Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane - Rilegato

Astala, Kari; Iwaniec, Tadeusz; Martin, Gaven

Rilegato

ISBN 10: 0691137773 ISBN 13: 9780691137773

Casa editrice: Princeton Univ Pr, 2009

Riassunto

This book explores the most recent developments in the theory of planar quasiconformal mappings with a particular focus on the interactions with partial differential equations and nonlinear analysis. It gives a thorough and modern approach to the classical theory and presents important and compelling applications across a spectrum of mathematics: dynamical systems, singular integral operators, inverse problems, the geometry of mappings, and the calculus of variations. It also gives an account of recent advances in harmonic analysis and their applications in the geometric theory of mappings.


The book explains that the existence, regularity, and singular set structures for second-order divergence-type equations--the most important class of PDEs in applications--are determined by the mathematics underpinning the geometry, structure, and dimension of fractal sets; moduli spaces of Riemann surfaces; and conformal dynamical systems. These topics are inextricably linked by the theory of quasiconformal mappings. Further, the interplay between them allows the authors to extend classical results to more general settings for wider applicability, providing new and often optimal answers to questions of existence, regularity, and geometric properties of solutions to nonlinear systems in both elliptic and degenerate elliptic settings.

Su questo libro

Informazioni sull?autore

Kari Astala is the Finnish Academy Professor of Mathematics at the University of Helsinki. Tadeusz Iwaniec is the John Raymond French Distinguished Professor of Mathematics at Syracuse University. Gaven Martin is the Distinguished Professor of Mathematics at Massey University.

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Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane

By Kari Astala Tadeusz Iwaniec Gaven Martin

Princeton University Press

Copyright © 2009 Princeton University Press
All right reserved.
ISBN: 978-0-691-13777-3

