624-642)
26. Basic Notation
(pp. 643-646)
27. Bibliography
(pp. 647-670)
28. Index
(pp. 671-677)