Taylor M.E. Partial Differential Equations III: Nonlinear Equations

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xx Preface

Appendix C, Connections and Curvature, contains material of a differential

geometric 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 connection

and Riemann curvature tensor on a Riemannian manifold. We discuss basic prop-

erties of the geometry of submanifolds, relating the second fundamental form to

curvature via the Gauss–Codazzi equations. We describe how vector bundles arise

from principal bundles, which themselves carry various connections and curvature

forms. We then discuss the Chern–Weil construction, yielding certain closed dif-

ferential forms associated to curvatures of connections on principal bundles. We

give several proofs of the classical Gauss–Bonnet theorem and some related re-

sults on two-dimensional surfaces, which are useful particularly in Chaps. 10 and

14. We also give a geometrical proof of the Chern–Gauss–Bonnet theorem, which

can 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 appendices

to 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 of

harmonic functions useful for the proof of the linear Cauchy–Kowalewsky theo-

rem, and Appendix B deals with the stationary phase formula, useful for the study

of geometrical optics in Chap. 6 and also for results later, in Chap. 9. There are

other 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 complementary

material.

Having described the scope of this work, we ﬁnd it necessary to mention a

number of topics in PDE that are not covered here, or are touched on only very

brieﬂy.

For example, we devote little attention to the real analytic theory of PDE. We

note that harmonic functions on domains in R

are real analytic, but we do not

discuss analyticity of solutions to more general elliptic equations. We do prove

the Cauchy–Kowalewsky theorem, on analytic PDE with analytic Cauchy data.

We derive some simple results on unique continuation from these few analyticity

results, but there is a large body of lore on unique continuation, for solutions to

nonanalytic PDE, neglected here.

There is little material on numerical methods. There are a few references to

applications of the FFT and of “splitting methods.” Difference schemes for PDE

are mentioned just once, in a set of exercises on scalar conservation laws. Finite

element methods are neglected, as are many other numerical techiques.

There is a large body of work on free boundary problems, but the only one

considered here is a simple one space dimensional problem, in Chap. 15.

While we have considered a variety of equations arising from classical

physics 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 scattering

mentioned in Chap. 9. However, the physical theories behind these equations are

not 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 are

not 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 on

listing important topics in this vast subject. The author can at best hope that the

reader will ﬁnd it easier to understand many of these topics with this book, than

without it.

Acknowledgments

I have had the good fortune to teach at least one course relevant to the material of

this 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, for

the supportive atmospheres at these institutions. Also, a number of individuals

provided valuable advice on various portions of the manuscript, as it grew over the

years. I particularly want to thank Florin David, David Ebin, Frank Jones, Anna

Mazzucato, Richard Melrose, James Ralston, Jeffrey Rauch, Santiago Simanca,

and James York. The ﬁnal touches were put on the manuscript while I was visiting

the 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 senior

thesis 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 over

the past dozen years, I have taken the opportunity to make some very signiﬁcant

changes, some of which broaden the scope of the work, some of which clarify

previous presentations, and a few of which correct errors that have come to my

attention.

There are seven additional sections in this edition, two in Volume 1, two in

Volume 2, and three in Volume 3. Chapter 4 has a new section, “Sobolev spaces

on 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 layer

phenomena 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 oscillator

type,” 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 the

study of Zygmund spaces. Finally, Chap. 17 has two new sections, “Vanishing

viscosity 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 books

over time.

Function Space and Operator Theory

for Nonlinear Analysis

Introduction

This chapter examines a number of analytical techniques, which will be applied

to diverse nonlinear problems in the remaining chapters. For example, we study

Sobolev spaces based on L

, rather than just L

. Sections 1 and 2 discuss the

deﬁnition of Sobolev spaces H

k;p

,fork 2 Z

, and inclusions of the form

k;p

 L

. Estimates based on such inclusions have reﬁned forms, due to

E. Gagliardo and L. Nirenberg. We discuss these in  3, together with results of

J. Moser on estimates on nonlinear functions of an element of a Sobolev space,

and on commutators of differential operators and multiplication operators. In  4

we establish some integral estimates of N. Trudinger, on functions in Sobolev

spaces for which L

-bounds just fail. In these sections we use such basic tools

as H¨older’s inequality and integration by parts.

The Fourier transform is not as effective for analysis on L

as on L

.One

result that does often serve when, in the L

-theory, one could appeal to the

Plancherel 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 of

-spectral theory of the Laplace operator, on compact manifolds, possibly with

boundary.

In  8 we study spaces C

of H¨older continuous functions, and their relation

with Zygmund spaces C



. We derive estimates in these spaces for solutions to

elliptic boundary problems.

The next two sections extend results on pseudodifferential operators, intro-

duced in Chap. 7. Section 9 considers symbols p.x; / with minimal regularity

in x. We derive both L

-andH¨older estimates. Section 10 considers paradiffer-

ential operators, a variant of pseudodifferential operator calculus particularly well

suited to nonlinear analysis. Sections 9 and 10 are largely taken from [T2].