Contents

Preface........................................................................................................xv1 Introduction................................................................................................11.1 Calculus of Variations, PDEs and Quasiconformal Mappings..................................................21.2 Degeneracy................................................................................................61.3 Holomorphic Dynamical Systems.............................................................................81.4 Elliptic Operators and the Beurling Transform.............................................................92 A Background in Conformal Geometry..........................................................................122.1 Matrix Fields and Conformal Structures....................................................................122.2 The Hyperbolic Metric.....................................................................................152.3 The Space S(2)............................................................................................172.4 The Linear Distortion.....................................................................................212.5 Quasiconformal Mappings...................................................................................222.6 Radial Stretchings........................................................................................282.7 Hausdorff Dimension.......................................................................................302.8 Degree and Jacobian.......................................................................................322.9 A Background in Complex Analysis..........................................................................342.10 Distortion by Conformal Mapping...........................................................................413 The Foundations of Quasiconformal Mappings..................................................................483.1 Basic Properties..........................................................................................483.2 Quasisymmetry.............................................................................................493.3 The Gehring-Lehto Theorem.................................................................................513.4 Quasisymmetric Maps Are Quasiconformal....................................................................583.5 Global Quasiconformal Maps Are Quasisymmetric.............................................................643.6 Quasiconformality and Quasisymmetry: Local Equivalence....................................................703.7 Lusin's Condition N and Positivity of the Jacobian........................................................723.8 Change of Variables.......................................................................................763.9 Quasisymmetry and Equicontinuity..........................................................................783.10 Hlder Regularity.........................................................................................803.11 Quasisymmetry and -Monotone Mappings......................................................................834 Complex Potentials..........................................................................................924.1 The Fourier Transform.....................................................................................984.2 The Complex Riesz Transforms [R.sup.k]....................................................................1024.3 Quantitative Analysis of Complex Potentials...............................................................1044.4 Maximal Functions and Interpolation.......................................................................1174.5 Weak-Type Estimates and [L.sup.p]-Bounds..................................................................1244.6 BMO and the Beurling Transform............................................................................1314.7 Hlder Estimates..........................................................................................1474.8 Beurling Transforms for Boundary Value Problems...........................................................1504.9 Complex Potentials in Multiply Connected Domains..........................................................1585 The Measurable Riemann Mapping Theorem: The Existence Theory of Quasiconformal Mappings.....................1615.1 The Basic Beltrami Equation...............................................................................1635.2 Quasiconformal Mappings with Smooth Beltrami Coefficient..................................................1655.3 The Measurable Riemann Mapping Theorem....................................................................1685.4 [L.sup.p]-Estimates and the Critical Interval.............................................................1725.5 Stoilow Factorization.....................................................................................1785.6 Factoring with Small Distortion...........................................................................1845.7 Analytic Dependence on Parameters.........................................................................1855.8 Extension of Quasisymmetric Mappings of the Real Line.....................................................1895.9 Reflection................................................................................................1925.10 Conformal Welding.........................................................................................1936 Parameterizing General Linear Elliptic Systems..............................................................1956.1 Stoilow Factorization for General Elliptic Systems........................................................1966.2 Linear Families of Quasiconformal Mappings................................................................1986.3 The Reduced Beltrami Equation.............................................................................2026.4 Homeomorphic Solutions to Reduced Equations...............................................................2047 The Concept of Ellipticity..................................................................................2107.1 The Algebraic Concept of Ellipticity......................................................................2117.2 Some Examples of First-Order Equations....................................................................2137.3 General Elliptic First-Order Operators in Two Variables...................................................2147.4 Partial Differential Operators with Measurable Coefficients...............................................2217.5 Quasilinear Operators.....................................................................................2227.6 Lusin Measurability.......................................................................................2237.7 Fully Nonlinear Equations.................................................................................2267.8 Second-Order Elliptic Systems.............................................................................2318 Solving General Nonlinear First-Order Elliptic Systems......................................................2358.1 Equations Without Principal Solutions.....................................................................2368.2 Existence of Solutions....................................................................................2378.3 Proof of Theorem 8.2.1....................................................................................2398.4 Equations with Infinitely Many Principal Solutions........................................................2488.5 Liouville Theorems........................................................................................2498.6 Uniqueness................................................................................................2538.7 Lipschitz H(z, w, [zeta]).................................................................................2569 Nonlinear Riemann Mapping Theorems..........................................................................2599.1 Ellipticity and Change of Variables.......................................................................2619.2 The Nonlinear Mapping Theorem: Simply Connected Domains...................................................2639.3 Mappings onto Multiply Connected Schottky Domains.........................................................26910 Conformal Deformations and Beltrami Systems.................................................................27510.1 Quasilinearity of the Beltrami System....................................................................27510.2 Conformal Equivalence of Riemannian Structures...........................................................27910.3 Group Properties of Solutions............................................................................28011 A Quasilinear Cauchy Problem................................................................................28911.1 The Nonlinear [??]