Applied Mathematical SciencesVolume 117

EditorsS.S. AntmanDepartment of MathematicsandInstitute for Physical

Science and TechnologyUniversity of MarylandCollege Park, MD 20742-4015USAssa@math.umd.edu

L. SirovichLaboratory of Applied MathematicsDepartment of Biomathematical

SciencesMount Sinai School of MedicineNew York, NY 10029-6574lsirovich@rockefeller.edu

J.E. MarsdenControl and Dynamical Systems,

107-81California Institute of TechnologyPasadena, CA 91125USAmarsden@cds.caltech.edu

AdvisorsL. Greengard P. Holmes J. KeenerJ. Keller R. Laubenbacher B.J. MatkowskyA. Mielke C.S. Peskin K.R. Sreenivasan A. Stevens A. Stuart

For further volumes:http://www.springer.com/series/34

Michael E. Taylor

Partial DifferentialEquations III

Nonlinear Equations

Second Edition

ABC

Michael E. TaylorDepartment of MathematicsUniversity of North CarolinaChapel Hill, NC 27599USAmet@math.unc.edu

ISSN 0066-5452ISBN 978-1-4419-7048-0 e-ISBN 978-1-4419-7049-7DOI 10.1007/978-1-4419-7049-7Springer New York Dordrecht Heidelberg London

Library of Congress Control Number: 2010937758

Mathematics Subject Classification (2010): 35A01, 35A02, 35J05, 35J25, 35K05, 35L05, 35Q30,35Q35, 35S05

c� Springer Science+Business Media, LLC 1996, 2011All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Usein connection with any form of information storage and retrieval, electronic adaptation, computersoftware, or by similar or dissimilar methodology now known or hereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks, and similar terms, even if theyare not identified as such, is not to be taken as an expression of opinion as to whether or not they aresubject to proprietary rights.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

To my wife and daughter, Jane Hawkinsand Diane Taylor

Contents

Contents of Volumes I and II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

13 Function Space and Operator Theory for Nonlinear Analysis . . . . . . . 11 Lp-Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Sobolev imbedding theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Gagliardo–Nirenberg–Moser estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Trudinger’s inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Singular integral operators on Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 The spaces H s;p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Lp-spectral theory of the Laplace operator . . . . . . . . . . . . . . . . . . . . . . . . 318 Holder spaces and Zygmund spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 Pseudodifferential operators with nonregular symbols . . . . . . . . . . . . 50

10 Paradifferential operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6011 Young measures and fuzzy functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7412 Hardy spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86A Variations on complex interpolation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

14 Nonlinear Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1051 A class of semilinear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072 Surfaces with negative curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1193 Local solvability of nonlinear elliptic equations . . . . . . . . . . . . . . . . . . . 1274 Elliptic regularity I (interior estimates) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1355 Isometric imbedding of Riemannian manifolds . . . . . . . . . . . . . . . . . . . . 1476 Minimal surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

6B Second variation of area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1687 The minimal surface equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1768 Elliptic regularity II (boundary estimates) . . . . . . . . . . . . . . . . . . . . . . . . . 1859 Elliptic regularity III (DeGiorgi–Nash–Moser theory) . . . . . . . . . . . . 196

10 The Dirichlet problem for quasi-linear elliptic equations . . . . . . . . . 20811 Direct methods in the calculus of variations . . . . . . . . . . . . . . . . . . . . . . . 22212 Quasi-linear elliptic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

12B Further results on quasi-linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24413 Elliptic regularity IV (Krylov–Safonov estimates) . . . . . . . . . . . . . . . . 258

viii Contents

14 Regularity for a class of completely nonlinear equations.. . . . . . . . . 27315 Monge–Ampere equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28216 Elliptic equations in two variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294A Morrey spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299B Leray–Schauder fixed-point theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

15 Nonlinear Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3131 Semilinear parabolic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3142 Applications to harmonic maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3253 Semilinear equations on regions with boundary . . . . . . . . . . . . . . . . . . . 3324 Reaction-diffusion equations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3355 A nonlinear Trotter product formula.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3536 The Stefan problem.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3627 Quasi-linear parabolic equations I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3768 Quasi-linear parabolic equations II (sharper estimates) . . . . . . . . . . . 3879 Quasi-linear parabolic equations III (Nash–Moser estimates) . . . . 396

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

16 Nonlinear Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4131 Quasi-linear, symmetric hyperbolic systems . . . . . . . . . . . . . . . . . . . . . . . 4142 Symmetrizable hyperbolic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4253 Second-order and higher-order hyperbolic systems. . . . . . . . . . . . . . . . 4324 Equations in the complex domain and the Cauchy–

Kowalewsky theorem.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4455 Compressible fluid motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4486 Weak solutions to scalar conservation laws; the viscosity method 4577 Systems of conservation laws in one space variable;

Riemann problems.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4728 Entropy-flux pairs and Riemann invariants. . . . . . . . . . . . . . . . . . . . . . . . . 4989 Global weak solutions of some 2 � 2 systems . . . . . . . . . . . . . . . . . . . . . 509

10 Vibrating strings revisited .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

17 Euler and Navier–Stokes Equations for Incompressible Fluids . . . . . . 5311 Euler’s equations for ideal incompressible fluid flow. . . . . . . . . . . . . . 5322 Existence of solutions to the Euler equations . . . . . . . . . . . . . . . . . . . . . . 5423 Euler flows on bounded regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5534 Navier–Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5615 Viscous flows on bounded regions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5756 Vanishing viscosity limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5867 From velocity field convergence to flow convergence . . . . . . . . . . . . . 599A Regularity for the Stokes system on bounded domains .. . . . . . . . . . . 605

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610

Contents ix

18 Einstein’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6151 The gravitational field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6162 Spherically symmetric spacetimes and the