In  11 we consider “fuzzy functions,” consisting of a pair .f; /,wheref is

afunctiononaspace and  is a measure on   R, with the property that

’

y'.x/ d.x;y/ D

'.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

 Springer Science+Business Media, LLC 1996, 2011

2 13. Function Space and Operator Theory for Nonlinear Analysis

smooth (“sharply deﬁned”) functions, and it is useful for analyses of nonlinear

maps that do not generally preserve weak convergence.

In  12 there is a brief discussion of Hardy spaces, subspaces of L

/ with

many desirable properties, only a few of which are discussed here. Much more on

this topic can be found in [S3], but material covered here will be useful for some

elliptic 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 of

this chapter. It turns out that slightly different complex interpolation functors are

better suited to the scale of Zygmund spaces.

1. L

-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

/ D

u 2 L

/ W D

u 2 L

/ for j˛jk



It is easy to see that S.R

/ is dense in each space H

k;p

/, with its natural

norm

(1.2) kuk

k;p

j˛jk

For p ¤ 2, we cannot characterize the spaces H

k;p

/ conveniently in terms of

the Fourier transform. It is still possible to deﬁne spaces H

s;p

/ by interpola-

tion; we will examine this in  6. Here we will consider only the spaces H

k;p

with k a nonnegative integer.

The chain rule allows us to say that if  W R

! R

is a diffeomorphism that

is linear outside a compact set, then 



W H

k;p

/ ! H

k;p

/. Also multi-

plication by an element ' 2 C

/ maps H

k;p

/ to itself. This allows us to

deﬁne H

k;p

.M / for a compact manifold M via a partition of unity subordinate

to a coordinate chart. Also, for compact M ,ifwedeﬁneDiff

.M / to be the set

of differential operators of order  k on M , with smooth coefﬁcients, then

(1.3) H

k;p

.M / Dfu 2 L

.M / W P u 2 L

.M / for all P 2 Diff

.M /g:

We can deﬁne H

k;p

/ as in (1.1), with R

replaced by R

. The exten-

sion operator deﬁned by (4.2)–(4.4) of Chap. 4 also works to produce extension

maps E W H

k;p

/ ! H

k;p

/. Similarly, if M is a compact manifold

with smooth boundary, with double N , we can deﬁne H

k;p

.M / via coordinate

charts and the notion of H

k;p

/,orby(1.3), and we have extension operators

E 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

/ ! H

kj˛j;p

for j˛jk,and

(1.5) P W H

k;p

.M / ! H

k`;p

.M / if P 2 Diff

.M /;

provided `  k.

Exercises

1. A Friedrichs molliﬁer on R

is a family of smoothing operators J

u.x/ D j

 u.x/

where

.x/ D "

n

j."

1

x/;

j.x/dx D 1; j 2 S.R

Equivalently, J

u.x/ D '."D/u.x/; ' 2 S.R

/; '.0/ D 1. Show that, for each

p 2 Œ1; 1/; k 2 Z

W H

k;p

/ !

`<1

`;p

for each ">0,and

u ! u in H

k;p

as " ! 0 if u 2 H

k;p

2. Suppose A 2 C

/, with kAk

D sup

j˛j1

. Show that when J

is a

Friedrichs molliﬁer as above, then

kŒA; J

vk

1;p

 C kAk

kvk

;

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

.x; y/

 y

/; jB

.x; y/jK, and, with q

.x/ D @j =@ x

ŒA; J

v.x/ D

.x; y/

n



x  y





 y

v.y/ dy;

with absolute value bounded by

n



1

.x  y/



jv.y/j dy;

where '

.x/ D x

.x/:)

3. Using Exercise 2, show that

kŒA; J

@

 C kAk

kvk

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 will

concentrate on the case M D R

. The discussion of  1 will give associated

results when M is a compact manifold, possibly with (smooth) boundary.

One technical tool useful for our estimates is the following generalized H¨older

inequality:

Lemma 2.1. If p

2 Œ1; 1;

1

D 1,then

(2.1)

u

j dx ku

.M /

ku

.M /

The proof follows by induction from the case m D 2, which is the usual H¨older

inequality.

Our ﬁrst Sobolev imbedding theorem is the following:

Proposition 2.2. Fo r p 2 Œ1; n/,

(2.2) H

1;p

/  L

np=.np/

In fact, there is an estimate

(2.3) kuk

np =. np/

 C kruk

;

for u 2 H

1;p

/, with C D C.p;n/.