-Equation..............................................................................28911.2 A Fixed-Point Theorem....................................................................................29011.3 Existence and Uniqueness.................................................................................29112 Holomorphic Motions.........................................................................................29312.1 The [lambda]-Lemma.......................................................................................29412.2 Two Compelling Examples..................................................................................29612.3 The Extended [lambda]-Lemma..............................................................................29812.4 Distortion of Dimension in Holomorphic Motions...........................................................30612.5 Embedding Quasiconformal Mappings in Holomorphic Flows...................................................30912.6 Distortion Theorems......................................................................................31012.7 Deformations of Quasiconformal Mappings..................................................................31313 Higher Integrability........................................................................................31613.1 Distortion of Area.......................................................................................31713.2 Higher Integrability.....................................................................................32713.3 The Dimension of Quasicircles............................................................................33313.4 Quasiconformal Mappings and BMO..........................................................................34313.5 Painlev's Theorem: Removable Singularities..............................................................34713.6 Examples of Nonremovable Sets............................................................................35714 [L.sup.p]-Theory of Beltrami Operators......................................................................36214.1 Spectral Bounds and Linear Beltrami Operators............................................................36514.2 Invertibility of the Beltrami Operators..................................................................36614.3 Determining the Critical Interval........................................................................36914.4 Injectivity in the Borderline Cases......................................................................37314.5 Beltrami Operators; Coefficients in V MO.................................................................38214.6 Bounds for the Beurling Transform........................................................................38515 Schauder Estimates for Beltrami Operators...................................................................38915.1 Examples.................................................................................................39015.2 The Beltrami Equation with Constant Coefficients.........................................................39115.3 A Partition of Unity.....................................................................................39215.4 An Interpolation.........................................................................................39415.5 Hlder Regularity for Variable Coefficients..............................................................39515.6 Hlder-Caccioppoli Estimates.............................................................................39815.7 Quasilinear Equations....................................................................................40016 Applications to Partial Differential Equations..............................................................40316.1 The Hodge Method.........................................................................................40416.2 Topological Properties of Solutions......................................................................41816.3 The Hodographic Method...................................................................................42016.4 The Nonlinear A-Harmonic Equation........................................................................43316.5 Boundary Value Problems..................................................................................44916.6 G-Compactness of Beltrami Differential Operators.........................................................45617 PDEs Not of Divergence Type: Pucci's Conjecture.............................................................47217.1 Reduction to a First-Order System........................................................................47517.2 Second-Order Caccioppoli Estimates.......................................................................47617.3 The Maximum Principle and Pucci's Conjecture.............................................................47817.4 Interior Regularity......................................................................................48117.5 Equations with Lower-Order Terms.........................................................................48317.6 Pucci's Example..........................................................................................48818 Quasiconformal Methods in Impedance Tomography: Caldern's Problem..........................................49018.1 Complex Geometric Optics Solutions.......................................................................49318.2 The Hilbert Transform [H.sub.[sigma]]....................................................................49518.3 Dependence on Parameters.................................................................................49718.4 Nonlinear Fourier Transform..............................................................................49918.5 Argument Principle.......................................................................................50218.6 Subexponential Growth....................................................................................50418.7 The Solution to Caldern's Problem.......................................................................51019 Integral Estimates for the Jacobian.........................................................................51419.1 The Fundamental Inequality for the Jacobian..............................................................51419.2 Rank-One Convexity and Quasiconvexity....................................................................51819.3 [L.sup.1]-Integrability of the Jacobian..................................................................52320 Solving the Beltrami Equation: Degenerate Elliptic Case.....................................................52720.1 Mappings of Finite Distortion; Continuity................................................................52920.2 Integrable Distortion; [W.sup.1,2]-Solutions and Their Properties........................................53420.3 A Critical Example.......................................................................................54020.4 Distortion in the Exponential Class......................................................................54320.5 Optimal Orlicz Conditions for the Distortion Function....................................................57020.6 Global Solutions.........................................................................................57620.7 A Liouville Theorem......................................................................................57920.8 Applications to Degenerate PDEs..........................................................................58020.9 Lehto's Condition........................................................................................58121 Aspects of the Calculus of Variations.......................................................................58621.1 Minimizing Mean Distortion...............................................................................58621.2 Variational Equations....................................................................................59921.3 Mean Distortion, Annuli and the Nitsche Conjecture.......................................................606Appendix: Elements of Sobolev Theory and Function Spaces.......................................................624A.1 Schwartz Distributions....................................................................................624A.2 Definitions of Sobolev Spaces.............................................................................627A.3 Mollification.............................................................................................628A.4 Pointwise Coincidence of Sobolev Functions................................................................630A.5 Alternate Characterizations...............................................................................630A.6 Embedding Theorems........................................................................................633A.7 Duals and Compact Embeddings..............................................................................636A.8 Hardy Spaces and BMO......................................................................................637A.9 Reverse Hlder Inequalities...............................................................................640A.10 Variations of Sobolev Mappings............................................................................640Basic Notation.................................................................................................643Bibliography...................................................................................................647Index..........................................................................................................671

Chapter One

Introduction

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, conformal dynamical systems and so forth is the very same as that which underpins existence, regularity, singular set structure and so forth for precisely the most important class of equations one meets in physical (and other) applications, namely, second-order divergence-type equations. All these theories are inextricably linked in two dimensions by the theory of quasiconformal mappings.

Because of these and other compelling applications, there has recently been considerable pressure to extend classical results from conformal geometry to more general settings, for instance, to obtain optimal bounds on the existence, regularity and geometric properties of solutions of quasilinear and general non-linear systems in the plane both in the classical elliptic setting and now in the degenerate elliptic setting. Here one moves from the established theory of quasiconformal mappings, through the theory of weakly quasiregular mappings, and comes to the more general class of Sobolev mappings of finite distortion. This progression is natural as one seeks greater knowledge about the fine properties of these mappings for implementation. Even for such well-known problems as the nonlinear [??]-problem, we find that precise [L.sup.2]-bounds lead to a simple and beautiful proof of the extension theorem for holomorphic motions. In the same vein, we use optimal regularity to prove Pucci's conjecture, as well as related precise results to give a solution to Caldern's problem on impedance tomography and also to Painlev's problem on the size and structure of removable singular sets for solutions to elliptic and degenerate elliptic equations.

These precise results are in a large part due to a new understanding of the relationship between quasiconformal mappings and holomorphic flows on the one hand, and, on the other, precise results on the [L.sup.p]-invertibility of classes of singular integral operators called Beltrami operators. However, there have been other recent developments in the theory of quasiconformal mappings-notably in the field of analysis on metric spaces principally established by Heinonen and Koskela. These advances could not leave a book such as ours untouched, for they clarify many of the basic facts and the precise hypotheses necessary to prove them and often provide elementary and clear proofs. Thus the reader will find novelty and simplicity here even for the foundations of the theory, which now go back more than half a century.

Another novelty in the approach of this book is the use of many of the significant advances in harmonic analysis made over the last few decades; these include [H.sup.1]-BMO duality, maximal function estimates, the theory of nonlinear commutators and integral estimates for Jacobian determinants both above and below their natural Sobolev domain of definition, all crucial for our studies of optimal regularity and nonlinear PDEs, as well as the Painlev problem on removable singularities. The reader will have ample opportunity to see these powerful modern techniques in diverse applications.

1.1 Calculus of Variations, PDEs and Quasiconformal Mappings

The strong interplay among the calculus of variations, partial differential equations and the geometric theory of mappings (which is what this book is all about) has a long and distinguished history-going back at least to d'Alembert who in 1746 first related the derivatives of the real and imaginary part of a complex function in his work on hydrodynamics [51, p. 497]. These equations came to be known as the Cauchy-Riemann equations.

Conservation laws and equations of motion or state in physics and mathematics are described by divergence-type second-order differential equations. This is no accident. It is a fundamental precept of physics that a system acts so as to minimize some action functional-Hamilton's principle of least action. Hamilton's principle applies quite generally to classical fields such as the electromagnetic, gravitational and even quantum fields. We are therefore naturally led to study the minima of energy functionals, regularity of minimizers and other aspects of the calculus of variations. We give a classical problem a review in the next section. Loosely, minima satisfy an associated Euler-Lagrange equation that appears in divergence form as a result of integration by parts in the derivation of the equation. Similar examples appear in continuum mechanics and materials science.

On the other hand, general conservation laws are described as follows. Suppose the flux density of a scalar quantity e, such as density, concentration, temperature or energy, is q = A(z, [nabla] e), a function of the gradient of e. A basic assumption of continuum physics is that the gain of the physical quantity in a domain [OMEGA] corresponds to the loss of this quantity across the boundary [partial derivative][OMEGA]. Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here f denotes the source density and v denotes the outer normal. The above identity is called the conservation law with respect to the flux q and leads (we describe how in Section 16.3) to the differential equation

[??] + div(q) = [??] + divA(z, [nabla]e) = f

This is a conservation law for the physical quantity e. In the steady-state case we obtain a second-order equation in divergence form for q.

With so many compelling applications in hand, it is no wonder that there is considerable interest in the topological and analytic properties of the minimizers of various functionals and also in solutions of second-order equations in divergence form. These topological and analytic properties describe, for instance, the flow lines of the field and the structure and size of any singular set.

Let us explain using an elementary example from the calculus of variations how related first- and second-order equations might arise. Consider deforming the unit disk D to another domain [OMEGA] minimizing energy. This was in fact Riemann's approach to his mapping theorem, and which he called the Dirichlet principle. He obtained the desired conformal mapping as an absolute minimizer of the Dirichlet energy. Weierstrass showed Riemann's argument was not generally valid, however Hilbert later ironed out the details-ultimately requiring some regularity of [partial derivative][OMEGA]. As this discussion suggests, the example is quite classical, but it's solution contains many key ideas and provides us with some important lessons.

Problem. Given a simply connected domain [OMEGA],

(a) find the homeomorphism of minimal energy mapping the disk to [OMEGA],

(b) find the minimizer subject to prescribed boundary values.

The energy of a mapping is defined as the Dirichlet integral, so we are asked to find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

over all homeomorphisms, with the possible restriction f|[partial derivative]D = [g.sub.o]. In order to solve this problem (if it is possible at all), we should consider the correct function space to start looking for a solution. For the minimum to be finite, we certainly need for there to be some mapping [f.sub.0] satisfying the hypotheses (the gradient of [f.sub.0] should be square-integrable with correct boundary values). Given this mapping, we can then assume the existence of a sequence tending to the minimum (a minimizing sequence). Then comes the difficult problem of proving this sequence has a convergent subsequence whose limit is sufficiently regular to satisfy the hypotheses (thus the need for a priori estimates). For the problem in hand, Hadamard's inequality for matrices A [member of] [R.sup.2x2](C) states [[parallel]A[parallel].sup.2] = tr ([A.sup.t]A) [greater than or equal to] 2 det A and therefore gives the pointwise almost everywhere estimate

[[parallel]Df(z)]parallel].sup.2] [greater than or equal to] 2 J(z, f) = 2 det Df(z)

(we consider only orientation-preserving homeomorphisms, meaning that the Jacobian J(z, f) [greater than or equal to] 0 almost everywhere in D.) Then for every homeomorphism of Sobolev class [W.sup.1,2](D), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

providing a lower bound on the minimum. Consequently, if there is to be an absolute minimizer f achieving this lower bound we must have it solving the first(!)-order equation for an absolute minimizer

[[parallel]Df(z)]parallel].sup.2] = 2 J(z, f)

Some linear algebra (we have equality in Hadamard's estimate) shows this to be equivalent to

[D.sup.t]f(z) Df(z) = J(z, f) I,

where I is the identity matrix. This is the equation for a conformal mapping, of course-in complex notation this system is the Cauchy-Riemann equations (which points to the virtue of complex notation). Back to our problem, if we prescribe the boundary values and they happen not to be those of a conformal mapping, then a minimizer cannot achieve our a priori lower bound. Another approach is to vary a supposed minimizer f by a parameterized family of homeomorphisms of D that are the identity near the boundary, say [[psi].sub.t] normalized so [psi]0(z) = z. Since f is a minimizer we must have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

leading to the second-order Euler-Lagrange equation for f, divDf = [DELTA]f = 0. Thus the minimum should be a harmonic mapping with the given boundary values, and the question boils down to whether our prescribed boundary values [g.sub.0] have a harmonic homeomorphic extension to D. The Poisson formula gives a harmonic function, and we are left to discuss the topological properties of this solution. A way forward here is to show that the Jacobian is continuous and does not vanish (so local injectivity) and use the monodromy theorem, but the geometry of the domain and the boundary values must come into play. For instance, without some convexity assumption on [OMEGA] the mean value of [g.sub.0] may lie outside [OMEGA]. It is a classical theorem of Choquet, Kneser and Rado that as soon as [OMEGA] is convex, one can solve the posed problem with homeomorphic boundary data and the solution is a smooth diffeomorphism.

We may consider the above problem in more general circumstances. For instance, if H : [OMEGA] [right arrow] [R.sup.2x2], symmetric and positive definite, is some measurable function describing some material property of [OMEGA], we could seek to minimize the new energy functional

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We use Hadamard's inequality in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and, as before, an absolute minimizer must satisfy the nonlinear PDE

[D.sup.t] fH(f)Df = [square root of det H(f)] J(z, f) I

It is only in two dimensions that such an equation is not overdetermined (this accounts for higher-dimensional rigidity), and we have the possibility of finding a solution in quite reasonable generality. For conformal geometry we are interested in the case det H [equivalent to] 1 yielding the nonlinear Beltrami equation

[D.sup.t] fH(f)Df = J(z, f) I

If in the above we consider a tensor field G : D [right arrow] [R.sup.2x2], det G [equivalent to] 1, we have

[D.sup.t]f D f = J(z, f) G,

equivalent to a linear (over C) equation called the complex Beltrami equation,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which we will spend quite a bit of time discussing. Finally, if we consider a constrained problem and look for the Euler-Lagrange equation we are quickly led to second-order equations in divergence form (arising from the necessary integration by parts) for the real and imaginary parts of f = u + iv,

div [G.sup.-1] [nabla]u = 0, div [G.sup.-1] [nabla]v = 0,

and ultimately to more general second-order equations in divergence form.

There are a few important points we would like to draw from this discussion regarding minima of variational problems:

Unconstrained or absolute minimizers of variational functionals are likely to satisfy first-order differential equations.

Constrained or stationary mappings will likely satisfy a second-order differential equation.

We may well find stationary solutions that are not minimizers. Indeed, there might not be a minimizer within the class of homeomorphisms.

Of course, in the most general setting of multiple connected domains, one would consider minimizers in a given homotopy class of maps between domains, or more generally, homotopy classes of maps between Riemann surfaces. Moreover we would seek to minimize more general functionals. Here we find clear connections with Teichmller theory, surface topology and so forth.

A significant portion of this book is given over to the study of the equations like those we have discovered above where we will seek existence, uniqueness and optimal regularity and so forth for their solutions-and also for the counterparts to these equations in other settings. Later we shall discuss recent developments in the study of existence and uniqueness properties for mappings between planar domains whose boundary values are prescribed and have the smallest mean distortion-this will bring the relevance of the first example discussed above back into focus because of a surprising connection with harmonic mappings and other surprises as well. Indeed, the analogy here with Teichmller theory is quite strong. This theory is partly concerned with extremal quasiconformal mappings in a homotopy class. These mappings minimize the [L.sup.[infinity]]-norm of the distortion. We investigate what happens when the [L.sup.1]-norm of the distortion is minimized instead. Further, in these studies we will find many new and unexpected phenomena concerning existence, uniqueness and regularity for these extremal problems where the functionals are polyconvex but typically not convex. These seem to differ markedly from phenomena observed when studying multi-well functionals in the calculus of variations. The phenomena observed concerning mappings between annuli present a case in point.

In two dimensions, the methods of complex analysis, conformal geometry and quasiconformal mappings provide powerful techniques, not available in other dimensions, to solve highly nonlinear partial differential equations, especially those in divergence form. Of course the relevance of divergence-type equations to quasiconformal mappings is not new. It has been evident to researchers for at least 70 years, beginning with M.A. Lavrentiev, C.B. Morrey, R. Caccioppoli, L. Bers and L. Nirenberg, B. Bojarski, Finn and Serrin, among many others. In the literature one finds concrete applications in materials science, particularly, nonlinear elasticity, gas flow and fluid flow, and in the calculus of variations going back generations.

One of the primary aims of this book is to give a thorough account of this classical theory from a modern perspective and connect it with the most recent developments.

(Continues...)

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Dettagli dell'edizione

• Editore: Princeton Univ Pr
• Data di pubblicazione: 2009
• Lingua: Inglese
• ISBN 10: 0691137773
• ISBN 13: 9780691137773
• Rilegatura: Copertina rigida
• Numero di pagine: 677
• Contatto del produttore: non disponibile
• Persona responsabile: non disponibile