Schwarzschild solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6263 Stationary and static spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6394 Orbits in Schwarzschild spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6495 Coupled Maxwell–Einstein equations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6566 Relativistic fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6597 Gravitational collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6708 The initial-value problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6779 Geometry of initial surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687

10 Time slices and their evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711

Contents of Volumes I and II

Volume I: Basic Theory

1 Basic Theory of ODE and Vector Fields

2 The Laplace Equation and Wave Equation

3 Fourier Analysis, Distributions, and Constant-Coefficient Linear PDE

4 Sobolev Spaces

5 Linear Elliptic Equations

6 Linear Evolution Equations

A Outline of Functional Analysis

B Manifolds, Vector Bundles, and Lie Groups

Volume II: Qualitative Studies of Linear Equations

7 Pseudodifferential Operators

8 Spectral Theory

9 Scattering by Obstacles

10 Dirac Operators and Index Theory

11 Brownian Motion and Potential Theory

12 The N@-Neumann Problem

C Connections and Curvature

Preface

Partial differential equations is a many-faceted subject. Created to describe themechanical behavior of objects such as vibrating strings and blowing winds, ithas developed into a body of material that interacts with many branches of math-ematics, such as differential geometry, complex analysis, and harmonic analysis,as well as a ubiquitous factor in the description and elucidation of problems inmathematical physics.

This work is intended to provide a course of study of some of the major aspectsof PDE. It is addressed to readers with a background in the basic introductorygraduate mathematics courses in American universities: elementary real and com-plex analysis, differential geometry, and measure theory.

Chapter 1 provides background material on the theory of ordinary differentialequations (ODE). This includes both very basic material–on topics such as theexistence and uniqueness of solutions to ODE and explicit solutions to equationswith constant coefficients and relations to linear algebra–and more sophisticatedresults–on flows generated by vector fields, connections with differential geom-etry, the calculus of differential forms, stationary action principles in mechanics,and their relation to Hamiltonian systems. We discuss equations of relativisticmotion as well as equations of classical Newtonian mechanics. There are alsoapplications to topological results, such as degree theory, the Brouwer fixed-pointtheorem, and the Jordan-Brouwer separation theorem. In this chapter we also treatscalar first-order PDE, via Hamilton–Jacobi theory.

Chapters 2–6 constitute a survey of basic linear PDE. Chapter 2 begins with thederivation of some equations of continuum mechanics in a fashion similar to thederivation of ODE in mechanics in Chap. 1, via variational principles. We obtainequations for vibrating strings and membranes; these equations are not necessarilylinear, and hence they will also provide sources of problems later, when nonlinearPDE is taken up. Further material in Chap. 2 centers around the Laplace operator,which on Euclidean space Rn is

(1) � D @2

@x21

C � � � C @2

@x2n

;

and the linear wave equation,

(2)@2u

@t2��u D 0:

xiv Preface

We also consider the Laplace operator on a general Riemannian manifold andthe wave equation on a general Lorentz manifold. We discuss basic consequencesof Green’s formula, including energy conservation and finite propagation speedfor solutions to linear wave equations. We also discuss Maxwell’s equations forelectromagnetic fields and their relation with special relativity. Before we canestablish general results on the solvability of these equations, it is necessary todevelop some analytical techniques. This is done in the next couple of chapters.

Chapter 3 is devoted to Fourier analysis and the theory of distributions. Thesetopics are crucial for the study of linear PDE. We give a number of basic ap-plications to the study of linear PDE with constant coefficients. Among theseapplications are results on harmonic and holomorphic functions in the plane,including a short treatment of elementary complex function theory. We derive ex-plicit formulas for solutions to Laplace and wave equations on Euclidean space,and also the heat equation,

(3)@u

@t��u D 0:

We also produce solutions on certain subsets, such as rectangular regions, usingthe method of images. We include material on the discrete Fourier transform, ger-mane to the discrete approximation of PDE, and on the fast evaluation of thistransform, the FFT. Chapter 3 is the first chapter to make extensive use of func-tional analysis. Basic results on this topic are compiled in Appendix A, Outline ofFunctional Analysis.

Sobolev spaces have proven to be a very effective tool in the existence the-ory of PDE, and in the study of regularity of solutions. In Chap. 4 we introduceSobolev spaces and study some of their basic properties. We restrict attentionto L2-Sobolev spaces, such as H k.Rn/; which consists of L2 functions whosederivatives of order � k (defined in a distributional sense, in Chap. 3) belong toL2.Rn/; when k is a positive integer. We also replace k by a general real numbers: The Lp-Sobolev spaces, which are very useful for nonlinear PDE, are treatedlater, in Chap. 13.

Chapter 5 is devoted to the study of the existence and regularity of solutions tolinear elliptic PDE, on bounded regions. We begin with the Dirichlet problem forthe Laplace operator,

(4) �u D f on ˝; u D g on @˝;

and then treat the Neumann problem and various other boundary problems, in-cluding some that apply to electromagnetic fields. We also study general boundaryproblems for linear elliptic operators, giving a condition that guarantees regu-larity and solvability (perhaps given a finite number of linear conditions on thedata). Also in Chap. 5 are some applications to other areas, such as a proof ofthe Riemann mapping theorem, first for smooth simply connected domains in thecomplex plane C; then, after a treatment of the Dirichlet problem for the Laplace

Preface xv

operator on domains with rough boundary, for general simply connected domainsin C:We also develop Hodge theory and apply it to DeRham cohomology, extend-ing the study of topological applications of differential forms begun in Chap. 1.

In Chap. 6 we study linear evolution equations, in which there is a “time”variable t , and initial data are given at t D 0: We discuss the heat and waveequations. We also treat Maxwell’s equations, for an electromagnetic field, andmore general hyperbolic systems. We prove the Cauchy–Kowalewsky theorem, inthe linear case, establishing local solvability of the Cauchy initial value problemfor general linear PDE with analytic coefficients, and analytic data, as long as theinitial surface is “noncharacteristic.” The nonlinear case is treated in Chap. 16.Also in Chap. 6 we treat geometrical optics, providing approximations to solu-tions of wave equations whose initial data either are highly oscillatory or possesssimple singularities, such as a jump across a smooth hypersurface.

Chapters 1–6, together with Appendix A and Appendix B, Manifolds, VectorBundles, and Lie Groups, make up the first volume of this work. The secondvolume consists of Chaps. 7–12, covering a selection of more advanced topics inlinear PDE, together with Appendix C, Connections and Curvature.

Chapter 7 deals with pseudodifferential operators ( DOs). This class of opera-tors includes both differential operators and parametrices of elliptic operators, thatis, inverses modulo smoothing operators. There is a “symbol calculus” allowingone to analyze products of DOs, useful for such a parametrix construction. TheL2-boundedness of operators of order zero and the Garding inequality for elliptic DOs with positive symbol provide very useful tools in linear PDE, which willbe used in many subsequent chapters.

Chapter 8 is devoted to spectral theory, particularly for self-adjoint ellipticoperators. First we give a proof of the spectral theorem for general self-adjointoperators on Hilbert space. Then we discuss conditions under which a differentialoperator yields a self-adjoint operator. We then discuss the asymptotic distribu-tion of eigenvalues of the Laplace operator on a bounded domain, making use ofa construction of a parametrix for the heat equation from Chap. 7. In the next foursections of Chap. 8 we consider the spectral behavior of various specific differ-ential operators: the Laplace operator on a sphere, and on hyperbolic space, the“harmonic oscillator”

(5) ��C jxj2;

and the operator

(6) �� � K

jxj ;

which arises in the simplest quantum mechanical model of the hydrogen atom.Finally, we consider the Laplace operator on cones.

In Chap. 9 we study the scattering of waves by a compact obstacle K in R3:

This scattering theory is to some degree an extension of the spectral theory of the

xvi Preface

Laplace operator on R3 nK; with the Dirichlet boundary condition. In addition tostudying how a given obstacle scatters waves, we consider the inverse problem:how to determine an obstacle given data on how it scatters waves.

Chapter 10 is devoted to the Atiyah–Singer index theorem. This gives a for-mula for the index of an elliptic operatorD on a compact manifoldM; defined by

(7) IndexD D dim kerD � dim kerD�:

We establish this formula, which is an integral over M of a certain differentialform defined by a pair of “curvatures,” when D is a first order differential oper-ator of “Dirac type,” a class that contains many important operators arising fromdifferential geometry and complex analysis. Special cases of such a formula in-clude the Chern–Gauss–Bonnet formula and the Riemann–Roch formula. We alsodiscuss the significance of the latter formula in the study of Riemann surfaces.

In Chap. 11 we study Brownian motion, described mathematically by Wienermeasure on the space of continuous paths in Rn: This provides a probabilisticapproach to diffusion and it both uses and provides new tools for the analysis ofthe heat equation and variants, such as

(8)@u

@tD ��u C V u;

where V is a real-valued function. There is an integral formula for solutions to (8),known as the Feynman–Kac formula; it is an integral over path space with respectto Wiener measure, of a fairly explicit integrand. We also derive an analogousintegral formula for solutions to

(9)@u

@tD ��u CXu;

where X is a vector field. In this case, another tool is involved in constructing theintegrand, the stochastic integral. We also study stochastic differential equationsand applications to more general diffusion equations.

In Chap. 12 we tackle the N@-Neumann problem, a boundary problem for an el-liptic operator (essentially the Laplace operator) on a domain ˝ � Cn, whichis very important in the theory of functions of several complex variables. From atechnical point of view, it is of particular interest that this boundary problem doesnot satisfy the regularity criteria investigated in Chap. 5. If ˝ is “strongly pseu-doconvex,” one has instead certain “subelliptic estimates,” which are establishedin Chap. 12.

The third and final volume of this work contains Chaps. 13–18. It is here thatwe study nonlinear PDE.

We prepare the way in Chap. 13 with a further development of function spaceand operator theory, for use in nonlinear analysis. This includes the theory ofLp-Sobolev spaces and Holder spaces. We derive estimates in these spaces on

Preface xvii

nonlinear functions F.u/, known as “Moser estimates,” which are very useful.We extend the theory of pseudodifferential operators to cases where the symbolshave limited smoothness, and also develop a variant of DO theory, the theoryof “paradifferential operators,” which has had a significant impact on nonlinearPDE since about 1980. We also estimate these operators, acting on the functionspaces mentioned above. Other topics treated in Chap. 13 include Hardy spaces,compensated compactness, and “fuzzy functions.”

Chapter 14 is devoted to nonlinear elliptic PDE, with an emphasis on secondorder equations. There are three successive degrees of nonlinearity: semilinearequations, such as

(10) �u D F.x; u;ru/;

quasi-linear equations, such as

(11)X

ajk.x; u;ru/@j @ku D F.x; u;ru/;

and completely nonlinear equations, of the form

(12) G.x;D2u/ D 0:

Differential geometry provides a rich source of such PDE, and Chap. 14 contains anumber of geometrical applications. For example, to deform conformally a metricon a surface so its Gauss curvature changes from k.x/ toK.x/; one needs to solvethe semilinear equation

(13) �u D k.x/ �K.x/e2u:

As another example, the graph of a function y D u.x/ is a minimal submanifoldof Euclidean space provided u solves the quasilinear equation

(14)�1C jruj2��u C .ru/ �H.u/.ru/ D 0;

called the minimal surface equation. Here,H.u/ D .@j @ku/ is the Hessian matrixof u: On the other hand, this graph has Gauss curvature K.x/ provided u solvesthe completely nonlinear equation

(15) detH.u/ D K.x/�1C jruj2�.nC2/=2

;

a Monge-Ampere equation. Equations (13)–(15) are all scalar, and the maximumprinciple plays a useful role in the analysis, together with a number of other tools.Chapter 14 also treats nonlinear systems. Important physical examples arise instudies of elastic bodies, as well as in other areas, such as the theory of liquid crys-tals. Geometric examples of systems considered in Chap. 14 include equations forharmonic maps and equations for isometric imbeddings of a Riemannian manifoldin Euclidean space.

xviii Preface

In Chap. 15, we treat nonlinear parabolic equations. Partly echoing Chap. 14,we progress from a treatment of semilinear equations,

(16)@u

@tD Lu C F.x; u;ru/;

where L is a linear operator, such as L D �; to a treatment of quasi-linear equa-tions, such as

(17)@u

@tDX

@jajk.t; x; u/@ku CX.u/:

(We do very little with completely nonlinear equations in this chapter.) We studysystems as well as scalar equations. The first application of (16) we consider isto the parabolic equation method of constructing harmonic maps. We also con-sider “reaction-diffusion” equations, ` � ` systems of the form (16), in whichF.x; u;ru/ D X.u/; where X is a vector field on R`, and L is a diagonal opera-tor, with diagonal elements aj�; aj � 0: These equations arise in mathematicalmodels in biology and in chemistry. For example, u D .u1; : : : ; u`/ might repre-sent the population densities of each of ` species of living creatures, distributedover an area of land, interacting in a manner described by X and diffusing in amanner described by aj�: If there is a nonlinear (density-dependent) diffusion,one might have a system of the form (17).

Another problem considered in Chap. 15 models the melting of ice; one hasa linear heat equation in a region (filled with water) whose boundary (where thewater touches the ice) is moving (as the ice melts). The nonlinearity in the probleminvolves the description of the boundary. We confine our analysis to a relativelysimple one-dimensional case.

Nonlinear hyperbolic equations are studied in Chap. 16. Here continuum me-chanics is the major source of examples, and most of them are systems, ratherthan scalar equations. We establish local existence for solutions to first order hy-perbolic systems, which are either “symmetric” or “symmetrizable.” An exampleof the latter class is the following system describing compressible fluid flow:

(18)@v

@tC rvv C 1

�gradp D 0;

@�

@tC rv�C � div v D 0;

for a fluid with velocity v; density �; and pressure p; assumed to satisfy a relationp D p.�/; called an “equation of state.” Solutions to such nonlinear systems tendto break down, due to shock formation. We devote a bit of attention to the studyof weak solutions to nonlinear hyperbolic systems, with shocks.

We also study second-order hyperbolic systems, such as systems for a k-dimensional membrane vibrating in Rn; derived in Chap. 2. Another topic coveredin Chap. 16 is the Cauchy–Kowalewsky theorem, in the nonlinear case. We use amethod introduced by P. Garabedian to transform the Cauchy problem for an an-alytic equation into a symmetric hyperbolic system.

Preface xix

In Chap. 17 we study incompressible fluid flow. This is governed by the Eulerequation

(19)@v

@tC rvv D � gradp; div v D 0;

in the absence of viscosity, and by the Navier–Stokes equation

(20)@v

@tC rvv D �Lv � gradp; div v D 0;

in the presence of viscosity. Here L is a second-order operator, the Laplace oper-ator for a flow on flat space; the “viscosity” � is a positive quantity. The equation(19) shares some features with quasilinear hyperbolic systems, though there arealso significant differences. Similarly, (20) has a lot in common with semilinearparabolic systems.

Chapter 18, the last chapter in this work, is devoted to Einstein’s gravitationalequations:

(21) Gjk D 8��Tjk :

HereGjk is the Einstein tensor, given byGjk D Ricjk �.1=2/Sgjk;where Ricjk

is the Ricci tensor and S the scalar curvature, of a Lorentz manifold (or “space-time”) with metric tensor gjk : On the right side of (21), Tjk is the stress-energytensor of the matter in the spacetime, and � is a positive constant, which can beidentified with the gravitational constant of the Newtonian theory of gravity. Inlocal coordinates, Gjk has a nonlinear expression in terms of gjk and its secondorder derivatives. In the empty-space case, where Tjk D 0; (21) is a quasilin-ear second order system for gjk : The freedom to change coordinates provides anobstruction to this equation being hyperbolic, but one can impose the use of “har-monic” coordinates as a constraint and transform (21) into a hyperbolic system.In the presence of matter one couples (21) to other systems, obtaining more elab-orate PDE. We treat this in two cases, in the presence of an electromagnetic field,and in the presence of a relativistic fluid.

In addition to the 18 chapters just described, there are three appendices, al-ready mentioned above. Appendix A gives definitions and basic properties ofBanach and Hilbert spaces (of which Lp-spaces and Sobolev spaces are exam-ples), Frechet spaces (such as C1.Rn/), and other locally convex spaces (such asspaces of distributions). It discusses some basic facts about bounded linear oper-ators, including some special properties of compact operators, and also considerscertain classes of unbounded linear operators. This functional analytic materialplays a major role in the development of PDE from Chap. 3 onward.

Appendix B gives definitions and basic properties of manifolds and vectorbundles. It also discusses some elementary properties of Lie groups, includinga little representation theory, useful in Chap. 8, on spectral theory, as well as inthe Chern–Weil construction.

xx Preface

Appendix C, Connections and Curvature, contains material of a differentialgeometric nature, crucial for understanding many things done in Chaps. 10–18.We consider connections on general vector bundles, and their curvature. We dis-cuss in detail special properties of the primary case: the Levi–Civita connectionand Riemann curvature tensor on a Riemannian manifold. We discuss basic prop-erties of the geometry of submanifolds, relating the second fundamental form tocurvature via the Gauss–Codazzi equations. We describe how vector bundles arisefrom principal bundles, which themselves carry various connections and curvatureforms. We then discuss the Chern–Weil construction, yielding certain closed dif-ferential forms associated to curvatures of connections on principal bundles. Wegive several proofs of the classical Gauss–Bonnet theorem and some related re-sults on two-dimensional surfaces, which are useful particularly in Chaps. 10 and14. We also give a geometrical proof of the Chern–Gauss–Bonnet theorem, whichcan be contrasted with the proof in Chap. 10, as a consequence of the Atiyah–Singer index theorem.

We mention that, in addition to these “global” appendices, there are appendicesto some chapters. For example, Chap. 3 has an appendix on the gamma function.Chapter 6 has two appendices; Appendix A has some results on Banach spaces ofharmonic functions useful for the proof of the linear Cauchy–Kowalewsky theo-rem, and Appendix B deals with the stationary phase formula, useful for the studyof geometrical optics in Chap. 6 and also for results later, in Chap. 9. There areother chapters with such “local” appendices. Furthermore, there are two sections,both in Chap. 14, with appendices. Section 6, on minimal surfaces, has a com-panion, Sect. 6B, on the second variation of area and consequences, and Sect. 12,on nonlinear elliptic systems, has a companion, Sect. 12B, with complementarymaterial.

Having described the scope of this work, we find it necessary to mention anumber of topics in PDE that are not covered here, or are touched on only verybriefly.

For example, we devote little attention to the real analytic theory of PDE. Wenote that harmonic functions on domains in Rn are real analytic, but we do notdiscuss analyticity of solutions to more general elliptic equations. We do provethe Cauchy–Kowalewsky theorem, on analytic PDE with analytic Cauchy data.We derive some simple results on unique continuation from these few analyticityresults, but there is a large body of lore on unique continuation, for solutions tononanalytic PDE, neglected here.

There is little material on numerical methods. There are a few references toapplications of the FFT and of “splitting methods.” Difference schemes for PDEare mentioned just once, in a set of exercises on scalar conservation laws. Finiteelement methods are neglected, as are many other numerical techiques.

There is a large body of work on free boundary problems, but the only oneconsidered here is a simple one space dimensional problem, in Chap. 15.

While we have considered a variety of equations arising from classicalphysics and from relativity, we have devoted relatively little attention to quan-tum mechanics. We have considered one quantum mechanical operator, given

Preface xxi

in formula (6) above. Also, there are some exercises on potential scatteringmentioned in Chap. 9. However, the physical theories behind these equations arenot discussed here.

There are a number of nonlinear evolution equations, such as the Korteweg–deVries equation, that have been perceived to provide infinite dimensional ana-logues of completely integrable Hamiltonian systems, and to arise “universally”in asymptotic analyses of solutions to various nonlinear wave equations. They arenot here. Nor is there a treatment of the Yang–Mills equations for gauge fields,with their wonderful applications to the geometry and topology of four dimen-sional manifolds.

Of course, this is not a complete list of omitted material. One can go on and onlisting important topics in this vast subject. The author can at best hope that thereader will find it easier to understand many of these topics with this book, thanwithout it.

Acknowledgments

I have had the good fortune to teach at least one course relevant to the material ofthis book, almost every year since 1976. These courses led to many course notes,and I am grateful to many colleagues at Rice University, SUNY at Stony Brook,the California Institute of Technology, and the University of North Carolina, forthe supportive atmospheres at these institutions. Also, a number of individualsprovided valuable advice on various portions of the manuscript, as it grew over theyears. I particularly want to thank Florin David, David Ebin, Frank Jones, AnnaMazzucato, Richard Melrose, James Ralston, Jeffrey Rauch, Santiago Simanca,and James York. The final touches were put on the manuscript while I was visitingthe Institute for Mathematics and its Applications, at the University of Minnesota,which I thank for its hospitality and excellent facilities.

Finally, I would like to acknowledge the impact on my studies of my seniorthesis and Ph.D. thesis advisors, Edward Nelson and Heinz Cordes.

Introduction to the Second Edition

In addition to making numerous small corrections to this work, collected overthe past dozen years, I have taken the opportunity to make some very significantchanges, some of which broaden the scope of the work, some of which clarifyprevious presentations, and a few of which correct errors that have come to myattention.

There are seven additional sections in this edition, two in Volume 1, two inVolume 2, and three in Volume 3. Chapter 4 has a new section, “Sobolev spaceson rough domains,” which serves to clarify the treatment of the Dirichlet prob-

xxii Preface

lem on rough domains in Chap. 5. Chapter 6 has a new section, “Boundary layerphenomena for the heat equation,” which will prove useful in one of the new sec-tions in Chap. 17. Chapter 7 has a new section, “Operators of harmonic oscillatortype,” and Chap. 10 has a section that presents an index formula for elliptic sys-tems of operators of harmonic oscillator type. Chapter 13 has a new appendix,“Variations on complex interpolation,” which has material that is useful in thestudy of Zygmund spaces. Finally, Chap. 17 has two new sections, “Vanishingviscosity limits” and “From velocity convergence to flow convergence.”

In addition, several other sections have been substantially rewritten, and nu-merous others polished to reflect insights gained through the use of these booksover time.

13

Function Space and Operator Theoryfor Nonlinear Analysis

Introduction

This chapter examines a number of analytical techniques, which will be appliedto diverse nonlinear problems in the remaining chapters. For example, we studySobolev spaces based on Lp , rather than just L2. Sections 1 and 2 discuss thedefinition of Sobolev spaces H k;p, for k 2 ZC, and inclusions of the formH k;p � Lq . Estimates based on such inclusions have refined forms, due toE. Gagliardo and L. Nirenberg. We discuss these in � 3, together with results ofJ. Moser on estimates on nonlinear functions of an element of a Sobolev space,and on commutators of differential operators and multiplication operators. In � 4we establish some integral estimates of N. Trudinger, on functions in Sobolevspaces for which L1-bounds just fail. In these sections we use such basic toolsas Holder’s inequality and integration by parts.

The Fourier transform is not as effective for analysis on Lp as on L2. Oneresult that does often serve when, in the L2-theory, one could appeal to thePlancherel theorem, is Mikhlin’s Fourier multiplier theorem, established in � 5.This enables interpolation theory to be applied to the study of the spaces H s;p,for noninteger s, in � 6. In � 7 we apply some of this material to the study ofLp-spectral theory of the Laplace operator, on compact manifolds, possibly withboundary.

In � 8 we study spaces C r of Holder continuous functions, and their relationwith Zygmund spaces C r� . We derive estimates in these spaces for solutions toelliptic boundary problems.

The next two sections extend results on pseudodifferential operators, intro-duced in Chap. 7. Section 9 considers symbols p.x; �/ with minimal regularityin x. We derive both Lp- and Holder estimates. Section 10 considers paradiffer-ential operators, a variant of pseudodifferential operator calculus particularly wellsuited to nonlinear analysis. Sections 9 and 10 are largely taken from [T2].

In � 11 we consider “fuzzy functions,” consisting of a pair .f; /, where f isa function on a space and is a measure on � R, with the property that’y'.x/ d.x; y/ D R

'.x/f .x/ dx. The measure is known as a Young mea-sure. It incorporates information on how f may have arisen as a weak limit of

M.E. Taylor, Partial Differential Equations III: Nonlinear Equations,Applied Mathematical Sciences 117, DOI 10.1007/978-1-4419-7049-7 1,c� Springer Science+Business Media, LLC 1996, 2011

1

2 13. Function Space and Operator Theory for Nonlinear Analysis

smooth (“sharply defined”) functions, and it is useful for analyses of nonlinearmaps that do not generally preserve weak convergence.

In � 12 there is a brief discussion of Hardy spaces, subspaces of L1.Rn/ withmany desirable properties, only a few of which are discussed here. Much more onthis topic can be found in [S3], but material covered here will be useful for someelliptic regularity results in � 12B of Chap. 14.

We end this chapter with Appendix A, discussing variants of the complex in-terpolation method introduced in Chap. 4 and used a lot in the early sections ofthis chapter. It turns out that slightly different complex interpolation functors arebetter suited to the scale of Zygmund spaces.

1. Lp-Sobolev spaces

Let p 2 Œ1;1/. In analogy with the definition of the Sobolev spaces in Chap. 4,we set, for k D 0; 1; 2; : : : ,

(1.1) H k;p.Rn/ D ˚u 2 Lp.Rn/ W D˛u 2 Lp.Rn/ for j˛j � k

�:

It is easy to see that S.Rn/ is dense in each space H k;p.Rn/, with its naturalnorm

(1.2) kukH k;p DX

j˛j�k

kD˛ukLp :

For p ¤ 2, we cannot characterize the spacesH k;p.Rn/ conveniently in terms ofthe Fourier transform. It is still possible to define spaces H s;p.Rn/ by interpola-tion; we will examine this in � 6. Here we will consider only the spacesH k;p.Rn/

with k a nonnegative integer.The chain rule allows us to say that if � W Rn ! Rn is a diffeomorphism that

is linear outside a compact set, then �� W H k;p.Rn/ ! H k;p.Rn/. Also multi-plication by an element ' 2 C1

0 .Rn/ mapsH k;p.Rn/ to itself. This allows us todefine H k;p.M/ for a compact manifold M via a partition of unity subordinateto a coordinate chart. Also, for compact M , if we define Diff k.M/ to be the setof differential operators of order � k onM , with smooth coefficients, then

(1.3) H k;p.M/ D fu 2 Lp.M/ W P u 2 Lp.M/ for all P 2 Diff k.M/g:

We can define H k;p.RnC/ as in (1.1), with Rn replaced by RnC. The exten-sion operator defined by (4.2)–(4.4) of Chap. 4 also works to produce extensionmaps E W H k;p.RnC/ ! H k;p.Rn/. Similarly, if M is a compact manifoldwith smooth boundary, with double N , we can define H k;p.M/ via coordinatecharts and the notion of H k;p.RnC/, or by (1.3), and we have extension operatorsE W H k;p.M/ ! H k;p.N /.

Exercises 3

We also note the obvious fact that

(1.4) D˛ W H k;p.Rn/ �! H k�j˛j;p.Rn/;

for j˛j � k, and

(1.5) P W H k;p.M/ �! H k�`;p.M/ if P 2 Diff `.M/;

provided ` � k.

Exercises1. A Friedrichs mollifier on Rn is a family of smoothing operators J"u.x/ D j" � u.x/

where

j".x/ D "�nj."�1x/;

Zj.x/dx D 1; j 2 S.Rn/:

Equivalently, J"u.x/ D '."D/u.x/; ' 2 S.Rn/; '.0/ D 1. Show that, for eachp 2 Œ1;1/; k 2 ZC,

J" W Hk;p.Rn/ �!\

`<1H `;p.Rn/;

for each " > 0, and

J"u ! u in Hk;p.Rn/

as " ! 0 if u 2 Hk;p.Rn/.2. Suppose A 2 C 1.Rn/, with kAkC 1 D supj˛j�1 kD˛AkL1 . Show that when J" is a

Friedrichs mollifier as above, then

kŒA; J"�vkH 1;p � CkAkC 1 kvkLp ;

with C independent of " 2 .0; 1�. (Hint: Write A.x/ � A.y/ D PBk.x; y/

.xk � yk/; jBk.x; y/j � K, and, with q`.x/ D @j=@x` ,

@`ŒA; J"�v.x/ DXZ

Bk.x; y/h"�nq`

�x � y

"

�� xk � yk

"

iv.y/ dy;

with absolute value bounded by

K "�nXZ ˇ

'k`

�"�1.x � y/�ˇ � jv.y/j dy;

where 'k`.x/ D xkq`.x/:)3. Using Exercise 2, show that

kŒA; J"�@j vkLp � CkAkC 1 kvkLp :

4 13. Function Space and Operator Theory for Nonlinear Analysis

2. Sobolev imbedding theorems

We will derive various inclusions of the type H k;p.M/ � H `;q.M/. We willconcentrate on the case M D Rn. The discussion of � 1 will give associatedresults when M is a compact manifold, possibly with (smooth) boundary.

One technical tool useful for our estimates is the following generalized Holderinequality:

Lemma 2.1. If pj 2 Œ1;1�;Pp�1

j D 1, then

(2.1)Z

M

ju1 � � � umj dx � ku1kLp1 .M/ � � � kumkLpm .M/:

The proof follows by induction from the casem D 2, which is the usual Holderinequality.

Our first Sobolev imbedding theorem is the following:

Proposition 2.2. For p 2 Œ1; n/,

(2.2) H 1;p.Rn/ � Lnp=.n�p/.Rn/:

In fact, there is an estimate

(2.3) kukLnp=.n�p/ � CkrukLp ;

for u 2 H 1;p.Rn/, with C D C.p; n/.

Proof. It suffices to establish (2.3) for u 2 C10 .Rn/. Clearly,

ju.x/j �Z 1

�1jDj uj dxj ;(2.4)

so

(2.5) ju.x/jn=.n�1/ �8<

:

nY

j D1

Z 1

�1jDj uj dxj

9=

;

1=.n�1/

:

We can integrate (2.5) successively over each variable xj ; j D 1; : : : ; n, andapply the generalized Holder inequality (2.1) with m D p1 D � � � D pm D n � 1

after each integration. We get

(2.6) kukLn=.n�1/ �8<

:

nY

j D1

Z

Rn

jDj uj dx9=

;

1=n

� CkrukL1 :

2. Sobolev imbedding theorems 5

This establishes (2.3) in the case p D 1. We can apply this to v D juj� ; > 1,obtaining

(2.7)��juj���

Ln=.n�1/ � C��juj��1jruj��

L1 � C��juj��1

��Lp0

��ru��

Lp :

For p < n, pick D .n�1/p=.n�p/. Then (2.7) gives (2.3) and the propositionis proved.

Given u 2 H k;p.Rn/, we can apply Proposition 2.2 to estimate theLnp=.n�p/-norm of Dk�1u in terms of kDkukLp , where we use the notation

(2.8) Dku D fD˛u W j˛j D kg; kDkukLp DX

j˛jDk

kD˛ukLp ;

and proceed inductively, obtaining the following corollary.

Proposition 2.3. For kp < n,

(2.9) H k;p.Rn/ � Lnp=.n�kp/.Rn/:

The same result holds with Rn replaced by a compact manifold of dimension n.If we take p D 2, then for the Sobolev spacesH k.Rn/ D H k;2.Rn/, we have

(2.10) H k.Rn/ � L2n=.n�2k/.Rn/; k <n

2:

Consequently, the interpolation theory developed in Chap. 4 implies

(2.11) H s.Rn/ � L2n=.n�2s/.Rn/;

for any real s 2 Œ0; k�; k < n=2 an integer. Actually, (2.11) holds for any reals 2 Œ0; n=2/, as will be shown in � 6. We write down some particular examples,for n D 2; 3; 4, which will play a role later in various nonlinear evolution equa-tions, such as the Navier–Stokes equations. The cases n D 3; 4 follow from theresults proved above, while the case n D 2 follows from the general case of (2.11)established in � 6.

H 1.R3/ � L6.R3/ H 1.R4/ � L4.R4/

H 3=4.R3/ � L4.R3/(2.12)

H 1=2.R2/ � L4.R2/ H 1=2.R3/ � L3.R3/

Note that interpolation of the R2-result with L2.R2/ D L2.R2/ yields

H 1=3.R2/ � L3.R2/:

6 13. Function Space and Operator Theory for Nonlinear Analysis

The next result provides a partial generalization of the Sobolev imbeddingtheorem,

H s.Rn/ � C.Rn/; s >n

2;

proved in Chap. 4. A more complete generalization is given in � 6.

Proposition 2.4. We have

(2.13) H k;p.Rn/ � C.Rn/\ L1.Rn/; for kp > n:

Proof. It suffices to obtain a bound on kukL1.Rn/ for u 2 H k;p.Rn/, if kp > n.In turn, it suffices to bound u.0/ appropriately, for u 2 C1

0 .Rn/. Use polar coor-dinates, x D r!; ! 2 Sn�1. Let g 2 C1.R/ have the property that g.r/ D 1

for r < 1=2 and g.r/ D 0 for r > 3=4. Then, for each !, we have

u.0/ D �Z 1

0

@

@rŒg.r/u.r; !/� dr

D .�1/k.k � 1/Š

Z 1

0

rk�n

(�@

@r

k

Œg.r/u.r; !/�

)rn�1dr;

upon integrating by parts k � 1 times. Integrating over ! 2 Sn�1 gives

ju.0/j � C

Z

B

rk�n

ˇˇˇ

�@

@r

k

Œg.r/u.x/�

ˇˇˇ dx;

where B is the unit ball centered at 0. Holder’s inequality gives

(2.14) ju.0/j � Ckrk�nkLp0

.B/

��@kr Œg.r/u.x/�

��Lp.B/

;

with 1=p C 1=p0 D 1. We claim that .@=@r/k is a linear combination ofD˛; j˛j D k, with L1-coefficients. To see this, note that @k

r annihilates x˛ forj˛j < k, so we get

(2.15)

�@

@r

k

DX

j˛jDk

a˛.x/@˛ ;

with a˛.x/ D .1=˛Š/@kr x

˛, for j˛j D k, or

a˛.r!/ D kŠ

˛Š!˛;

so a˛.x/ is homogeneous of degree 0 in x and smooth on Rn n 0.

Exercises 7

Returning to the estimate of (2.14), our information on .@=@r/k implies thatthe last factor on the right side is bounded by the H k;p-norm of u. The factorkrk�nkLp0

.B/ is finite provided kp > n, so the proposition is proved.

To close this section, we note the following simple consequence ofProposition 2.2, of occasional use in analysis. Let M.Rn/ denote the spaceof locally finite Borel measures (not necessarily positive) on Rn. Let us assumethat n � 2.

Proposition 2.5. If we have u 2 M.Rn/ and ru 2 M.Rn/, then it follows thatu 2 Ln=.n�1/

loc .Rn/.

Proof. Using a cut-off in C10 , we can assume u has compact support. Applying

a mollifier, we get uj D �j � u 2 C10 .Rn/ such that uj ! u and ruj ! ru

in M.Rn/. In particular, we have a uniform L1-norm estimate on ruj . By (2.3)we have a uniform Ln=.n�1/-norm estimate on uj , which gives the result, sinceLn=.n�1/.Rn/ is reflexive.

Exercises1. If pj 2 Œ1;1� and uj 2 Lpj , show that u1u2 2 Lr provided r�1 D p1

�1 C p2�1 2

Œ0; 1�. Show that this implies Lemma 2.1.2. Use the containment (which follows from Proposition 2.2)

Hk;p.Rn/ � H1;np=.n�.k�1/p/.Rn/ if .k � 1/p < n

to show that if Proposition 2.4 is proved in the case k D 1, then it follows in general.Note that the proof in the text of Proposition 2.4 is slightly simpler in the case k D 1

than for k � 2.3. Suppose k D 2` is even. Suppose u 2 S 0.Rn/ and

.��C 1/`u D f 2 Lp.Rn/:

Show thatu D Jk � f; bJ k.�/ D h�i�k :

Using estimates on Jk.x/ established in Chap. 3, � 8, show that

kp > n H) u 2 C.Rn/ \ L1.Rn/:

Show that this gives an alternative proof of Proposition 2.4 in case k is even.4. Suppose k D 2`C 1 is odd, kp > 1. Use the containment

Hk;p.Rn/ � Hk�1;np=.n�p/.Rn/ if p < n;

which follows from Proposition 2.2, to deduce from Exercise 3 that Proposition 2.4holds for all integers k � 2.

5. Establish the following variant of the k D 1 case of (2.14):

(2.16) ju.0/ � u.x/j � CkrukLp.B/; p > n; x 2 @B:

8 13. Function Space and Operator Theory for Nonlinear Analysis

(Hint: Suppose x D e1. If z is the line segment from 0 to z, followed by the linesegment from z to e1, write

u.e1/ � u.0/ DZ

˙

�Z

�z

du�dS.z/; ˙ D

x 2 B W x1 D 1

2

�:

Show that this gives u.e1/ � u.0/ D RB ru.z/ � '.z/ d z, with ' 2 Lq.B/; 8 q <

n=.n� 1/:)6. Show that Hn;1.Rn/ � C.Rn/ \ L1.Rn/.

(Hint: u.x/ D R 0�1 � � � R 0

�1D1 � � �Dnu.x C y/ dy1 � � � dyn:)

3. Gagliardo–Nirenberg–Moser estimates

In this section we establish further estimates on various Lp-norms of derivativesof functions, which are very useful in nonlinear PDE. Estimates of this sort arosein work of Gagliardo [Gag], Nirenberg [Ni], and Moser [Mos]. Our first suchestimate is the following. We keep the convention (2.8).

Proposition 3.1. For real k � 1; 1 � p � k, we have

(3.1) kDj uk2L2k=p � CkukL2k=.p�1/ � kD2

j ukL2k=.pC1/ ;

for all u 2 C10 .Rn/, hence for all u 2 Lq2.Rn/ \H 2;q1 , where

(3.2) q1 D 2k

p C 1; q2 D 2k

p � 1:

Proof. Given v 2 C10 .Rn/; q � 2, we have vjvjq�2 2 C 1

0 .Rn/ and

Dj .vjvjq�2/ D .q � 1/.Dj v/jvjq�2:

Letting v D Dj u, we have

jDj ujq D Dj .u Dj ujDj ujq�2/ � .q � 1/u D2j ujDj ujq�2:

Integrating this, we have, by the generalized Holder inequality (2.1),

(3.3) kDj ukqLq � jq � 1j � kukLq2 kD2

j ukLq1 kDj ukq�2Lq ;

where q D 2k=p and q1 and q2 are given by (3.2). Dividing by kDj ukq�2Lq gives

the estimate (3.1) for u 2 C10 .Rn/, and the proposition follows.

If we apply (3.1) to D`�1u, we get

(3.4) kD`uk2L2k=p � CkD`�1ukL2k=.p�1/kD`C1ukL2k=.pC1/ ;