Proof. It sufﬁces to establish (2.3)foru 2 C

/. Clearly,

ju.x/j

1

uj dx

;(2.4)

(2.5) ju.x/j

n=.n1/



j D1

1

ujdx

;

1=.n1/

We can integrate (2.5) successively over each variable x

;j D 1;:::;n,and

apply the generalized H¨older inequality (2.1) with m D p

DDp

D n  1

after each integration. We get

(2.6) kuk

n=.n1/



j D1

ujdx

;

1=n

 C kruk

2. Sobolev imbedding theorems 5

This establishes (2.3) in the case p D 1. We can apply this to v Djuj



;>1,

obtaining

(2.7)



juj





n=.n1/

 C



juj

1

jruj



 C



juj

1



For p<n,pick D .n 1/p=.n p/.Then(2.7)gives(2.3) and the proposition

is proved.

Given u 2 H

k;p

/, we can apply Proposition 2.2 to estimate the L

np=.np/

norm of D

k1

u in terms of kD

, where we use the notation

(2.8) D

u DfD

u Wj˛jDkg; kD

j˛jDk

;

and proceed inductively, obtaining the following corollary.

Proposition 2.3. Fo r kp < n,

(2.9) H

k;p

/  L

np=.nkp /

The same result holds with R

replaced by a compact manifold of dimension n.

If we take p D 2, then for the Sobolev spaces H

/ D H

k;2

/,wehave

(2.10) H

/  L

2n=.n2k/

/; k <

Consequently, the interpolation theory developed in Chap.4 implies

(2.11) H

/  L

2n=.n2s/

for any real s 2 Œ0; k; k < n=2 an integer. Actually, (2.11) holds for any real

s 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 the

results proved above, while the case n D 2 follows from the general case of (2.11)

established in  6.

/  L

3=4

/  L

/(2.12)

1=2

/  L

1=2

/  L

Note that interpolation of the R

-result with L

/ D L

/ yields

1=3

/  L

6 13. Function Space and Operator Theory for Nonlinear Analysis

The next result provides a partial generalization of the Sobolev imbedding

theorem,

/  C.R

/; s >

;

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

Proposition 2.4. We have

(2.13) H

k;p

/  C.R

/ \ L

/; for kp > n:

Proof. It sufﬁces to obtain a bound on kuk

for u 2 H

k;p

/,ifkp > n.

In turn, it sufﬁces to bound u.0/ appropriately, for u 2 C

/. Use polar coor-

dinates, x D r!; ! 2 S

n1

.Letg 2 C

.R/ have the property that g.r/ D 1

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

u.0/ D

Œg.r/u.r; !/ dr

.1/

.k  1/Š

kn

(





Œg.r/u.r; !/

)

n1

dr;

upon integrating by parts k  1 times. Integrating over ! 2 S

n1

gives

ju.0/jC

kn





Œg.r/u.x/

dx;

where B is the unit ball centered at 0.H¨older’s inequality gives

(2.14) ju.0/jC kr

kn

.B/



Œg.r/u.x/



.B/

;

with 1=p C 1=p

D 1. We claim that .@=@r/

is a linear combination of

; j˛jDk, with L

-coefﬁcients. To see this, note that @

annihilates x

for

j˛j <k,soweget

(2.15)





j˛jDk

.x/@

;

with a

.x/ D .1=˛Š/@

,forj˛jDk,or

.r!/ D

kŠ

˛Š

;

so a

.x/ is homogeneous of degree 0 in x and smooth on R

n 0.

Exercises 7

Returning to the estimate of (2.14), our information on .@=@r/

implies that

the last factor on the right side is bounded by the H

k;p

-norm of u. The factor

kn

.B/

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

To close this section, we note the following simple consequence of

Proposition 2.2, of occasional use in analysis. Let M.R

/ denote the space

of locally ﬁnite Borel measures (not necessarily positive) on R

. Let us assume

that n  2.

Proposition 2.5. If we have u 2 M.R

/ and ru 2 M.R

/, then it follows that

u 2 L

n=.n1/

loc

Proof. Using a cut-off in C

, we can assume u has compact support. Applying

a molliﬁer, we get u

D 

 u 2 C

/ such that u

! u and ru

!ru

in M .R

/. In particular, we have a uniform L

-norm estimate on ru

.By(2.3)

we have a uniform L

n=.n1/

-norm estimate on u

, which gives the result, since

n=.n1/

/ is reﬂexive.

Exercises

1. If p

2 Œ1; 1 and u

2 L

, show that u

2 L

provided r

1

D p

1

C p

1

Œ0; 1. Show that this implies Lemma 2.1.

2. Use the containment (which follows from Proposition 2.2)

k;p

/  H

1;np=.n.k1/p/

/ 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

/ and

. C1/

u D f 2 L

Show that

u D J

 f;

./ Dhi

k

Using estimates on J

.x/ established in Chap. 3,  8, show that

kp > n H) u 2 C.R

/ \ L

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

k;p

/  H

k1;np=.np/

/ if p<n;

which follows from Proposition 2.2, to deduce from Exercise 3 that Proposition 2.4

holds 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C kruk

.B/

;p>n;x2 @B: