Methods on Nonlinear Elliptic Equations (Aims Series on Differential Equations & Dynamical Systems)

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Methods on Nonlinear Elliptic Equations

Wenxiong Chen Congming Li A

M S American Institute of Mathematical Sciences

AIMS Series on Differential Equations & Dynamical Systems

Volume 4

Wenxiong Chen Yeshiva University

Congming Li University of Colorado at Boulder

Methods on Nonlinear Elliptic Equations

A!h AIMS

American Institute of Mathematical Sciences

EDITORIAL COMMITTEE Editor in Chief: Jianhong Wu (Canada) Jerry L. Bona (USA), Hans G. Othmer (USA) Jeff Webb (UK), Messoud Efendiev (Germany) Yacine Chitour (France)

Wenxiong Chen

Department of Mathematics Yeshiva University New York, NY 10033, USA E-mail: wchen@yu. edu Congming Li

Department of Applied Mathematics University of Colorado Boulder, CO 80309, USA E-mail: [email protected] AMS 2010 subject classifications: 31A10, 31B35, 31C15, 35-02, 35A01, 35A02, 35A08, 35A15, 35A23, 35A25, 35B06, 35B09, 35B33, 35B38, 35B40, 35B45, 35B50, 35B51, 35B53, 35B65, 35D30, 35D35, 35E05, 35G20, 35J05, 35J15,

35J20, 35J25, 35J60, 35J61, 35J91, 45G05, 45G10, 45M20, 45P05, 46E35, 46F25, 46N20, 47H10, 47J30, 49-02, 49J35, 53-02, 53B21, 53C21, 58J05. ISBN-10: 1-60133-006-5 ISBN-13: 978-1-60133-006-2

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Preface

In this book we present basic concepts as well as real research examples to young researchers interested in the field of non-linear analysis of partial differential equations (PDEs); in particular, the text focuses on the analysis of semi-linear elliptic PDEs. We hope that graduate students consider our text good reading material and that professors consider it a handy textbook for use in a topics course on non-linear analysis. After necessary preparations for basic knowledge have been made, a series of typical methods in non-linear analysis will be introduced, some of which are well known while others are relatively new. We will first illustrate these ideas and techniques using simple examples; we will then lead readers to the research front and explain how these methods can be applied to solve practical problems through careful analysis of a series of recent research articles. Roughly speaking, in applying these commonly used methods, there are

usually two aspects: i) A general scheme (more or less universal) and, ii) Key points for each individual problem.

By understanding these research examples, readers should then be able to apply these general schemes to solve their own research problems by discovering their own key points. In Chapter 1, we introduce basic concepts of Sobolev spaces and some commonly used inequalities. These are the major spaces in which we will seek weak solutions to PDEs. Chapter 2 shows how to find weak solutions for some typical linear and semi-linear PDEs by using functional analysis methods, mainly, the calculus of variations and critical point theories, including the well-known Mountain Pass Lemma.

VI

Preface

In Chapter 3, we establish W2,P a priori estimates and regularity. We prove that, in most cases, weak solutions are actually differentiable and hence are classical solutions. Our approach here is quite different from the traditional approach. We will also present two Regularity Lifting Theorems. The first Theorem uses an operator which is contracting in both spaces. It is a simple method to boost the regularity of solutions and has been used extensively in various forms in the authors' previous works. The essence of the approach is well known in the analysis community, while the version here contains some new developments. The second Theorem employs an operator which is contracting in one space and shrinking in the other space. It is a brand new idea, and as we believe, it will find broad applications in a variety of nonlinear problems. We will use examples to show how these Theorems can be applied to systems of PDEs and integral equations including a fully nonlinear system of Wolff type.

Chapter 4 is a preparation for chapters 5 and 6. We introduce Riemannian manifolds, curvatures, covariant derivatives, and Sobolev embedding on manifolds. Chapter 5 deals with semi-linear elliptic equations arising from prescribing Gaussian curvature on both positively and negatively curved manifolds. We

show the existence of weak solutions in both subcritical and critical cases via variational approaches. We also introduce the method of lower and upper solutions.

Chapter 6 focuses on the well-known Yamabe problem and its general-

ization, prescribing scalar curvature on S' for n > 3. The latter is in the critical case where the corresponding variational functional is not compact at any level sets. To recover the compactness, we construct a max-mini variational scheme. The outline is clearly presented; however, the detailed proofs are rather complex, and beginners are welcome to skip these proofs. Chapter 7 is devoted to the study of various Maximum Principles, in particular, the ones based on comparisons. Besides classical ones, we also introduce a version of the Maximum Principle at infinity and a Maximum Principle for integral equations that basically depends on the absolute continuity of a Lebesgue integral. It is a preparation for the Method of Moving Planes. In Chapter 8, we introduce the Method of Moving Planes and its variantthe Method of Moving Spheres-and apply them to obtain the symmetry, monotonicity, a priori estimates, and even non-existence of solutions. We also introduce an integral form of the Method of Moving Planes, a relatively new idea that is quite different from the traditional ones for PDEs. Instead of using local properties of a PDE, global norms of solutions for integral equations will be exploited. Many research examples are illustrated, including the wellknown one by Gidas, Ni, and Nirenberg, as well as a few from the authors' recent papers.

Preface

VII

Chapters 7 and 8 function as a self-contained group. Readers who are only interested in the Maximum Principles and the Method of Moving Planes can skip the first six chapters and start directly with Chapter 7. The authors thank Professor Allison Smith at Yeshiva College Writing Center for editing some chapter introductions.

Wenxiong Chen and Congming Li

Contents

1

2

Introduction to Sobolev Spaces ............................ Distributions ............................................

3

1.2

Sobolev Spaces ..........................................

8

.......................

1.3

Approximation by Smooth Functions

1.4

Sobolev Embeddings ..................................... 22

1.5

Compact Embedding ..................................... 33

1.6

Other Basic Inequalities .................................. 36

10

1.6.1

Poincare'sInequality ............................... 36

1.6.2

The Classical Hardy-Littlewood-Sobolev Inequality ....

38

Existence of Weak Solutions ............................... 43 2.1

Second Order Elliptic Operators ...........................

44

2.2

Weak Solutions .........................................

45

2.3

Methods of Linear Functional Analysis .....................

46

2.3.1

Linear Equations ..................................

46

2.3.2 2.3.3

Some Basic Principles in Functional Analysis .........

47

Existence of Weak Solutions ........................

51

Variational Methods .....................................

53

2.4

2.4.1 2.4.2 2.4.3 2.4.4 2.4.5 2.4.6 3

1

1.1

Semi-linear Equations .............................. 53 Calculus of Variations .............................. 53 Existence of Minimizers ............................ 55 Existence of Minimizers Under Constraints ........... 58

Mini-max Critical Points ........................... 62 Existence of a Mini-max via the Mountain Pass Theorem 68

Regularity of Solutions .................................... 77

3.1

W2' a Priori Estimates ..................................

79

3.1.1

Newtonian Potentials .............................. 79

3.1.2

Uniform Elliptic Equations ......................... 86

X

Contents

3.2

W2'P Regularity of Solutions .............................. 91 3.2.1

The Case p > 2 ................................... 93

3.2.2

The Case 1 < p < 2 .

3.2.3

Other Useful Results Concerning the Existence,

.............................. 98

Uniqueness, and Regularity ......................... 100 3.3

Regularity Lifting .......................................101 3.3.1

Bootstrap ........................................ 101

3.3.2

Regularity Lifting by Contracting Operators .......... 102

3.3.3 Applications to PDEs .............................. 103 3.3.4 Applications to Integral Equations ................... 109 3.3.5 Regularity Lifting by Combinations of Contracting and Shrinking Operators ........................... 111

3.3.6 3.3.7

4

Applications to Integral Equations ................... 115 Applications to Fully Nonlinear Systems of Wolff Type. 119

Preliminary Analysis on Riemannian Manifolds ............ 123 4.1

Differentiable Manifolds .................................. 124

4.2

Tangent Spaces ......................................... 125

4.3

Riemannian Metrics ..................................... 128

4.4

Curvature .............................................. 131

4.5

4.4.1

Curvature of Plane Curves .......................... 131

4.4.2

Curvature of Surfaces in R3 ........................ 132

4.4.3

Curvature on Riemannian Manifolds ................. 133

Calculus on Manifolds .................................... 137

4.5.1

Higher Order Covariant Derivatives and the Laplace-Beltrami Operator .........................137

4.6

5

4.5.2

Integrals ......................................... 139

4.5.3

Equations on Prescribing Gaussian and Scalar Curvaturel40

Sobolev Embeddings ..................................... 141

Prescribing Gaussian Curvature on Compact 2-Manifolds.. 145 5.1

Variational Methods in General ........................... 145

5.2

Introduction to Prescribing Gaussian Curvature ............148

5.3

The Negative Case ...................................... 150

5.3.1

Kazdan and Warner's Results-Method of Lower and Upper Solutions ................................... 150

5.3.2 5.4

The Limiting Situation ............................. 156

The Non-negative Case-Variational Approaches ............. 162

5.4.1 5.4.2 5.4 3

Obstructions ...................................... 162

The Variational Approach and Key Inequalities ....... 163 Existence of Weak Solutions in Subcritical Case ....... 166 5.4.4 A Remedy for Critical Case-Recovering Coerciveness .. 167

Contents

5.4.5 Existence of Weak Solutions in the Critical Case 6

XI

...... 171

The Yamabe Problem and Prescribing Scalar Curvature on S", for n. > 3 ........................................... 179 6.1

Introduction ............................................ 179 6.1.1

The Yamabe Problem .............................. 179

6.1.2

Prescribing Scalar Curvature on S^ ................... 18

6.2 The Variational Approach for the Yamabe Problem .......... 186 6.3 The Variational Approach for Prescribing Scalar Curvature ... 192 6.3.1 Estimate the Values of the Functional ................ 192 6.3.2 The Variational Scheme The a priori Estimates for Prescribing Scalar Curvature ...... 203 6.4

............................ 202

7

6.4.1

In the Region Where R < 0 ......................... 204

6.4.2

In the Region Where R is Small

6.4.3

In the Regions Where R > 0 ........................206

..................... 204

Maximum Principles .......................................

209 ............................................ 209 ............................... 214

Introduction Weak Maximum Principles The Hopf Lemma and Strong Maximum Principles .......... 218 7.4 Maximum Principles Based on Comparisons ................ 224 7.5 A Maximum Principle for Integral Equations ................ 227 7.1 7.2 7.3

8

Methods of Moving Planes and Moving Spheres 8.1 8.2

........... 231

Outline of the Method of Moving Planes ................... Applications of the Maximum Principles Based on Comparisons

233

............................................ 235

Symmetry of Solutions in a Unit Ball ............... 235 Symmetry of Solutions of -Du = un in R" ........... 238 Symmetry of Solutions for -Au = e" in R2 ........... 246 8.3 Method of Moving Planes in a Local Way .................. 251 8.2.1 8.2.2 8.2.3

8.4

8.5

8.3.1

The Background .................................. 251

8.3.2

The A Priori Estimates

............................ 253

Method of Moving Spheres ............................... 260 8.4.1

The Background .................................. 260

8.4.2

Necessary Conditions .............................. 262

Method of Moving Planes in Integral Forms ................. 266

A Appendices ................................................275 A.1 Notation ...............................................275 A.1.1 Algebraic and Geometric Notation ................... 275 A.1.2 Notation for Functions and Derivatives ............... 276

XII

Contents A.1.3 Function Spaces ...................................277 A.1.4 Notation for Estimates ............................. 279

A.2 Notation and Basic Facts from Riemannian Geometry ........

279

A.3 Common Inequalities and Their Proofs ..................... 282 A.4 Calderon-Zygmund's Decomposition ....................... 284 A.5 The Contracting Mapping Principle ........................286

A.6 The Arzela-Ascoli Theorem ............................... 288 A.7 The Proof of Lemma 5.2.1 ................................ 288 References ..................................................... 291

Index ..........................................................297

1

Introduction to Sobolev Spaces

1.1

Distributions

1.2

Sobolev Spaces

1.3

Approximation by Smooth Functions

1.4

Sobolev Embeddings

1.5

Compact Embedding

1.6

Other Basic Inequalities 1.6.1

1.6.2

Poincare's Inequality The Classical Hardy-Littlewood-Sobolev Inequality

People use numbers every day to quantify surrounding objects. In mathematics, the absolute value la-bi is used to measure the difference between two numbers a and b. Functions are used to describe physical states. For example, temperature is a function of time and place. Very often, we use a sequence of approximate solutions to approach a real one; and how close these solutions are to the real one depends on how we measure them (i.e., which metric we are choosing). Hence, not only must we develop suitable metrics to measure different states (functions), but we must also study relationships among different metrics. For these purposes, the Sobolev Spaces were introduced. They have many applications in various branches of mathematics, in particular, in the theory of partial differential equations. The role of Sobolev Spaces in the analysis of PDEs is somewhat similar to the role of Euclidean Spaces in the study of geometry. The fundamental research on the relations among various Sobolev Spaces (Sobolev norms) was first carried out by G. Hardy and J. Littlewood in the 1910s and then by S. Sobolev in the 1930s. More recently, many well known mathematicians, such as H. Brezis, L. Caffarelli, A. Chang, E. Lieb, L. Nirenberg, J. Serrin, and

2

1 Introduction to Sobolev Spaces

E. Stein have worked in this area. The main objectives are to determine if and how the norms dominate each other, what the sharp estimates are, which functions achieve these sharp estimates, and which functions are `critically' related to these sharp estimates. To find the existence of weak solutions for partial differential equations, especially for nonlinear partial differential equations, the method of functional analysis, in particular, the calculus of variations, has seen increasing application.

To roughly illustrate this kind of application, let's start with a simple example. Let .2 be a bounded domain in R" and consider the Dirichlet problem associated with the Laplace equation:

-Du=f(x), xE,(l u=0,

XEOQ

(1.1)

To prove the existence of solutions, one may view -4 as an operator acting on a proper linear space and then apply some known principles of functional analysis, such as the `fixed point theory' or `the degree theory,' to derive the existence. One may also consider the corresponding variational functional

J(u) = 1 L IVU12dx- J f(x)udx 2

n

n

(1.2)

in a proper linear space and seek critical points of the functional in that space. This kind of variational approach is particularly powerful in dealing with nonlinear equations. For example, in equation (1.1), instead of f(x), we consider f (x, u). Then it becomes a semi-linear equation. Correspondingly, we have the functional

J(u) = 1

2Ln

IpuI2dx-L F(x,u)dx, n

(1.3)

where

F(x, u) = f f (x, s) d x 0

is an anti-derivative of f (x, ). From the definition of the functional in either (1.2) or (1.3), one can see that the function u in the space need not be second order differentiable as is required by classical solutions of (1.1). Hence the critical points of the functional are solutions of the problem only in the `weak' sense. However, by an appropriate regularity argument, one may recover the differentiability of the solutions so that they can still satisfy equation (1.1) in the classic sense. In general, given a PDE problem, our intention is to view it as an operator A acting on some proper linear spaces X and Y of functions and to write the equation symbolically as

1.1 Distributions

Au= f

3

(1.4)

We can then apply the general and elegant principles of linear or nonlinear functional analysis to study the solvability of various equations involving A, the result of which can then be applied to a broad class of partial differential equations. We may also associate this operator with a functional J(.), whose critical points are the solutions of the equation (1.4). In this process, the key is to find an appropriate operator `A' and appropriate spaces `X' and `Y'. As we shall see later, the Sobolev spaces are designed precisely for this purpose and will work out properly. In solving a partial differential equation, in many cases it is natural to first find a sequence of approximate solutions and proceed to investigate the convergence of the sequence. The limit function of a convergent sequence of approximate solutions represents the desired, exact solution to the equation. As we shall see in the next few chapters, there are two basic stages in showing convergence:

i) In a reflexive Banach space, every bounded sequence has a weakly convergent subsequence, and then

ii) By the compact embedding from a "stronger" Sobolev space into a "weaker" one, the weak convergent sequence in the "stronger" space becomes a strong convergent sequence in the "weaker" space.

Before going into the details of this chapter, the reader may take a glance at the introduction of the next chapter to gain motivation for studying Sobolev spaces.

In Section 1.1, we will introduce the distributions, mainly the notion of the weak derivatives, which are the elements of the Sobolev spaces. We then define Sobolev spaces in Section 1.2. In deriving many useful properties in Sobolev spaces, it is inconvenient to work directly with weak derivatives. Hence, in Section 1.3, we show that these weak derivatives can be approximated by smooth functions. The three sections that follow then focus on smooth functions in establishing a series of important inequalities.

1.1 Distributions As we saw in the introduction, the functional J(u) in (1.2) or (1.3) involved only the first derivatives of u rather than the second derivatives as is required for classical second order equations; moreover, these first derivatives need not be continuous nor even defined everywhere. Therefore, by using a functional analysis approach one can substantially weaken the notion of partial derivatives. The advantage is that it divides the task of finding "suitable" smooth solutions for a PDE into two major steps:

4

1 Introduction to Sobolev Spaces

Step 1. Existence of Weak Solutions. One seeks solutions that are less differentiable but easier to obtain. It is very common to use "energy" minimization or conservation, or sometimes to use finite dimensional approximation, to show the existence of such weak solutions. Step 2. Regularity Lifting. One uses various analysis tools to boost the differentiability of the known weak solutions and try to show that they are actually classical solutions. Both the existence of weak solutions and regularity lifting have become two major branches of today's PDE analysis. Various function spaces and related embedding theories are basic tools in both analyses, among which Sobolev spaces are the most frequently used. In this section, we introduce the notion of `weak derivatives,' which will be the elements of the Sobolev spaces. Let R" be the n-dimensional Euclidean space and 1? be an open connected subset in R". Let D(12) = Co (,fl) be the linear space of infinitely differentiable functions with compact support in Q. This is called the space of test functions on Q.

Example 1.1.1 Assume BR(x°) :_ {x E R" I Ix - x°I < R} C 9, then for any r < R, the following function f (x)

exp{

to

I__

r2

} for Ix - x° I < r elsewhere

is in Co (fl). Example 1.1.2 Assume p E Co (R"), it E LP(.f2), and suppu C K CC Q. Let uE(x)

PE * u:=

J/

JR" E

P(x

E

y )u(y)dy.

Then uE E Co (.fl) for e sufficiently small.

Now, let Lloc(Q) be the space of pth-power locally summable functions for 1 < p < oo. Such functions are Lebesgue measurable functions f defined on ,(2 and with the property that 1/p

IfILPcxl U If(x)IPdx)

0, JE f is a C°° function, and as e-40,

1.3 Approximation by Smooth Functions

11

JEf-*f in Wk'P Actually, notice that (JEf)(x) = J

B

(s)nn

9E(x - y)f(y)dy,

and hence in order for J,f(x) to approximate f(x) well, we need B,(x) to be

completely contained in JZ to ensure that JB,(:)nn

j,(x - y)dy = 1

(an important property of the delta function). Equivalently, we need x to be in the interior of .fl. For this reason, we first prove a local approximation

theorem.

Theorem 1.3.1 (Local Approximation by Smooth Functions).

For any f E Wk'P(s7), JE f E C°°(R") and JEf i f in W`oP(Jl) as e- O. To extend this result to the entire .2, we will choose infinitely many open sets Oi, i = 1, 2, - , each of which has a positive distance to the boundary of B and whose union is the whole Q. Based on the above theorem, we are able to approximate a Wk'P(q) function on each Oi by a sequence of smooth functions. Combining this with a partition of unity, and a cut-off function if JZ is unbounded, we will then prove: - -

Theorem 1.3.2 (Global Approximation by Smooth Functions). For any f E Wc'P(fl), there exists a sequence of functions {fm} C C°°(0) f1Wk'P(,fl) such that fm -* f in Wk'P(Q) as m oo.

Theorem 1.3.3 (Global Approximation by Smooth Functions up to the Boundary).

Assume that Bl is bounded with C' boundary 80. Then for any f e Wk'P(fl), there exists a sequence of functions {f,,,} C C°°(fl) = C°°(0) n W',P(dl) such that fm -> f in Wk'P(JZ) as moo.

When .2 is the entire space R", the approximation by C°° or by Co functions are essentially the same. We have

Theorem 1.3.4 Wk'P(R") = Wk,,(R"). In other words, for any f E W',P(R"), there exists a sequence of functions {fm} C Co (R"), such that

fm 4f,

as m->oo;

in Wk'P(R").

12

1 Introduction to Sobolev Spaces

Proof of Theorem 1.3.1 We prove the theorem in three steps. In step 1, we show that JEf E C°°(R") and IIJcfIILP(ra) ! IIf IILP(a).

From the definition of JEf (x), we can see that it is well defined for all x E R'°, and it vanishes if x is of e distance away from Q. Here and in the following, for simplicity of argument, we extend f to be zero outside of Q.

In step 2, we prove that if f is in L1(Q),

(JEf)-if in Ltoc(Q). We first verify this for continuous functions and then approximate LP functions by continuous functions. In step 3, we reach the conclusion of the Theorem. For each f E W"P(Q)

and Ial < k, D' f is in LP(Q). Then from the result in Step 2, we have

JE(D"f)-rD"f in Lloc(r). Hence, what we need to verify is

D"(JEf)(x) - JE(D"f)(x). As the reader will notice, the arguments in the last two steps only work in any compact subset of ,(l. Step 1.

Let e; = (0, , 0) be the unit vector in the xi direction. , 0, 1, 0, Fix e > 0 and x E R. By the definition of JEf we have, for Ihi < e, -

(JEf)(x +hei) - (J 0, choose a a continuous function g, such that -

If - 91ILP(n)
0. Choose ei > 0 so small that fi

JE; (ni f) satisfies

11Ifi-11ifllWk.a(n) 2+Tiri-01 1' i = 1, 2, . .. l suPPfi C (fli+a \ fli),

1

(1.16) .

Set g = E°Oo fi. Then g E C°°(fl) because each fi is in C°°(0), and for each open set 0 cc fl, there are at most finitely many nonzero terms in the sum. To see that g E Wk,p(fl), we write 00

g - f - E(A - nif)i=0

It follows from (1.16) that

16

1 Introduction to Sobolev Spaces

m

0o

1

= S.

2i+1

From here we can see that the series E2'=0(fi -rti f) converges in W k'P (.f2) (see

Exercise 1.3.1 below); hence, (g - f) E W'"P(Q) and therefore g E Wk,P(fl). Moreover, from the above inequality, we have 119 - f llwk.P(o) < S.

Since b > 0 is any number, we complete Step 1.

Remark 1.3.1 In the above argument, neither E fi nor > r)i f converge, but their difference converges. Although each partial sum of E fi is in Co ([2), the infinite series E fi does not converge to g in Wk,P(fl). Then how did we prove that g E Wk=P(Q)? We showed that the difference (g- f) is in Wk,P(12).

Exercise 1.3.1

.

Let X be a Banach space and vi E X. Show that the series E°° vi converges in X if E°° IIvill < oo. Hint: Show that the partial sum is a Cauchy sequence. Step 2. For Unbounded Region S?.

Given any S > 0, since f E Wk-P(d2), there exist R > 0, such that Ilfllwk.P(o\Ba_3(o)) yl in Y. Then Ilyi

you = urn IITx( -TxiII S

nil = 0.

Now, for x c D, define Tx = Tx; and for other x E X, define Tx by (1.25). Obviously, T is a linear operator. Moreover

IITxoII = lim ITx:Ij < ilim IITIIIIxiIl = IITIIIIxoll. Hence T is also bounded. This completes the proof.

1-1

As a direct application of this Operator Completion Theorem, we prove the following Extension Theorem. Theorem 1.3.6 Assume that S2 is bounded and OS? is G. Then for any open W0.P(0) set 0 D (2, there exists a bounded linear operator E : Wk,P(S2) --> such that Eu = it a. e. in Q.

Proof. To define the extension operator E, we first work on functions it E Ck(Q). Then we can apply the Operator Completion Theorem to extend E to Wk.P(S2), since C'(.(2) is dense in Wk,P(0) (see Theorem 1.3.3). From this density, it is easy to show that, for it E Wk,P(0),

E(u)(x) = u(x) almost everywhere on d2 based on

E(u)(x) =u(x) on ,2 Vu E Ck(S2). Now we prove the theorem for u E Ck(Q). We define E(u) in the following two steps. Step 1. The special case when S2 is a half ball

Br(0):=Ix

0}.

Assume a E Ck(B, (0)). We extend it to be a Ck function on the whole ball Br(0). Define

1.3 Approximation by Smooth Functions

21

xn > 0 u(x,xn)_ u(x',xn), E 0aiu(x',-Aixn),xn 0, such that Bro(0) CC 4'(U), Dyo :_ ¢-1(Bro(0)) is an open set containing x°, and 95 E Ck(Dyo). Choose r° sufficiently small so that Dz0 C O. All such Dyo and U forms an open covering of U, hence there exists a finite sub-covering

Do := U, Dl, ..., D. and a corresponding partition of unity ,rlm,

770,771,...

such that

77iEC0 (Di) i=0,1,--- ,m and

E17i(x) = 1 V X E Q. i=o

Let 4'i be the mapping associated with Di as described above, and let iii(y) = u(¢ 1(y) be the function defined on the half ball Br (0) = Wi (Di n fl)

,

i = 1, 2, - - m. -

From Step 1, each ui(y) can be extended as a Ck function ui(y) onto the whole ball Br, (0).

1 Introduction to Sobolev Spaces

22

We can now define the extension of it from .f2 to its neighborhood 0 as E(u)

_

rji(x)ui(6i(x)) + rfou(x).

i.1

Obviously, E(u) E Co (O), and

«llullwk,ylal' Moreover, for any x E .(2, we have

E(u)(x)

- i=1 1Ji(x)tii(01(x)) +'7o(x)u(x) _i=1Eij (x)u'i(0i(x)) +no(x)u(x) m

>n

E??i(x)ti'(q

- i=1

1(0i(x))) +7]o(x)u(x) = Y:gi(x)u(x) +77o(x)u(x) i=1

(t) 71'(x) = n(x). This completes the proof of the Extension Theorem.

Remark 1.3.2 Notice that the Extension Theorem actually implies an improved version of the Approximation Theorem under stronger assumption on 80 (be Ck instead of C1). From the Extension Theorem, one can immediately derive:

Corollary 1.3.1 Assume that .f2 is bounded and 80 is Ck. Let 0 be an open set containing D. Let 0E :_ {x E Rn I dist(x, 0) < e}. Then the linear operator JE(E(u)) : Wk,P(n)-4Co (O-*O ,

as - 0.

Here, as compared to the previous approximation theorems, the improvement is that one can write out the explicit form of the approximation.

1.4 Sobolev Embeddings When we seek weak solutions of partial differential equations, we start with functions in a Sobolev space W°,P. Naturally, we would like to know whether

1.4 Sobolev Embeddings

23

or not the functions in this space also automatically belong to some other spaces. The following theorem answers the question and at the same time provides inequalities among the relevant norms.

Theorem 1.4.1 (General Sobolev Inequalities). Assume .(2 is bounded and has a Cl boundary. Let it E Wk,P(S2). (i) If k < p, then it E L7(f2) with 4 = n - f-a and there exists a constant C such that IIuIIL9(a) p, then it E

Ck-[pl

GIIuIIWi,P(n)

r'7(Sl), and there exists a constant C, such

that 1lullck_Iyi_1.7(71) 0 ask -> oc. Applying Theorem 1.4.2, we obtain IIu=-njIILP*(Rn)SCIIRi-ujIIw=.P(Rn)->0,

asi,,j

oo.

Thus {uk} also converges to u in Lp*(R"). Consequently, we arrive at IIuIILP'(R') =limIIUkIILP'(Rt) 5limCllukIIW ,P(Rr)

This completes the proof of the Corollary.

The Proof of Theorem 1.4.3. Now for functions in W1'p(.Q), to apply inequality (1.27), we first extend them to be functions with compact supports in R. More precisely, let 0 be an open set that covers I?. By the Extension Theorem (Theorem 1.3.6), for every u in W1'P(.Q), there exists a function u in Wd"3(O), such that

u = u, almost everywhere in [2; moreover, there exists a constant Cl = Cr (p, n, .f2, O), such that IIRIIWI,P(O) 5 C]IIu1IW1.P(n).

(1.30)

Now we can apply the Gagliardo-Nirenberg-Sobolev inequality to u to derive IIkIILP'(U) < IIRIILP'(0) 5 CIIuIIw1.P(o) 5 CC,IIUIIWI.P(n).

This completes the proof of the Theorem.

The Proof of Theorem 1.4.2. We first establish the inequality for p = 1, i.e., we prove

\JRn

LRn

IDulda.

(1.31)

Then we will apply (1.31) to Iul' for a properly chosen y > 1 to extend the inequality to the case when p > 1. We need

1.4 Sobolev Embeddings

27

Lemma 1.4.1 (General Holder Inequality). Assume that u; E LP' (Q) for i = 1, 2, ... , m. and

1

1

1

P1

P2

Pm

Then

Ja

Iu1u2... umldx
1 and 12 is bounded. Now, by Lemma 1.5.1, there is a subsequence (still denoted by {uk}) that converges strongly to uo in L1(Q). Applying the Holder inequality link - UOIILc(n) v(a)

u(c) is the L2 inner product. where < Consequently, IIT9IIp=

sup 0, such that

2.2 Weak Solutions

45

n

E

j > bj£12

i,9=1

for almost every x E 2 and for all E Rn. In the simplest case when ai5 (x) = bi bi (x) 0, and c(x) 0, L reduces to -0. And, as we will see later, the solutions of the general second order elliptic equation Lu = 0 share many similar properties with the harmonic functions.

2.2 Weak Solutions Let the operator L be in the divergence form as defined in (2.1). Consider the second order elliptic equation with Dirichlet boundary condition

JLu =.f,xEQ,

(2.3)

U-0, xE8f2.

When f = f (x), the equation is linear; and when f = f (x, u), semi-linear. Assume that f (x) is in L2 (2), or for a given solution u, f (x, u) is in L2(2). Multiplying both sides of the equation by a test function v E Co (.2), and integrating by parts on ,fl, we have aii(x)uy:vz; +I: bi(x)us;v + c(x)uv I dx = Jn fvdx.

Jn

(2.4)

J

i_1

9=l

There are no boundary terms because both u and v vanish on 499. One can see that the integrals in (2.4) are well defined if u, v, and their first derivatives are square integrable. Write Hl(S2) = W1,2(2), and let H01(d2) be the completion

of Cp (,fl) in the norm of H'(2): 1/2

hull =

Dub2

LJa(I

+ IuI2)dx]

Then (2.4) remains true for any v E Ho (,2). One can easily see that Hp (f2) is also the most appropriate space for u. On the other hand, if u E Hu (0) satisfies (2.4) and is second order differentiable, then through integration by parts, we have

(Lu - f)vdx = 0, Vv E Ha (.2). 12

This implies that

Lu=f, VxEQ,

and therefore u is a classical solution of (2.3). The above observation naturally leads to the following:

46

2 Existence of Weak Solutions

Definition 2.2.1 The weak solution of problem (2.3) is a function u E Ho (.2) that satisfies (2.4) for all v E HH(Q).

Remark 2.2.1 Actually, here the condition on f can be relaxed a little. By the Sobolev embedding

H'(r) - L.'-2 (9), we see that v is in L ^a (,2), and hence by duality, f only needs to be in L +_ (2). In the semi-linear case, if f (x, u) = up for any p < generally,

nu

- 222 , or more

z

f(x,u) < Cr + C2IuI

then for anyu E H1(fl), f(x,u) is in L+(.2).

2.3 Methods of Linear Functional Analysis 2.3.1 Linear Equations For the Dirichlet problem with linear equation

Lu=f(x),xE.Q xEafl, u=0,

(2.5)

to find its weak solutions, we can apply representation type theorems in Linear Functional Analysis, such as the Riesz Representation Theorem or the LaxMilgram Theorem. To this end, we introduce the bilinear form r

n

B[u,v] =J

n

aij(x)ux,v., +bi(x)uz,v+c(x)uv I dx i,j=1

%=1

1

defined on Ho(2), which is the left hand side of (2.4) in the definition of the weak solutions, while the right hand side of (2.4) may be regarded as a linear functional on Ho (2):

< f,v>:= J fvdx. Now, to find a weak solution of (2.5), it is equivalent to show that there exists a function u E HO '(9), such that B[u, v] =< f, v >, V v E Hp (fl).

(2.6)

This can be realized by representation type theorems in Functional Analysis.

2.3 Methods of Linear Functional Analysis

47

2.3.2 Some Basic Principles in Functional Analysis Here we list briefly some basic definitions and principles of functional analysis, which will be used to obtain the existence of weak solutions. For more details, please see, for example, [Schl].

Definition 2.3.1 A Banach space X is a complete, normed linear space with norm 11

-

11 that satisfies

(1) Ilu+vII < IIuII + 1lvII, Vu,v E X, (ii) IIAull _ JAIVIuII, Vu, v E X,AE R,

(iii) hull = 0 if and only if u = 0. The examples of Banach spaces are: (i) LP(fl) with norm JuIIyn(n), (ii) Sobolev spaces Wk,P(fl) with norm lluIIwk,v(n), and (iii) Holder spaces C°''r(Q) with norm lluhlcon(n).

Definition 2.3.2 A Hilbert space H is a Banach space endowed with an inner product (., .) which generates the norm (u,u)112

IIuIJ

with the following properties (i) (u, v) = (v, u), Vu, v E H,

(ii) the mapping u H (u, v) is linear for each v E H, (iii) (u, u) > 0, Vu E H, and (iv) (u, u) = 0 if and only if u = 0. Examples of Hilbert spaces are (i) L2(fl) with inner product u(x)v(x)dx,

(u,v) =

J

and

(ii) Sobolev spaces H1(0) or Ho (fl) with inner product (u, v) =

J [u(x)v(x) + Du(x) . Dv(x)]dx.

Definition 2.3.3 (i) A mapping A : X-->Y is a linear operator if A[au + bv] = aA[u] + bA[u] V u, v E X, a, b E R1.

2 Existence of Weak Solutions

48

(ii) A linear operator A : X-+Y is bounded provided IIAII :=

sup

IIAully < oo.

IuIIx= (u, v), Vv E H. The mapping u* -+ it is a linear isomorphism from H* onto H.

2.3 Methods of Linear Tinctional Analysis

49

This is a well-known theorem in functional analysis. The reader may see the book [Schl] for its proof or do it as the following exercise.

Exercise 2.3.2 Prove the Riesz Representation Theorem in 4 steps. Let T be a linear operator from H to Rl and T # 0. Show that i) K(T) := {u E H I Tu = 0} is closed. ii) H = K(T) (D K(T)1. iii) The codimension of K(T) is 1. iv) There exists v E K(T)1, such that To = 1. Then

Tv=(u,v), VuEH. From Riesz Representation Theorem, one can derive the following:

Theorem 2.3.2 (Lax-Milgram Theorem). Let H be a real Hilbert space with norm 11

-

11. Let

B:HxH-rR1

be a bilinear mapping. Assume that there exist constants M, m > 0, such that IB[u,vlj < MIlullIIvII, Vu,v E H rnIIuII2 < B[u,u], Vu E H.

Then for each bounded linear functional f on H, there exists a unique element u E H, such that B[u, v] =< f, v >, Vv E H. Proof. 1. If B[u,v] is symmetric, i.e., B[u, v] = B[v,u], then the conditions

(i) and (ii) ensure that B[u,v] can be made an inner product in H, and the conclusion of the theorem follows directly from the Riesz Representation Theorem.

2. In the case B[u, v] may not be symmetric, we proceed as follows.

On the one hand, by the Riesz Representation Theorem, for a given bounded linear functional f on H, there is an f E H, such that

=(f,v), VvEH.

(2.8)

On the other hand, for each fixed element is E H, by condition (i), B[u, ] is also a bounded linear functional on H, and hence there exist a u E H, such that B[u, v] = (u, v), Vv E H. (2.9)

50

2 Existence of Weak Solutions

Denote this mapping from it to ii by A, i.e.,

A : H-+H, Au = w.

(2.10)

From (2.8), (2.9), and (2.10), it suffices to show that for each f E H, there exists a unique element it E H, such that Au

f.

We carry this out in two parts. In part (a) we show that A is a bounded linear operator, and in part (b) we prove that it is one-to-one and onto. (a) For any u1, u2 E H, any al, a2 E Rl, and each v E H, we have

(A(alul + a2u2), v) = B[(alul + a2u2), v] = a1 B [ul, v] + a2B [u2, v]

= a,(Aul,v) + a2(Au2,v) = (aiAul +a2Au2,v) Hence

A(alul + a2u2) = a1Au1 + a2Au2. A is linear. Moreover, by condition (i), for any it e H, IIAuII2 = (Au, Au) = B[u, Au] < MIIuII IIAuII,

and consequently IlAull 5 MIIuII Vu E H.

(2.11)

This verifies that A is bounded. (b) To see that A is one-to-one, we apply condition (ii): MIIuII2 = 0 Yv E X. iii) We call J'(u) = 0 the Euler-Lagrange equation of the functional J.

Remark 2.4.1 One can verify that, if J is Frechet differentiable at it, then

limoJ(u+tv) -J(u) <J'(u),v>= t-+ t

=

dtJ(u+tv)

It=o.

In the previous subsection we observed that the critical points of the functional

J(u) = 1 2

Jn

IDuI2dx - L f (x)udx n

are weak solutions of the Dirichlet problem (2.15). Our next question is this: Does the functional J actually possess a critical point? We will show that there exists a minimizer of J. To this end, we verify that J possesses the following three properties: i) boundedness from below, ii) coercivity , and iii) weak lower semi-continuity. i) Boundedness from Below.

First we prove that J is bounded from below in H Ho (dl) if f E L2 (Q). Again, for simplicity, we use the equivalent norm in H:

2 Existence of Weak Solutions

56

Dul2dx.

IIuIIH

fn

By the Poincare and Holder inequalities, we have

J(u) ?

2IIUIIH

- CIIUIIHIIfIIL2(Q) 2

= 2 (IIuIIH-CIIfIIL2(a))2-

-2

If 11L2(f2)

2

(2.17)

2 IIfIIL2(n)

Therefore, the functional J is bounded from below. ii) Coercivity. However, a functional that is bounded from below does not guarantee that it has a minimum. A simple counterexample is

g(x)=1+x2, xERr. Obviously, the infimum of g(x) is 0, but it can never be attained. If we take a minimizing sequence {xk} such that g(xk)->0, then xk goes away to infinity. This suggests that in order to prevent a minimizing sequence from "leaking" to infinity, we need to have some sort of "coercive" condition on our functional J. That is, if a sequence {uk} goes to infinity, i.e., if IIuIIH--boo, then J(uk) must also become unbounded. Therefore, a minimizing sequence would be retained in a bounded set. And this is indeed true for our functional J. Actually, from the second part of (2.17), one can see that if IIukIIH-*00, then

This implies that any minimizing sequence must be bounded in H. iii) Weak Lower Semi-continuity. If H is a finite dimensional space, then a bounded minimizing sequence {uk} would possesses a convergent subsequence,

and the limit would be a minimizer of J. This is no longer true in infinite dimensional spaces. We therefore turn our attention to weak topology.

Definition 2.4.2 Let X be a real Banach space. If a sequence NO C X satisfies

< f, uk > -r < f, uo > as k-roo for every bounded linear functional f on X, we say that {uk} converges weakly

to uo in X and write uk

U0.

Definition 2.4.3 A Banach space is called reflexive if (X*)* = X, where X* is the dual of X.

2.4 Variational Methods

57

Theorem 2.4.1 (Weak Compactness.) Let X be a reflexive Banach space. Assume that the sequence {uk} is bounded in X. Then there exists a subsequence of {uk}, which converges weakly in X.

Let us now return to our Hilbert space H := Ho (,2). By the Riesz Representation Theorem, for every linear functional f on H, there is a unique v E H, such that < f, u >= (v, u), Vu E H. It follows that H* = H, and therefore H is reflexive. By the Coercivity and Weak Compactness Theorem, now a minimizing sequence {uk} is bounded and hence possesses a subsequence {uk, } that converges weakly to an element U. in H: (uku v)-Y(uo, v), Vv E H.

Naturally, we wish the functional J we consider to be continuous with respect to this weak convergence, that is,

llim. J(uk.) = J(uo), then uo would be our desired minimizer. However, this is not the case with our

functional, nor is it true for most other functionals of interest. Fortunately, we do not really need such a continuity but instead need only a weaker one:

J(uo) < l minf J(uk,). Since on the other hand, by the definition of a minimizing sequence, we obviously have

J(u0) >

J(uk,),

and hence

J(uo) = lief J(uk; ) uo is therefore a minimizer.

Definition 2.4.4 We say that a functional J() is weakly lower semicontinuous on a Banach space X, if for every weakly convergent sequence

uk - U. ,

in X,

we have

ki.f J(u0) < liM

J(uk)

Now we show that our functional J is weakly lower semi-continuous in H. Assume that {uk} C H, and Uk - uo

in H.

58

2 Existence of Weak Solutions

Since for each fixed f E L +2- (12), fn fudx is a linear functional on H, it follows immediately

f

r fukdx-+

fuodx,

as k->oo.

(2.18)

Jn

Jn

From the algebraic inequality a2 + b2 > 2ab, we have

IDuk12dx>J IDu0I2dx+2J Du,-(Duk-Duo)dx. Jf n

n

n

Here the second term on the right goes to zero due to the weak convergence Uk -' uo. Therefore liminf fn IDukI2dx > Jn IDuo12dx. k-4w

(2.19)

Now (2.18) and (2.19) imply the weakly lower semi-continuity. So far, we have proved the following:

Theorem 2.4.2 Assume that ,12 is a bounded domain with smooth boundary. Then for every f e L +_ (d2) with n > 2, the functional

J(u) = 2 L IDuI2dx - f fudx n

n

possesses a minimum uo in Ho (fl), which is a weak solution of the boundary value problem

'Lu=f(x),xE12 1U(X) = 0,

x E 4912.

2.4.4 Existence of Minimizers Under Constraints Now we consider the semi-linear Dirichlet problem

Du = uj' 1u, x E 57, u(x) = 0,

(2.20)

x E 49n,

with 1g(x, u(x)) belongs to C(L'(.Q), Ls(0)). Proof. By (92),

f

n

Ig(x,u(x))I'dx


In

(2.35)

I¢rrdx > mrIQ2I.

?

Since it E Lr(,f2), we can make ffla JuIrdx as small as we wish by letting 10221 be small. Now by (2.35), we can choose 6 so small, such that 1021 is sufficiently small, and hence the right hand side of (2.34) is less than (2)8. Consequently, by (2.33), for such a small 6,

I1+I2 3. ThenI eC'(H,R/) and < 1'(U), V >= / f (x, u)vdx, Vv E H. n is weakly continuous and I'() is compact, i.e.,

Moreover,

I(uk)-rI(u) and I'(uk)-*I'(u), whenever uk - it in H.

Proof. We will first show that I is Frechet differentiable on H and then prove that F(u) is continuous. 1. Let u,v E H. We want to show that given any c > 0, there exists a

6 = 6(e, u), such that

Q(u, v) := II(u+v) -I(U) -

r

Jn

f (x, u)vdxI < ellvii

(2.36)

whenever iIvII < J. Here evil denotes the H'(12) norm of v. In fact, Q(u,v)

IF(x,u(x) + v(x)) - F(x,u(x)) - f(x,u(x))v(x)ldx

a

=

Jn

If(x,u(x) + e(x)) - f(x,u(x))I.Iv(x)Idx

If(-,u+e)- f(, u)I1L

(.R)HullL

,(f2JKIIvII.

(Q)

(2.37)

72

2 Existence of Weak Solutions

Here we have applied the Mean Value Theorem (with Ii;(x)I < Iv(x)I), the Holder inequality, and the Sobolev inequality IIvIIL

W)

< KIIvII-

In Proposition 2.4.1, let r = n^2^2 and s = +2. Then by (f2), we deduce that the map 'a(x)-+f(x, u(x))

is continuous from L- (2) to L +2 (fl). Hence for any given e > 0, we can choose sufficiently small S > 0, such that whenever IIvII < 8, we have < KS, and hence IIeiiL; (Q) < KS, and therefore IIV IIL "z (fl)

(0)
ppJ(Uk)- < J'(uk),uk >

µ 2

- 1)

IDuk I2dx +

fn

J

[f(x, uk)uk - jLF(x, Uk)1 dx

IDukI2dx+J 2

> (2

1)

= CoIIUk

II2-Ca,

Jn

[f(x, uk)uk- AF(x, Uk)] dx Iukfr)I= J Du Dvdx. n

Consequently,

A-'J'(u) = u - A-'f (-, u).

(2.44)

From Proposition 2.4.2,

f (, uk)-> f (., u) strongly in H*. It follows that

uk = A-'J'(uk)+A-'f(.,uk)

as k-roo.

This verifies the (PS). To see there is a "mountain range" surrounding the origin, we estimate JJ F(x, u)dx. By (f3), for any given e > 0, there is a 6 > 0, such that, for all x E ,fl,

IF(x, s)I < elsl2, whenever Isl < 5.

(2.45)

2.4 Variational Methods

75

While by (f2), there is a constant M = M(6), such that for all x E S2, IF(x, s)I < MIsIr+r

(2.46)

Combining (2.45) and (2.46), we have, IF(x, s)I < EISI2 +MIsI1+l,

for all x E £2 and for all s E R.

It follows, via the Poincare and the Sobolev inequality, that

f F(x,u)dxI < E

L u2dx+M f

lul"+1dX < C(E+MIIuIIp-1)IIuII2 (2.47)

And consequently,

J(u) > [2 - C(E+MIIuhIp-1)]

(2.48)

IIuPI2.

Choose c = &C. For this E, we fix M; then choose IIull = p small, so that

CMPP-r < $. Then from (2.48), we deduce JIaB,(o) ?

P2

To see the existence of a point e E H, such that J(e) < 0, we fix any u $ 0 in H, and consider IDuI2dx -

J(tu) = 2

n in

J

F(x,tu)dx.

(2.49)

By (f4), we have for IsI > r, d

ds)){>0

ifs r,

F(x, s) > a3sI"; and it follows that

F(x,s) > a3IsV" -a4-

(2.50)

Combining (2.49) and (2.50), and taking into account that p > 2, we conclude

that

J(tu)i - co, as t-*co. Now J satisfies all the conditions in the Mountain Pass Theorem, hence it possesses a mini-max critical point u,, which is a weak solution of (2.31). Moreover, J(uo) > 0, therefore it is a nontrivial solution. This completes the proof of the Theorem 2.4.6.

3

Regularity of Solutions

3.1

W2,r a priori Estimates 3.1.1

3.1.2 3,2

Newtonian Potentials Uniform Elliptic Equations

W2.v Regularity of Solutions 3.2.1

3.2.2 3.2.3

The Case p > 2 The Case 1 < p < 2 Other Useful Results Concerning the Existence, Uniqueness, and

Regularity 3.3

Regularity Lifting 3.3.1

3.3.2 3.3.3

3.3.4 3:3.5

Bootstraps Regularity Lifting by Contracting Operators Applications to PDEs Applications to Integral Equations Regularity Lifting by Combinations of Contracting and Shrinking

Operators 3.3.6 3.3.7

Applications to Integral Equations Applications to Fully Nonlinear Systems of Wolff Type

In the previous chapter, we used functional analysis, mainly the calculus of variations, to seek the existence of weak solutions for second order linear or semi-linear elliptic equations. These weak solutions we obtained were in the Sobolev space W1,2 and hence, by Sobolev embedding, in LP space for p < 2. Usually, the weak solutions are easier to obtain than the classical ones, and this is particularly true for non-linear equations. However, in practice, we

78

3 Regularity of Solutions

are often required to find classical solutions. Therefore, one would naturally want to know whether these weak solutions are actually differentiable, so that they can become classical ones. These questions will be answered here. In this chapter, we will introduce methods to show that, in most cases, a weak solution is in fact smooth. This is called the regularity argument. Very often, the regularity is equivalent to the a priori, estimate of solutions. To see the difference between the regularity and the a priori estimate, we take the equation

-Au = f (x) for example. Roughly speaking, the W2,P regularity theory implies that If u E W1"p is a weak solution and if f E LP, then u E W2,p.

While the W2'P a priori estimate says If u c Wo'p n W2,p is a weak solution and if f E LP, then IIUIIW=.n G CIIfIILP.

In the a priori estimate, we presumed that it is in W2'P. It might seem a bit surprising at this moment that the a priori estimates turn out to be powerful tools in deriving regularities. The readers will see in Section 3.2 how the a priori estimates and uniqueness of solutions lead to the regularity. Let I? be an open bounded set in Rn. Let aid (x), bi(x), and c(x) be bounded functions on .fl with aij (x) = aji(x). Consider the second order partial differential equation n

n

Lu :_ - E ai1(x)us;x; + E bi(x)uy, + c(x)u = f(x). i,)=1

(3.1)

i=1

We assume L is uniformly elliptic, that is, there exists a constant S > 0, such that n

E atj(x)ei£9 > ERn. In Section 3.1, we establish W2,p a priori estimate for the solutions of 6

for all

(3.1). We show that if the function f (x) is in L"(92), and it c W2'p(.f2) is a strong solution of (3.1), then there exists a constant C, such that IIuIIW?..P ')1. By (iii) in the Calderon-Zygmund Decomposition Lemma, we have

(3.16)

3.1 W2'' a priori Estimates 0o

k =1

f

00

co

Qkl-
n.

Proof of Theorem 3.1.2. We divide the proof into two major parts. Part 1. Interior Estimate 11o2UIILP(K) 6}.

Combining the interior and the boundary estimates, we obtain

IIu1IW2.P(.n) S

C(IIVuIILP(n) +I IUIILP(O) +I If IILP(n)).

(3.25)

Theorem 3.1.2 is then a simple consequence of (3.25) and the following well-known Gargaliado-John-Nirenberg type interpolation estimate

IIVUIILP(n) S CIIuIIi (n)Ilo2ullL (n) S EI102uIILP(n) + 4EIIu11LP(n). Substituting this estimate of I IVUI I LP(n) into our inequality (3.25), we get: IIUIIW2.P(n) 0}. The equation becomes

ya -1 aii (y)'av:v, + z=1 bi(y)nv:

{u(y)=0, foryEBB,+(0)with

c(y) a =

1(y), for y E Br (0) (3.31)

where the new coefficients are computed from the original ones via the chain rule of differentiation. For example,

aij(y) =

8xt('

If necessary, we make a linear transformation so that ai5(0) = hip. Since a plane is still mapped to a plane, we may assume that equation (3.31) is valid for some smaller r. Applying the method of frozen coefficients (with vl).u(y)) we get: w(y) = 0(2 Aw(y) = F(y) on Br (0). Now let w(y) and F(y) be the odd extension of w(y) and F(y) from B,: (0) to B,.(0), i.e., 'D(y)=w(y1,...,Yn-1,yn)w(i,

Similarly for F(y). Then one can check that:

,, Yyni,Yn),

-Y.), if V, < 0.

3.2 W2'P Regularity of Solutions

91

Lw(y) = F(y) , for y E B,(0). Through the same calculations as in Part 1, we derive the basic interior estimate II 02 uIILP(B,(s°)) < C(II fII LP(B2,(s°)) + I I VUIILP(B,r(z°)) +

< Cl(IIfIILP(B,, (xo)nn) +

IIUIILP(B2r(z°)))

Ilnl ILP(n2r(so)nn))

for any x° on 82 and for some small radius r. The last inequality in the above was due to the symmetric extension of w to w from the half ball to the whole ball.

These balls form a covering of the compact set 81 and thus have a finite covering Br; (x') (i = 1, 2. ..... k). These balls also cover a neighborhood of 8.12

including .2\05 for some 5 small. Summing the estimates up, and following the same steps as we did for the interior estimates, we get:

IIUIIW2.P(n\na) < C(IIouIILP(n) +I U11LP(Q) +I If IILP(n)).

(3.32)

This establishes the boundary estimates and hence completes the proof of the theorem.

Exercise 3.1.1 Assume that B is bounded, and w E WW ''(Q). Show that if

-Aw = F, then w = F * F. Hint: (a) Show that r * F(x) = 0 for x 0 (2. (b) Show that it is true for w E Co (Rn). (c) Show that

A(JEW) = JE(Aw)

and thus

J,w = F * (JF). Let fi0 to derive w = F * F.

3.2 W2" Regularity of Solutions In this section, we will use the a priori estimates established in the previous section to derive the W2'p regularity for the weak solutions. Let L be a second order uniformly elliptic operator in divergence form n

n

Lu = -

bi(x)u1, + c(x)u.

(a,j (x)us,)y. +

i,j=1

i=1

(3.33)

3 Regularity of Solutions

92

Definition 3.2.1 We say that u E WJ'P(fl) is a weak solution of

Lu=f(x),xE0

{u(x)=0, xEafl

(3.34)

if for any t E- W01, (p),

I J

>bi(x)uxiv

Y

aij(x)ux,vi, +

+ c(x)uv dx = J f(x)vdx,

(3.35)

where 1 + q = 1.

Remark 3.2.1 Since Co (.2) is dense in WJ'P((l), we only need to require (3.35) be true for all v E Co (fl); and this is more convenient in many applications.

The main result of the section is this:

Theorem 3.2.1 Assume fl is a bounded domain in R'°. Let L be a uniformly elliptic operator defined in (3.33) with aij(x) Lipschitz continuous and bi(x) and c(x) bounded. Assume that u E WJ'P(Q) is a weak solution of (3.34). If f(x) e LP (S?), then u E W2'P(Q) .

Outline of the Proof. The proof of the theorem is quite complex and will be accomplished in two stages.

In stage 1, we first consider the case when p > 2, because one can rather easily show the uniqueness of the weak solutions by multiplying both sides of the equation by the solution itself and integrating by parts. As the reader will see, this uniqueness plays a key role in deriving the regularity. The proof of the regularity is based on the following fundamental proposition on the existence, uniqueness, and regularity for the solution of the Laplace equation on a unit ball.

Proposition 3.2.1 Assume f E LP(Bl(0)) with p > 2. Then the Dirichlet problem

J u = f (x) , x E Bl (0) l u(x) = 0 ,

x E aBr(0)

(3.36)

has a unique solution u E W2"P(Bt(0)) satisfying IIuIIW=.P(B,) «IIlIILP(B1)-

(3.37)

Regularity of Solutions

3.2

93

We will prove this proposition in subsection 3.2.1, and then use the "frozen coefficients" method to derive the regularity for general operator L.

In stage 2 (subsection 3.2.2), we consider the case 1 < p < 2. The main difficulty lies in the uniqueness of the weak solution. To show the uniqueness, we first prove a W2,' version of the Redholm Alternative for p > 2 based on the regularity result in stage 1, which will be used to deduce the existence of solutions for equation

- E(aii(x)ws:)x; = F(x). ij

Then we use this equation with a proper choice of F(x) to derive a contradiction if there exists a non-trivial solution of equation

- E(aij(x)vs,)s,

= 0.

23

After proving the uniqueness, to derive the regularity, everything else is the same as in stage 1. Remark 3.2.2 Actually, the conditions on the coefficients of L can be weakened. This will use the Regularity Lifting Theorem in Section 3.3, and hence will be elaborated there.

3.2.1 The Case p > 2 We will prove Proposition 3.2.1 and use it to derive the regularity of weak solutions for general operator L. To this end, we need

Lemma 3.2.1 (Better A Priori Estimates at Presence of Uniqueness.) Assume that

i) for solutions of

Lu = f (x) , x E ,2

(3.38)

lull W2(Q) 2, we have proved the uniqueness. Now assume 1 < p < 2. Suppose there exists a non-trivial weak solution u. Let {uk} be a sequence of Co (0) functions such that

Uk-fl in W(, "(.fl)

,

as k-ioo.

For each k, consider the equation IVUklp/q'7U,k

(aij(x)v.,)xj = Fk(x) := 0

1+IVUkl2

(3.53)

where 1+P1 =1. Obviously, for p < 2, q is greater than 2, and Fk(x) is in Lq(Q). Since we already have uniqueness for W.1" weak solution, Lemma 3.2.2 guarantees the existence of a solution vk for equation (3.53). Through integration by parts several times, one can verify that 0

J vk(aij(x)us.).,dx= n

n

1+Ivukl2

(3.54)

Since uk-#u in W''P, it is easy to see that IDUkIPIgvUk

1+Ivuk12

IVUIP/qVU

1+IVUI2

in Lq(fl), and therefore, by (3.54, we must have vu = 0 almost everywhere. Taking into account that u E W0'P(0), we finally deduce that u = 0 almost everywhere. This completes the proof of the proposition.

100

3 Regularity of Solutions

After proving the uniqueness, the rest of the arguments are entirely the same as in stage 1. This completes stage 2 of the proof for Theorem 3.2.1-

3.2.3 Other Useful Results Concerning the Existence, Uniqueness, and Regularity Theorem 3.2.2 Given any pair p, q > 1 and f E L9(Il), if u E WW'P (.fl) is a weak solution of

Lu = f (x) , x E ,(l, then u E WZ^Q(fl) and the a priori estimate

(3.55)

(3.56)

Ilullw2,a(n) q. El From this Theorem, we can derive immediately the equivalence between the uniqueness of weak solutions in any two spaces Wo'p and W0I'4:

Corollary 3.2.1 For any pair p, q > 1, the following are equivalent i) If u (= Wo'"(fl) is a weak solution of Lu = 0, then u = 0. ii) If u e WJ'°(fl) is a weak solution of Lu = 0, then u = 0. Theorem 3.2.3 (W2. p Version of the Fredholm Alternative). Let 1 < p < co. If u = 0 whenever Lu = 0 (in the sense of Wo'"(0) weak solution), then for any f E L"(2), there exist a unique u E Wo'"(,fl) fl W2'"(2), such that

Lu = f(x) and IIuIIwa.P(n)
n, i.e., if

(1 -6)[n+2-6(n- 2)] - 4 n

-a

,

IK(x)I < M, and u E L a (R"),

(3.71)

then conditions (3.69) and (3.70) are satisfied. ii) The last condition in (3.71) is somewhat sharp in the sense that if it is violated, then equation (3.68) when K(x) - 1 possesses singular solutions such as

c

u(x) =

IxI

Proof of Theorem 3.3.4: Define the linear operator

f K(y)Iv(y)IP-wdy yI"-a JRT

Ix -

For any real number a > 0, define

f u"(x) = u(x) l ua(x) = 0 ,

,

if Iu(x)I > a, or if Ixl > a otherwise

.

Let ub(x) = u(x) - ua(x).

Then since it satisfies equation (3.68), one can verify that ua satisfies the equation U. = Tuaua + g(x)

(3.72)

with the function g(x) =

f JRn

K(y)Iub(y)IP-lub(y)dy

Ix - yI"-o

- ub(x)-

Under the second part of the condition (3.70), it is obvious that g(x) E L°° fl Lq°.

For any q > "n to obtain

we first apply the Hardy-Littlewood-Sobolev inequality

3.3 Regularity Lifting

IITu°tIILq nn-. From here, one can derive immediately that

Theorem 3.3.6 Every positive solution of equation (3.81) is uniformly bounded in Rn.

Proof. Divide the integral into two parts u(x)

= J B, (z) Ix - yIn-°

u' (y)dy +

1

JRn\B, (z) I x - YIn-IX

u' (y)dy

I1+I2. Use the Holder inequality on the first part

r/ Il < I /

1/q

11/p

1n_a)pdy1

u74(y)dyI

[Ll(X)

(3.82)

with 1/p + 1/q = 1. Choosing 1 < p < n=;, both integrals in (3.82) are uniformly bounded.

For the second part, we simply have

116

3 Regularity of Solutions

uT(y)dy 1, q > 1, and 1 < 7 < 2. Then (u, v) is uniformly bounded. Based on this proposition, [CLM] used Regularity Lifting Theorem II to lift the positive solution pair from L°°(R") to C°-1(R").

Theorem 3.3.8 Under the same conditions as in Proposition 3.3.1, (u, v) is Lipschitz continuous.

Due to the complexity of the estimates, we only present the idea and outline of the proof here. Interested readers can find the details in [CLM].

Outline of the Proof of Theorem 3.3.8. Fix the positive solution pair (u, v). Let 11 = {g E L°°(R") : II91ILw < 2IIuhILoc +2IIvIILm} and

Q) = {9 E C°''(R") : II9IIL- < 2IIuJIL= + 2IIvIIL°°},

3.3 Regularity Lifting

121

is L°°(R") norm of g. One can see that X and 2J are closed subsets of Banach spaces equipped with L°°(R'") norm and Lipschitz norm where 1191IL°°

IgllC°,,(R..) respectively.

Define on X and r

fa

tn-7A

)

r (fB±()v(z)dz\1 t^'-7T, dt

and

(fB, (y) v,"(z) dz "-' dt

G(x) c

to-7Q

)

t

Equip the product X x X with norm II (f, 9) II Lm = IIf IIL°° + 11911L00

and 2,J x 2 with norm 11Y, 011c--= = IIfllco,l +IIgIIco,=.

Consider the equation (f,g) = S(f,g) ° T(f,g) + (F, G)

(3.93)

where T(f, g) = (Tig, T2f). Obviously, (u, v) satisfies equation (3.93). We can prove that (See [CLM]), for sufficiently small e,

(i) T is a contracting operator from X x X to X x X, (ii) T is a shrinking operator from 23 x 2J to Y x Y, and (iii)

F,Gc(XxX)n(2)x23), Then by Regularity Lifting Theorem II, we can conclude that (u, v) is Lipschitz continuous.

preliminary Analysis on Riemannian Manifolds

4.1

Differentiable Manifolds

4.2

Tangent Spaces

4.3

Riemannian Metrics

4.4

Curvature

4.4.1 4.4.2

4.4.3 4.5

Curvature of Plane Curves Curvature of Surfaces in Rs Curvature on Riemannian Manifolds

Calculus on Manifolds 4.5.1

Higher Order Covariant Derivatives and the Laplace-Beltrami Op-

crater

4.6

4.5.2

Integrals

4.5.3

Equations on Prescribing Gaussian and Scalar Curvature

Sobolev Embeddings

According to Albert Einstein, the Universe we live in is a curved space. We call this curved space a manifold, which is a generalization of the flat Euclidean space. Roughly speaking, each neighborhood of a point on a manifold is homeomorphic to an open set in the Euclidean space, and the manifold as

a whole is obtained by pasting together pieces of the Euclidean spaces. In this section, we will introduce the most basic concepts of differentiable manifolds, tangent space, Riemannian metrics, and curvatures. We will try to be as intuitive and brief as possible. For more rigorous arguments and detailed discussions, please see, for example, the book Riemannian Geometry by M. do Carmo [Ca].

124

4 Preliminary Analysis on Riemannian Manifolds

4.1 Differentiable Manifolds A simple example of a two-dimensional differentiable manifold is a regular surface in R3. Intuitively, it is a union of open sets of R2, organized in such a way that at the intersection of any two open sets, the change from one to the other can be made in a differentiable manner. More precisely, we have

Definition 4.1.1 (Differentiable Manifolds) A differentiable n-dimensional manifold M is (i) a union of open sets, M = Ua Va and a family of homeomorphisms ¢a from Vn to an open set Oa(Va) in Rn, such that

(ii) For any pair a and Q with V0 fl V1 = U $ 0, the images of the intersection 0, (U) and Op (U) are open sets in R' and the mappings 013 o ¢Q 1 are differentiable.

(iii) The family { V0, 0,} are maximal relative to the conditions (i) and (ii).

A manifold M is a union of open sets. In each open set U C M, one can define a coordinate chart. In fact, let q5U be the homeomorphism from U to Ou(U) in Rn; for each p E U, if ¢u(p) = (x1, x2, , xn) in Rn, then we can define the coordinates of p to be (x1,x2, .xn). This is called the local coordinates of p, and (U, ¢u) is called a coordinate chart. If a family of coordinates charts that covers M are well made so that the transition from one to the other is in a differentiable way as in condition (ii), then it is called a differentiable structure of M. Given a differentiable structure on M, one can easily complete it into a maximal one by taking the union of all the coordinate charts of M that are compatible with (in the sense of condition (ii)) the ones in the given structure. A trivial example of n-dimensional manifolds is the Euclidean space Rn, with the differentiable structure given by the identity. The following are two non-trivial examples.

Example 4.1. The n-dimensional unit sphere

Sn = {x E Rn+1 I xl +... + xn+1 =

1}.

Take the unit circle S1 for instance, we can use the following four coordinate charts: V1 = {x E S1 I x2 > 0), 01(x) = x1, V2 = {x E S1 I x2 < 0}, ,02(x) = x1,

V3 = {x E S1 I x1 > 0}, 03(x) = x2, V4 = {x E S1 I x1 < 0}, 04(x) = x2.

Obviously, S1 is the union of the four open sets V1, V2, V3, and V4. On the intersection of any two open sets, say on V1 n V3, we have

4.2 Tangent Spaces ' J

}

x2 =

1

x1=

1-x2, x2>0.

125

- x1, x1 > 0;

They are both differentiable functions. The same is true on the other intersections. Therefore, with this differentiable structure, S1 is a 1-dimensional differentiable manifold.

Example 4.2. The n-dimensional real projective space Pn(R). It is the set of straight lines of Rn+1 which pass through the origin (0, . , 0) E Rn+1 or the quotient space of Rn+1 \ {0} by the equivalence relation

(xl...xn+1)-(Axl,...,Axn+1), AER, A$0. We denote the points in Pn(R) by [x]. Observe that if xi # 0, then

[xi...,xn+1] _

[xi/xi,...xi-i/xi,1,xi+1/xi,...xn+1/xi]-

Fori=1,2,- ,n+1, let Vi={[x]Ixi#0}. Apparently,

n+1

Pn(R) = U Vi. i=1

Geometrically, Vi is the set of straight lines of Rn+1 which pass through the origin and which do not belong to the hyperplane xi = 0. Define

¶i([x])=(S1, ..,Si-i' i+1' ,Sn+1),

where k=xk/xi(i#k)and1M, with y(0) = p and y'(0) = v. Let 9 = f o y. Define the mapping dfP : TpM->Tf(P)N by dff(v) = 0'(0).

We call dfP the differential of f. Obviously, it is a linear mapping from one tangent space TPM to the other Tf(P)N; and one can show that it does not

depend on the choice of y (See [Ca]). When N = R1, the collection of all such differentials form the cotangent space. From the definition, one can derive

that

dfP((axi)P) = (axj)Pf' In particular, for the differential (dxi)P of the coordinates function xi, we have

(dxi)P((aax;)) = 6i;. P

And consequently, dfP

of

=

ax a

t

(dxi)P. P

128

4 Preliminary Analysis on Riemannian Manifolds

From here we can see that (dxl)p, (d' 2)p, ... , (dx-)P form a basis of the cotangent space TT M at point p.

Definition 4.2.3 The tangent space of M is

TM = U TM. pEM

And the cotangent space of M is

UTpM.

T'M

pEM

4.3 Riemannian Metrics To measure the arc length, area, or volume on a manifold, we need to introduce some kind of measurements or `metrics'. For a surface S in the three dimensional Euclidean space R3, there is a natural way of measuring the length of vectors tangent to S, which are simply the length of the vectors in the ambient

space R3. These can be expressed in terms of the inner product in R3. Given a curve 7(t) on S for t E [a, b], its length is the integral b

ly (t) ldt,

where the length of the velocity vector y'(t) is given by the inner product in R3:

1ti (t) l =

< y' (t), y'(t) >.

The definition of the inner product enables us to measure not only the lengths of curves on S, but also the area of domains on S, as well as other quantities in geometry. Now we generalize this concept to differentiable manifolds without using the ambient space. More precisely, we have

Definition 4.3.1 A Riemannian metric on a differentiable manifold M is a correspondence which associates to each point p of M an inner product < >p on the tangent space TTM, which varies differentiably, that is, 95.7 (q) =< (C72;)q' (a!j)q >q is a differentiable function of q for all q near p. A differentiable manifold with a given Riemannian metric will be called a Riemannian manifold.

4.3 Riemannian Metrics

129

It is clear that this definition does not depend on the choice of coordinate system. Also one can prove the following (see [Ca]):

Proposition 4.3.1 A differentiable manifold M has a Riemannian metric. Now we show how a Riemannian metric can be used to measure the length of a curve on a manifold M.

Let I be an open interval in R1, and let y : I-+M be a differentiable mapping ( curve) on M. For each t E I, dry is a linear mapping from TtI to dy(dt) is a tangent vector of Try(t)M. The restriction Try(t)M, and y'(t) dt of the curve ry to a closed interval [a, b] C I is called a segment. We define the

length of the segment by b

1/2 dt, Here the are length element is

ds =< y (t),y (t) >1/2 dt.

As we mentioned in the previous section, let (xi, nates in a local chart (U, 0) that covers p = 7(t) (xl (t),

be the coordi,

x,(t)). Then

y(t)=Exe(t)(&) Consequently,

ds2

= < y'(t), ry'(t) >
p xi(t)dtxj(t)dt

W=1

_

gij(p)dxidxj. i,7=1

This expresses the length element ds in terms of the metric gij. To derive the formula for the volume of a region (an open connected subset) Din M in terms of metric gij, let's begin with the volume of the parallelepiped formed by the tangent vectors. Let {el, en} be an orthonormal basis of TPM. Let

(')

aij ej,

Then

gik(p) _
p= E aijaki < ej, el >= jd

aijakj. j

(4.2)

4 Preliminary Analysis on Riemannian Manifolds

130

We know the volume of the parallelepiped formed by as , as is the determinant det(aij), and by virtue of (4.2), it is the same as det(gij). Assume that the region D is covered by a coordinate chart (U,0) with coordinates (xi, - xn). Ftom the above arguments, it is natural to define the volume of D as vol(D) = det(gj)dx1 dx,. J ¢(U)

A trivial example is M = R", the n-dimensional Euclidean space with

a axi

The metric is given by

gij =< ei, ej >= bij. A less trivial example is Sz, the 2-dimensional unit sphere with standard metric, i.e., with the metric inherited from the ambient space R3 = {(x, y, z)}. We will use the usual spherical coordinates (B, 0) with

x=sinBcoso y = sin B sin 3/i

z=cos0 to illustrate how we use the measurement (inner product) in R3 to derive the expression of the standard metric gij in terms of the local coordinate (0,i) on Sz.

Let p be a point on Sz covered by a local coordinate chart (U,0) with coordinates (G4). We first derive the expression for ( ae)P andalp (a) , the tangent vectors at point p = 0-1(0, 0) of the coordinate curves and B = constant respectively.

P

= constant

In the coordinates of R3, we have,

a aB

and

7a

ax ay Oz (aB, aB, aB) =

(cos0cos0,cos0sinsin B),

ax, ay, az

a _ (av, a, a) _ (

sin B sin 3/i,sinecos /i,0).

It follows that gii (P) _< aB aB >P- cosz B cosz

t cost B sinz V) + sinz B = 1,

a

a

80 W

4.4 Curvature and

a

922(p) =
p=

131

sing 0.

Consequently, the length element is

ds =

d02 + sine d ?p2,

g11d02 + 2g12dOdili + g22d1G2 =

and the area element is dA =

det(gjj)dOd

=sinOdOdi/'.

These conform with our knowledge of the length and area in the spherical coordinates.

4.4 Curvature 4.4.1 Curvature of Plane Curves Consider curves on a plane. Intuitively, we feel that a straight line does not bend at all, and a circle of radius one bends more than a circle of radius 2. To measure the extent of bending of a curve, or the amount it deviates from straightness, we introduce curvature. Let p be a point on a curve, and T(p) be the unit tangent vector at p. Let q be a nearby point. Let da be the amount of change of the angle from T(p) to T(q), and As the arc length between p and q on the curve. We define the curvature at p as

Aa k(p) = q-+p lim As

_

ds

(P).

If a plane curve is given parametrically as c(t) _ (x (t), y(t)), then the unit tangent vector at c(t) is

T(t) _

(x'(t),y (ft)) [x'(t)]2 + [y'(t)]2 1

and

ds = [x'(t)]2 + [y'(t)]2dt. By a straightforward calculation, one can verify that y'(t)x"(t)

k(c(t)) = IdTI =

x'(t)y"(t) {[x'(t)]2 + [y'(t)72}3/2

If a plane curve is given explicitly as y = f (x), then in the above formula, replacing t by x and y(t) by f (x), we find that the curvature at point (x, f(x)) is f"(x)

k(x,f(x)) =

{1 + [f'(x)]2}3/2

.

(4.3)

4 Preliminary Analysis on Riemannian Manifolds

132

4.4.2 Curvature of Surfaces in R3 1. Principal Curvature

Let S be a surface in R3, p be a point on S, and N(p) a normal vector of S at p. Given a tangent vector T(p) at p, consider the intersection of the surface with the plane containing T(p) and N(p). This intersection is a plane curve, and its curvature is taken as the absolute value of the normal curvature, which is positive if the curve turns in the same direction as the surface's chosen normal, and negative otherwise. The normal curvature varies as the tangent vector T(p) changes. The maximum and minimum values of the normal curvature at point p are called principal curvatures and are usually denoted by kl(p) and k2(p).

2. Gaussian Curvature. One way to measure the extent of bending of a surface is by Gaussian curvature. It is defined as the product of the two principal curvatures

K(p) = ki(p) . k2(P)

From this, one can see that no matter on which side normal is chosen (say, for a closed surface, one may chose outer normals or inner normals), the Gaussian curvature of a sphere is positive, of a one sheet hyperboloid is negative, and of a torus or a plane is zero. A sphere of radius R has Gaussian curvature R . Let f (x, y) be a differentiable function. Consider a surface S in R3 defined

as the graph of x3 = f (x1, x2). Assume that the origin is on S, and the x-y plane is tangent to S, that is,

f(0,0) = 0 and fl(0, 0) =0 = f2(0,0), where, f o r simplicity, we write f; = a and will use f 7 to denote a We calculate the Gaussian curvature of S at the origin 0 = (0, 0, 0). Let u = (u1, u2) be a unit vector on the tangent plane. Then by (4.3), the normal curvature k (u) in u direction is

U [(a 2

) 2 + 1]3/2 (O)

where a and a are first and second directional derivative of f in u direction. Under the assumption that f1(0) = 0 = f2(0), we arrive at kn(n) = (fiin + 2f12u1u2 + f22n2)(0) = uTF u,

where

4.4 Curvature

133

F = (fir f12 l (0) f21 f22 J

,

and uT denotes the transpose of u.

Let Al and A2 be two eigenvalues of the matrix F with corresponding unit eigenvectors el and e2, i.e.,

Fei=Aiei, i=1,2.

(4.4)

Since the matrix F is symmetric, the two eigenvectors el and e2 are orthogonal, and it is obvious that

eTFei=Ai, i=1,2. Let 0 be the angle between u and el. Then we can express u = cos 0 el + sin 0 e2. It follows that

Assume that Al < A2. Then Al < Al + sine 0 (A2 - A1) =

cost 0 (A1 - A2) + A2 < A2.

Hence Al and A2 are minimum and maximumvalues of normal curvature, and therefore they are principal curvatures k1 and k2.

From (4.4), we see that Al and A2 are solutions of the quadratic equation

fu-Af12 f2l

f22-A

_ '\2 - (fll + f22)A+ (fllf22 - f12f21) = 0.

Therefore, by definition, the Gaussian curvature at 0 is

K(0) = klk2 = A1A2 = (fllf22 - f12f21)(0)

4.4.3 Curvature on Riemannian Manifolds On a general n-dimensional Riemannian manifold M, there are several kinds of curvatures. Before discussing them, we need to introduce a few preparatory concepts.

1. Vector Fields and Brackets

A vector field X on M is a correspondence that assigns each point p E M a vector X (p) in the tangent space TpM. It is a mapping from M to T M. If

134

4 Preliminary Analysis on Riemannian Manifolds

this mapping is differentiable, then we say the vector field X is differentiable. In local coordinates, we can express X (p)

a ai (p)

.

ag2y

When applied to a smooth function f on M, it is a directional derivative operator in the direction X (p): (X f) (p) _

ai (P)

Definition 4.4.1 For two differentiable vector fields X and Y, define the Lie Bracket as

[X, Y] = XY - YX. If X (p) _

° ai (p) aa; and Y(p) _ E' b; (p) as , then

\ of bia' [X,Y]f=XYf-YXf= I (aiab' ax; axi / I

ax;.

i,;=1

2. Covariant Derivatives and Riemannian Connections

Intuitively, we begin with a surface S in R3. Let c : I-+S be a curve on S, and let V(t) be a vector field along c(t), t E I. One can see that Ts is a vector in the ambient space R3, but it may not belong to the tangent plane of S. To consider the rate of change of V(t) restricted to the surface S, we make an orthogonal projection of dL onto the tangent plane, and denote it DV by We call this the covariant derivative of V(t) on S - the derivative viewed by the two-dimensional creatures on S. On an n-dimensional differentiable manifold M, the covariant derivative is defined by affine connections. Let D(M) be the set of all smooth vector fields.

Definition 4.4.2 Let X, Y, Z E D(M), and let f and g be smooth functions on M. An affine connection D on a differentiable manifold M is a mapping from D(M) x D(M) to D(M) which maps (X, Y) to DxY and satisfies

1. Dfx+gyZ= fDxZ+gDYZ, 2. Dx(Y + Z) = DxY + DxZ, 3. Dx(fY) = fDxY+X(f)Y.

4.4 Curvature

135

Given an affine connection D on M, for a vector field V(t) = Y(c(t)) along a differentiable curve c : I-4M, define the covariant derivative of V along c(t) as

DV dt

dt

In local coordinates, let c(t)=(xi(t),...x.(t))

and V(t)

vi(t) -.

Then by the properties of the affine connection, one can express

DV dt

dvi 8 dt 8xi

+ ij

dt vD ax'8 . dxi

(

4.5)

,i

Recall that in Euclidean spaces with inner product , a vector field V(t) along a curve c(t) is parallel if only if dl = 0; and for a pair of vector fields X and Y along c(t), we have < X, Y >= constant. Similarly, we have the following:

Definition 4.4.3 Let M be a differentiable manifold with an affine connecDV = 0. tion D. A vector field V (t) along a curve c(t) E M is parallel if Definition 4.4.4 Let M be a differentiable manifold with an affine connection D and a Riemannian metric . If for any pair of parallel vector fields X and Y along any smooth curve c(t), we have < X, Y >c(t)= constant

then we say that D is compatible with the metric .

The following fundamental theorem of Levi and Civita guarantees the existence of such an affine connection on a Riemannian manifold.

Theorem 4.4.1 Given a Riemannian manifold M with metric , there exists a unique connection D on M that is compatible with . Moreover this connection is symmetric, i.e.,

DXY - DyX = [X, Y1. We skip the proof of the theorem. Interested readers may see the book of do Carmo [Cal (page 55). In local coordinates (xr, , x.), this unique Riemannian connection can be expressed as

4 Preliminary Analysis on Riemannian Manifolds

136

ka

Daxj

k'riiaxk'

where

I,k

1

ax,

z

09mi

a9ii

ax;

axm

g ink

is called the Christoffel symbols, gij =< as., as. >, and (gig) is the inverse matrix of (g,i). 3. Curvature

Definition 4.4.5 Let M be a Riemannian manifold with the Riemannian connection D. The curvature R is a correspondence that assigns every pair of smooth vector field X, Y C D(M) a mapping R(X, Y) D(M) -*D(M) defined by

R(X,Y)Z=DyDXZ-DXDYZ+D(X,Y1Z, VZED(M). Definition 4.4.6 The sectional curvature of a given two-dimensional subspace E C TyM at point p is

K(E)

< R(X,Y)X,Y > IXI2IYI2-<X,Y>2'

where X and Y are any two linearly independent vectors in E, and XI2IYI2- < X,Y >2 is the area of the parallelogram spanned by X and Y. One can show that K(E) so defined is independent of the choice of X and Y. It is actually the Gaussian curvature of the section. Example. Let M be the unit sphere with standard metric, then one can verify that its sectional curvature at every point is K(E) = 1.

Definition 4.4.7 Let X = Y. be a unit vector in TyM, and let {Y1,.. ,Yn_1} be an orthonormal basis of the hyperplane in TyM orthogonal to X. Then the Ricci curvature in the direction X is the average of the sectional curvature of the n -1 sections spanned by X and Yl, X and Y2, and X and Yn-1: n-1

Ricy(X) = n ll E < R(X,Y,)X,Y; > =r

The scalar curvature at p is

R(p) _ -1 j: Ricy(Y:) :=r

.

4.5 Calculus on Manifolds

137

One can show that the above definition does not depend on the choice of the orthonormal basis. On two-dimensional manifolds, Ricci curvature is the same as sectional curvature, and they both depend only on the point p.

4.5 Calculus on Manifolds In the previous chapter, we introduced the concept of affine connection D on a differentiable manifold M. It is a mapping from 1'(M) x r(M) to T(M), where T(M) is the set of all smooth vector fields on M. Here we will extend this definition to the space of differentiable tensor fields and thus introduce higher order covariant derivatives. For convenience, we adapt the summation convention and write, for instance, a xi a'.= j ikdxk

rk dxk k

and so on. and

4.5.1 Higher Order Covariant Derivatives and the Laplace-Beltrami Operator Let p and q be two non-negative integers. At a point x on M, as usual, we

denote the tangent space by TxM. Define Tp(TxM) as the space of (p,q)-

tensors on TM, that is, the space of (p + q)-linear forms

rl:TMx .xTMxTMx...xT.M-rR1. P

q

In local coordinates, we can express

ajxj

T=V1.'°dx"®...®dxiP®a ®...0

49

Recall that, for the affine connection D, in local coordinates, if we set

then

Vi

axe

(x) = rij aak x,

4 Preliminary Analysis on Riemannian Manifolds

138

where I ..' are the Christoffel symbols. For smooth vector fields X,Y E P(M) with

y = (y1'. ..

X = (Xl, ... , Xn) and

, y"),

by the bilinear properties of the connection, we have

DxY =

X'DiY

X'

\88Yj

Xi (

09

+Yj1Oak/ll

xi axi

xi + q,akYk )

/x

(4.6)

Now we naturally extend this affine connection D to differentiable (p, q)-

tensor fields T on M. For a point x on M, and a vector X E TxM, DxT is defined to be a (p, q)-tensor on TM by

DxT(x) = X'(viT)(x), where '7i...Jq

P

bL,

(17J)(x)41...2P

axi

X

))I'7r-7y rri.(x)T l (x

k=1

q

Jk-1a7k+'j'.

+

ri'.k (x)T(x)it

-i'

(4.7)

k=1

Notice that (4.6) is a particular case when (4.7) is applied to the (0, 1)tensor Y. If we apply it to a (1,0)-tensor, say dx1, then

,7i(dx1) = _rikdxk.

(4.8)

Given two differentiable tensor fields T and T, we have

Dx(T®T)=DxT®T+T®DxT. For a differentiable (p, q)-tensor field T, we define VT to be the (p + 1, q)tensor field whose components are given by

Similarly, one can define 72T, 73T, - - -.

For example, a smooth function f (x) on M is a (0, 0)-tensor. Hence 7f is a (1, 0)-tensor defined by

dxi. of = Difdxi = of axi

4.5 Calculus on Manifolds

139

And v2f is a (2, 0)-tensor:

2f

of dx') ® dxj

\ ax-i of (7j raxi a

=

I

_ \aaaj

I dx' + of 7j (dx )) dxj allai

I'ak/ dx'0dxj.

Here we have used (4.7) and (4.8). In the Riemannian context, v2 f is called the Hessian of f and denoted by Hess(f). Its ij'h component is a2

gf

axiaxj

The trace of the Hessian matrix ((v2f)ij) is defined to be the LaplaceBeltrami operator A acting on f,

of=ga'( axi asx

-r'af1 ark

where (g'j) is the inverse matrix of (gij). Through a straightforward calculation, one can verify that

I=1

191 i,j=1

8

axi (VI-Igi gii axj

where gl is the determinant of (gij).

4.5.2 Integrals On a smooth n-dimensional Riemannian manifold (M, g), one can define a natural positive Radon measure on M, and the theory of Lebesgue integral applies.

Let (Ui, ¢i)i be a family of coordinate charts that covers M. We say that a family (Ui, Oi, 77i)i is a partition of unity associated with (Ui, Oi)i if

(i) (77j)i is a smooth partition of unity associated with the covering (U1)j, and

(ii) for any i, supp 77i C U. One can show that, for each family of coordinate chart (U1,01)1 of M, there exists a corresponding family of partition of unity (Us' Oi, 770i. Let f be a continuous function on M with compact support, and given a family of coordinate charts (Ui, 0i)i of M, we define its integral on M as the sum of the integrals on open sets in Rn:

140

4 Preliminary Analysis on Riemannian Manifolds

f mfdVg= I 1

(r1if)°0irdx,

0:(Ut)

where (Ui, Oj, r)i)i is a family of partition of unity associated with (Ui, bi)i, gj is the determinant of (gij) in the charts (U1,Oi)i, and dx stands for the volume element of R. One can prove that such a definition does not depend on the choice of the charts (Ui, ¢i)i and the partition of unity (Ui, ti, rli)i. For smooth functions it and v on a compact manifold M, one can verify the following integration by parts formula

-f LguvdVg= f dV

(4.9)

M

where

is the Laplace-Beltrami operator associated with the metric g, and

< 7u' pv >= g11 V/ up v =

9ji

an On axi axe

is the scalar product associated with g for 1-forms

4.5.3 Equations on Prescribing Gaussian and Scalar Curvature A Riemannian metric g on M is said to be pointwise conformal to another metric go if there exists a positive function p, such that 9(x) = P(x)9o(x), V X E M.

Suppose M is a two-dimensional Riemannian manifold with a metric go and the corresponding Gaussian curvature K0(x). Let g(x) = e2"(x)go(x) for some smooth function u(x) on M. Let K(x) be the Gaussian curvature associated with the metric g. Then by a straightforward calculation, one can verify that

-Aou+K0(x) = K(x)e2u(x) V X E M.

(4.10)

Here oa is the Laplace-Beltrami operator associated with go. Similarly, for conformal relation between scalar curvatures on dimensional manifolds (n > 3), say on standard sphere S", we have

+n(n4

2) u

4(n-1))R(x)u

,

VxES",

n-

(4.11)

where go is the standard metric on S" with the corresponding LaplaceBeltrami operator gyp, R(x) is the scalar curvature associated with the pointwise conformal metric g = u^4=go. In Chapter 5 and 6, we will study the existence and qualitative properties of the solutions it for the above two equations, respectively.

4.6 Sobolev Embeddings

141

4.6 Sobolev Embeddings Let (M, g) be a smooth Riemannian manifold. Let it be a smooth function k on M and k be a non-negative integer. We denote 7 u the kth covariant derivative of it as defined in the previous section, and its norm Ipkul as defined in a local coordinate chart by ik(,7kn)ji...jk.

I okul2 = Oil ... 9ikik (V ku)

In particular, we have Ip°ul = Jul and 49E 49u

Io1u12 = I VuI2

= 9"(7u)i(VU)j = s Ot nofu = 9ij 49xi 49xj

For an integer k and a real number p > 1, let Ck(M) be the space of C°° functions on M such that

f

M

Definition 4.6.1 The Sobolev space H1,P(M) is the completion of CE(M) with respect to the norm IIUIIHk.P(M) _

j=o

(f M IYulPdVg )

'iP

Many results concerning the Sobolev spaces on R" introduced in Chapter 1 can be generalized to Sobolev spaces on Riemannian manifolds. For application purposes, we list some of them in the following. Interested readers may find the proofs in [Hel] or [Au]. Theorem 4.6.1 For any integer k, Hk,2(M) is a Hilbert space when equipped with the equivalent norm. 112

k

Null =

f Io'ul2dVg)

s-1 M

The corresponding scalar product is defined by k

=Ef

oo; however, {xk} possesses no convergent subsequence because this sequence is not bounded. The functional has no compactness at level J = 0. We know that in finite-dimensional space, a bounded sequence has a convergent subsequence, whereas in infinite dimensional space, even a bounded sequence may not possess a convergent subsequence. For instance, {sin kx} is a bounded sequence in L2([0,1]) but does not have a convergent subsequence.

In the situation where the lack of compactness occurs, to find a critical point of the functional, one would try to recover the compactness through various means. Take again the one dimensional example J(x) = (x2 - 1)e'', which has no compactness at level J = 0. However, if we restrict it to levels less than 0. we can recover the compactness. That is, if we pick a sequence {xk} satisfying J(xk) < 0 and J'(xk)--*O, then one can verify that it possesses a subsequence which converges to a critical point x0 of J. Actually, this xo is a minimum of J. To further illustrate the concept of compactness, we consider some examples in infinite-dimensional space. Let 11 be an open bounded subset in R", and let Ho (1?) be the Hilbert space introduced in previous chapters. Consider the Dirichlet boundary value problem ZSu. = up(x), x E !2

U(X)=0,

xEBfl.

To seek weak solutions, we study the corresponding functional

JP(u) = )o lulp+rdx on the constraint

H = In EHa(Q) I

LIvul2dx=1}.

(5.1)

5.1 Variational Methods in General

147

When p < +2, it is in the subcritical case. As we have seen in Chapter 2, the functional satisfies the (PS) condition. Let MP = sup JP(u).

UEH

Consider a maximizing sequence {uk} with

JP(ak)-4MP and Jp'(7lk)iO. Then one can show that {uk} possesses a convergent subsequence in Ho(.(?), and the limiting function uo would be the maximum of J. in H, hence it is a desired weak solution. Here an alternative and equivalent way is to minimize the functional

IP(u) =

Jn

Ipul2dx

under the constraint

IuIP+'dx = 1,

u E Ho (fl) and Jn

as we have seen in Chapter 2. When p = T := n±2, it is in the critical case. One can find a maximizing

sequence of Jr that does not converge. For example, when I? = B1(0) is a ball, consider [n(n - 2)k]"
0. In Case ii), we choose a > °'2 and e° = a, where a is a positive number to be determined later. Then (5.14) becomes f (a, b, x) > -

2 - aR(x) (eau - 1)

>(-R a = aIJR11Lm

-IIRIIL;Ieav-1I)

R

(2

- leav - )

IIRIILQQ

Again by the continuity of v and the compactness of M, we see that uniformly on M as a-*0. Therefore, we can make eav(x)

-

11


a2 This completes the proof of the Lemma. .

Lemma 5.3.3 (Existence of Lower Solutions). Given any R E LP(M) and a function u+ E W2'P(M), there is a lower solution u_ E W2"P(M) of (5.9) such that u_ < u+.

Proof. If R(x) is bounded from below, then one can clearly use any sufficiently large negative constant as u_. For general R(x) E LP(M), let

K(x) = max 1, -R(x)}. Choose a positive constant a so that aK = -a. Then aK(x) + a = 0 and (aK(x) + a) E LP(M). Thus there is a solution to of

,Lw = aK(x) + a.

By the L. regularity theory, to E W2^P(M) and hence is continuous. Let

u- =w-A. Then -Au_ + a - R(x)ev- = -Aw + a = -aK(x) - R(x)e"'-.\

R(x)e'°_a

< -K(x)(a - eW_A).

(5.15)

5.3 The Negative Case

153

Choosing A sufficiently large so that a - ew

a

> 0,

and noticing that K(x) > 0, by (5.15), we arrive at

-,Lu_+a-R(x)e"- 2. If there exist upper and lower solutions, a+, u_ E w2,P(M) of (5.9), and if u_ < u+, then there is a solution u E W2,p(M). Moreover, u_ < u < u+, and u is C°° in any open set where R(x) is C°°. proof. We will rewrite the equation (5.9) as

Lu = f(x,u), then start from the upper solution u+ and apply iterations:

Lu1=f(x,u+) ,Lui+r=f(x,u:), i=1,2," In order that for each i, u_ < ui < u+, we need operator L to obey some Maximum Principle and f (x, ) to possess some monotonicity. The Laplace operator in the original equation does not obey Maximum Principles; however, if we let

Lu = -Du + k(x)u, where k(x) is any positive function on M, then the new operator L obeys the Maximum Principle:

IfLu>0, then u(x)>0, forallxEM. To see this, we suppose on the contrary that there is some point on M where it < 0. Let x° be a negative minimum of u; then, -Au(x°) < 0, and it follows that -Au(x°) + k(x°)u(x°) < k(x°)u(x°) < 0,

a contradiction. Hence the Maximum Principle holds for L. We now write equation (5.9) as Lu = -emu + k(x)u = R(x)eu - a + k(x)u := f(x,u).

Based on the Maximum Principle on L, keeping u_ < ui < u+ requires that f (x,) be monotone increasing, that is au = R(x)eu + k(x) > 0.

154

5 Prescribing Gaussian Curvature on Compact 2-Manifolds

To this end, we choose k(x) = kl(x)e"+lsl,

where

ki(x) = max{1, -R(x)}.

Based on this choice of L and f, we show by mathematical induction that the iteration sequence so constructed is monotone decreasing:

ui+1

&k = R(x)e''Pk(-)

i.e., bk is an upper solution of the equation

-emn + ak = R(x)e°(z).

(5.25)

Inspired by an idea of Ding and Liu [DL], we minimize the functional Jk (u) in a class of functions

H = {uECI(M) IA a> 0 , V x E 0

B2E (x° ),

for some constant a. Let vk be a solution of

Ovk-ak=OxE,(2

vk(x)1

xEt3(2.

Let wk = uk + vk. Then wk satisfies

-pwk = R(x)e- °ke'°'. By the Maximum Principle, the sequence {vk} is bounded from above and from below in the small ball R. Since {uk} is uniformly bounded from below, {wk} is also bounded from below. Locally, the metric is pointwise conformal to the Euclidean metric, hence one can apply Lemma 5.3.5 to {wk} to conclude that the sequence {wk} is also uniformly bounded from above in the smaller ball BE(x°). Since x° is an arbitrary point where R is positive, we conclude that

{wk} is uniformly bounded in the region where R(x) is positively bounded away from 0. Now the uniform bound for {uk} in the same region follows accordingly.

Step 3. We show that the sequence {uk} is bounded in H'(M). Choose a small 6 > 0, such that the set

D = {xEMIR(x)>6} is non-empty. It is open since R(x) is continuous. From the previous step, we know that {uk} is uniformly bounded in D. Let K(x) be a smooth function such that

K(x) < 0, for x E D; and K(x) = 0 , elsewhere. For each ak, let vk be the unique solution of

-tvk + ak = K(x)e°k , x E M. Then since

5.3 The Negative Case

161

-Ovk + ak < 0 , and ak-+a0 The sequence {vk} is uniformly bounded. Let wk = Uk - Vk, then

-AWk = R(x)e"k - K(x)e'4.

(5.32)

We show that {wk} is bounded in H'(M). On the one hand, multiplying both sides of (5.32) by ewk and integrating on M, we have

R(x)e°k+wkdA-J K(x)e°kdA.

JIowk12ewkdA=J

M

M

(5.33)

D

On the other hand, since each uk is a minimizer of the functional Jk, the second derivative Jk (uk) is positively definite. It follows that, for any function

¢ E H'(M), we have 2

11

rrI

LWt-2

J(uk + t¢)J

> 0, L=0

or equivalently,

- R(x)e"k¢2]dA > 0. f[Io0I2 M Choosing 0 = eWk, we derive 4

Ipwkl2ewkdA > f R(x)e"k+wkdA.

JM

(5.34)

M

Combining (5.33) and (5.34), we arrive at

-J K(x)e"kdA> 3J Ipwkl2ewkdA. 4M

D

(5.35)

Noticing that {uk} is uniformly bounded in region D and {wk} is bounded

from below due to the fact that {uk} is bounded from below and {vk} is bounded, (5.35) implies that fM 1ewkl2dA is bounded, and so is fm JpukI2dA.

Consider uk = Uk - Uk, where ilk is the average of uk. Obviously ukdA = 0, IM

hence we can apply the Poincare inequality to ilk and conclude that {ilk} is bounded in H1(M). Apparently, we have

JM

u2dAoo, D

for some subsequence, a contradiction with the fact that {v,k} is bounded in H1(M). Therefore, {uk} must also be bounded in H1(M). Consequently, there exists a subsequence of {uk} that converges to a function uo in H' (M), and uo is the desired weak solution for

-Auo + ao = R(x)eu' This completes the proof of the Theorem.

5.4 The Non-negative Case-Variational Approaches Now we assume a > 0 and consider the equation

-Au + a = R(x)e°l'>,

x E M,

(5.36)

where A is the Laplace-Beltrami operator associated with the base metric go. 5.4.1 Obstructions

Integrating both sides of the equation (5.36), and using the fact that

-JM AudA=JM 7u.V1dA=0, we have

R(x)e°dA = aIMI. IM

It follows that, in the case a = 0, R(x) must change signs (We assume R(x) 0 0); and in the case a > 0, R(x) must be positive somewhere. In any case, we have derived that a necessary condition for equation (5.36) to have a solution is as follows:

R(x) > 0 somewhere.

(5.37)

In the critical case when X(M) = 2, besides this obvious necessary condition, there are other necessary conditions. For simplicity, we may just consider M = S2, the two-dimensional sphere with standard metric. In this case equation (5.36) becomes

5.4 The Non-negative Case-Variational Approaches

-,Lu + 2 = R(x)e'('),

X E S2.

163

(5.38)

It is known as the Nirenberg problem.

Kazdan and Warner [KW1] proved that, if u is a solution of equation (5.38), then

VoiVR(x)e" dx = 0, i = 1, 2, 3.

(5.39)

Js

Here Oi are the first spherical harmonic functions, i.e.,

20i (x), i=1,2,3; and for simplicity, we write S = S2 and dA = dx.

Actually, 4i (x) = xi in the coordinates x = (X1, X2, X3) of R. Later, Bouguignon and Ezin generalized this condition to X (R)e" dx = 0.

(5.40)

is

where X is any conformal Killing vector field on standard S2. Condition (5.39) gives rise to many examples of R(x) for which the equation (5.38) has no solution. In particular, a monotone rotationally symmetric function admits no solution. Then for which kinds of functions R(x) can one solve (5.38)? This has been an interesting problem in geometry.

5.4.2 The Variational Approach and Key Inequalities To obtain the existence of a solution for equation (5.36), people usually use a variational approach. First prove the existence of a weak solution; then, by a regularity argument, one can show that the weak solution is smooth and hence is the classical solution. To obtain a weak solution, we let

H1(M) := H1,2(M) be the Hilbert space with norm 11/2

muli1= 11M (IVuI2+u2)dAJ Let

H(M) = {u E H1(M) I fm udA = 0,

f

JM

R(x)e"dA> 0}.

164

5 Prescribing Gaussian Curvature on Compact 2-Manifolds

Then by the Poincare inequality, we can use /2

dull = [ f 1,7U12 dAl J

as an equivalent norm in H(M). Consider the functional

r

J(u) = 2 J IpuI2 dA - 4irX In / R(x)eu dA M

in H(M), where X = x(M). To estimate the value of the functional J, we introduce some useful inequalities.

Lemma 5.4.1 (Moser-Trudinger Inequality) Let M be a compact, two-dimensional Riemannian manifold. Then there exists constant C, such that the inequality

M

e4,ru2dA

0.

Let 0 < ao < 2. Then for any e > 0, there exists a constant C = C(a", bo, e), such that the inequality

J e"dA

2IIull2

- 8ir fln(C maax R) + 132_ + el 1Iuii21 J

-8irln(CmaLxL

R) + \4 -87re)

>

IIull2.

/

Choosing c small enough, we arrive at

J(u) > g lltII2 - Ci, for all even functions it in H(S).

(5.53)

This stronger inequality guarantees the boundedness of a minimizing sequence, and hence it converges to a weak limit u, in H'(S). As we will show in the next section, ua so obtained is a weak solution of equation (5.38). In many articles, authors use the "center of mass" analysis, i.e., consider the center of mass of the sphere with density e"lsl at point x:

P(u) - fs x e"dA fs e"dA

and use this to study the behavior of a minimizing sequence (See [CD], [CD1], [CY], and [CYI]). Both analyses are equivalent on the sphere S2. However,

since "the center of mass" P(u) lies in the ambient space R3, on general two dimensional compact Riemannian surfaces, this kind of analysis can not be applied, while the "Distribution of Mass" analysis can be applied to any compact surfaces-even surfaces with conical singularities. Interested readers may see the author's paper [CL7] [CL8] for a more general version of inequality (5.51).

Proof of Lemma 5.4.4. Let gi, g2 be two smooth functions, such that

1>9i>-0, 9i(x)=1, forxEfli, i=1,2 and supp gin supp 92 = 0. It suffices to show that for it E H1(S) with fs u dA = 0, (5.52) implies e"dA < C ex p p

I (327r+E)ul2

Case i) If I19iuJI < II92uIl, then by (5.43)

(5.54)

170

5 Prescribing Gaussian Curvature on Compact 2-Manifolds

f eudA < s

1 f e" dA < 1 f ao n, ao Js C

ao

C

< ao
inf

J.

He(S)

As we have seen in the previous section, the functional J is weakly lower semi-continuous:

J(ua) < hmJ(uk) = inf J. H.(S)

Therefore, uo is a minimum of J in He(S). It follows that, for any v E He(S), we have

0 =< J'(uo), v >=

Js

7uovyv dA - fs R(8x )eu-dA is R(x)e"°vdA. s

For any w E H(S), let

v(x) = 2(w(x)+w(-x))

2(w(x)+"w-(x))-

172

5 Prescribing Gaussian Curvature on Compact 2-Manifolds

Then obviously, v E H,(S), and it follows that

0 =< J'(uo), v >= j(< J'(uo), w > + < J'(uo), w- >) _< X(uo), w >

.

Here we have used the fact that uo(-x) = uo(x). This implies that uo is a critical point of J in H1(S) and hence a constant multiple of uo is a weak solution of equation (5.38). p

General Symmetric Functions Chen and Ding [CD] generalized Moser's result to a broader class of symmetric functions, as we will describe next.

Let 0(3) be the group of orthogonal transformations of R. Let G be

a

closed finite subgroup of 0(3). Let

FG={xES2Igx=x,VgEG} be the set of fixed points under the action of G. The action of G on S2 induces an action of G on the Hilbert space HI(S): 9 [u] (x) " u(9x)

under which the fixed point subspace of H'(S) is given by

X = {u E H'(S) I u(gx) = u(x), Vg E G}. Assume that R(x) is G-symmetric, or G-invariant, i.e.,

R(gx) = R(x), Vg E G.

(5.61)

Let x. = X f l H(S). Recall that

H(S) = {u e HI (S) f udA =0, J R(x)e'dA > 01. S

Under the assumption (5.61), the functional

J(u) = l

Js

1,7UI2

dx - 8rr 1,

JS R(x)e,- dx

is G-invariant in X i.e.,

J(g[u])=J(u), VgEG, VuEX,. We seek a minimum of J in X.. The following lemma guarantees that such a minimum is the critical point we desired.

5.4 The Non-negative Case-Variational Approaches

173

Lemma 5.4.5 Assume that uo is a critical point of J in X.; then it is also a critical point of J in H(S). proof. First, for any g E G and any u, v E H(S), we have

< X(U), g['v] >=< J'(9-1[u]), v >,

(5.62)

where g-' is the inverse of g. This can easily be seen from

< J'(u),v >=

s

7u,7vdA-

8ir

fs ReudA s

Re' vdA.

Assume

G = {gl, ... gm}.

For any w E H(S), let

v=

1

(gi[w]+.+gm[w]).

Then for any g E G g [VI = I (ggi [w] + .

+ ggm [w]) = v.

Hence v E X.. It follows that

Y<J'(uo),gi[w]> 0=<J'(uo),v>= 1 m m

=

m

<J'(ga1[u,]),w>=-<J'(uo),w>

=<X(vo),w>. Therefore, uo is a critical point of J in H(S).

Theorem 5.4.3 (Chen and Ding [CDJ). Suppose that R E C°(S2) and satisfies (5.61) with maxs R > 0. If FG # 0, let in = maxFF R. Then equation (5.38) has a solution provided one of the following conditions holds:

(i) Fa = 0, (ii) in < 0, or (iii) in > 0 and c. = infx. J < -87r In 47rm.

Remark 5.4.1 In the case when the function R(x) is even, the group G = {id, -id}, where id is the identity transform, i.e.,

id(x) = x and - id(x) _ -x V x E S2. Obviously, in this case FG is empty. Hence the above theorem contains Moser's existence result as a special case.

5 Prescribing Gaussian Curvature on Compact 2-Manifolds

174

To prove this theorem, we need another two lemmas.

Lemma 5.4.6 (Onofri [Onj) For any it E H1(S) with fs u dA = 0, holds e"dA < 47r exp { is

ll

167r

f IpuI2dA } . s

(5.63)

11

This lemma provides another best constant 47r besides 167r, and it is needed in the finer analysis in the critical case. Interested readers may find its proof in [On] or [Ho].

Lemma 5.4.7 Suppose that {ui} is a sequence in H(S) with J(ui) < Q. if lluijj-*oo, then there exists a subsequence (still denoted by {u;}) and a point E S2 with R(e) > 0, such that

iR(x)e"idA=(R(e)+o(1))ieuidA,

(5.64)

lim J(ui) > -87rln47rR(e).

(5.65)

is

is

and

Proof. We first show that there exists a subsequence of {ui} and a point L; E S2, such that for any c > 0, we have

e"idA-40.

(5.66)

Otherwise, there exists two subsets of S2, 01 and 122, with dist(S7i, .22) > eo > 0 such that {u;} satisfies all the conditions in Lemma 5.4.4. Hence the inequality

f e"idA 0. Then by the previous argument, we conclude that IIuiII is bounded. This contradicts our assumption and hence verifies (5.66). Now (5.66) and the continuity assumption on R(x) implies (5.64). Roughly speaking, the above argument shows that if a minimizing sequence

of J is unbounded, then passing to a subsequence, the mass of the surface S with density e"i(x) at point x will be concentrating at one point e on S. To verify (5.65), we use (5.64) and Onofri's Lemma:

5.4 The Non-negative Case-Variational Approaches

1vnI2dA - 87rln(R(4) +o(1))

J(ui) = 2

>

r

2

r e',dAr

Ipu12dA - 8aln(R(£) +o(1)) J-

=2

f

175

8rlnJ e"'dA rs

Ipu12dA - 8irln(R(4) +o(1)) - 87r(ln47r+ 16 J Ipu12dA)

_ -87r ln(47R(4) + o(1)).

Letting i->oo, we arrive at (5.65). Since the right hand side of (5.65) is bounded above by 0, we must have R(4) > 0. This completes the proof of the Lemma.

The Proof of Theorem 5.4.3. Define

c,=infJ. X. As we argued in the previous section (see (5.47)), we have c, > -oc. Let {ui} C X. be a minimizing sequence, i.e., J(ui)ic., as i-+eo. Similar to the reasoning at the beginning of this section, one can see that we only need to show that {ui} is bounded in H'(S). Assume on the contrary that IIuiII-*oo. Then from the proof of Lemma 5.4.7, there is a point £ E S2, such that for any e > 0, we have e"'dA->0.

(5.67)

s\BE(C)

Since ui is invariant under the action of G, we must also have e'dA->0.

(5.68)

S'\BE(gf)

Because a is arbitrary, we deduce that gf = f, that is 4 E FG. Now under condition (i), Fc is empty. Therefore, {ui} must be bounded. Moreover, by Lemma 5.4.7,

R(4)>O and c , > -87rln47rR(4). Noticing that m > R(4) by definition, this again contradicts with conditions (ii) and (iii). Therefore, under any one of the conditions (i), (ii), or (iii), the minimizing 'sequence {ui} must be bounded and hence possesses a subsequence converging

weakly to some u" E X. Thus a constant multiple of uo is the desired weak solution of equation (5.38). Here conditions (i) and (ii) in Theorem 5.4.3 can be easily verified through the group G or the given function R. However, one may wonder under what

5 Prescribing Gaussian Curvature on Compact 2-Manifolds

176

conditions on R do we have condition (iii). The following theorem provides a sufficient condition.

Theorem 5.4.4 (Chen and Ding ]CDI]). Suppose that R E C2(S2) soinvariant, and in = maxF0 R > 0. Let D = {x E S2 I R(x) = m}.

If AR(x") > 0 for some x0 E D, then C. < -8ir In 4irm,

(5-69)

and therefore (5.38) possesses a weak solution.

Proof. Choose a spherical polar coordinate system on S := S2:

(9, 0), - 2 < 9 < 2 , 0 < ¢ < 21r, with xo = (2,

as the north pole.

Consider the following family of functions:

_ U), (0, ¢) = In

(1

2

1 a sin9)2' VA = ua - 4n J uadA, 0 < A < 1.

One can verify that this family of functions satisfy the equation -Aua + 2 = 2e"a.

(5.70)

As a-+1, the family {u,\} `blows up' (tends to oo) at the one point x,, while approaching -oo everywhere else on S2. Therefore, the mass of the sphere with density e°,(y) is concentrating at point xo as A-41. It is easy to see that vA E H(S) for A close to 1. Moreover, we claim that va E X, i.e., v,\(gx) = va(x), Vg E G.

In fact, since g is an isometry of S2, and gx, = x0, we have d(9x, xo) = d(9x, gxo) = d(x, xo),

where

is the geodesic distance on S2. But vA depends only on 0 =

d(x, x.), so V,\ (9x) = V,\ (X), V9 E G.

It follows that va E X. for A sufficiently close to 1. By a direct computation, one can verify that S

Ipua12dA+87ru,\

=0.

5.4 The Non-negative Case-Variational Approaches

177

Consequently,

J(va)=-8trlnJ R(x)e'AdA.

is

Notice that c, < J(va) by the definition of c,. Thus, in order to show (5.69), we only need to verify that, for A sufficiently close to 1 R(x)e"adA> 47rm.

(5.71)

Js

Let b > 0 be small, we have r2,r

r Re"'dA = (1 - A2) If

s

m]

a-s

o

+J

f,-a

cos O

(1-ain0)2

Cos 9

[R(O,¢) - m]

J a

(1 - A sin B)2

d&dO

dodo + 47m

= (1 - A2){I + 77} + 4am. Here we have used the fact

I J

e"a dA = 4tr,

which can be derived directly from equation (5.70). For each fixed 5, it is easy to check that the integral II remains bounded as A tends to 1. Thus (5.71) will hold if we can show that the integral I -4 +oo as A-r1. Notice that xo is a maximum point of the restriction of R on Fc. Hence by the principle of symmetric criticality, xo is a critical point of R, that is, VR(xo) = 0. Using this fact and the second order Taylor expansion of R(x) near x0, we can derive

[AR(xa) Cos20+0 (10 -

7=T r

(1-Asin0)2 z

COS3 B

c°j _a (1-Asin0)2

z 12) ] cos0do

dB.

z

Here co is a positive constant. We have chosen b to be sufficiently small and have used the fact that AR(xo) is positive. From the above inequality, one

-do

can easily verify that as A--+ 1, I-+ + oo, since the integral Cos3 0

_a (1-sin0)2 diverges to +oo.

This completes the proof of the Theorem.

The Yamabe Problem and Prescribing Scalar Curvature on S'E, for n > 3

6.1

Introduction 6.1.1

6.1.2 6.2 6.3

The Variational Approach for the Yamabe Problem The Variational Approach for Prescribing Scalar Curvature 6.3.1

6.3.2 6.4

The Yamabe Problem Prescribing Scalar Curvature on Sn

Estimate the Values of the Functional The Variational Scheme

The a priori Estimates for Prescribing Scalar Curvature 6.4.1

6.4.2 6.4.3

In the Region Where R < 0 In the Region Where R is small In the Region Where R > 0

6.1 Introduction 6.1.1 The Yamabe Problem Given a compact, smooth, and orientable manifold M of dimension n > 3 with a Riemannian metric go, does there exist a metric g conformal to go for which the scalar curvature of g is constant? In other words, does there exist a smooth positive function p(x) on M for which the metric g = P(x)g, has constant scalar curvature? This is the well-known Yamabe problem. It was named after the mathematician Hidehiko Yamabe, who claimed to have a solution in 1960 [Y], while a critical error in his proof was discovered in 1968

6 The Yamabe Problem and Prescribing Scalar Curvature on S^, for n > a

180

by Neil Trudinger [T]. The combined work of Neil Trudinger [T], Thierry Aubin [Aul], and Richard Schoen [Sc] provided a complete solution to the problem as of 1984.

As we introduced in Chapter 4, if we let g = u^^==go, then

4n - 1) R0(x)u = 4(n - 1) Rx)u

x E M,

,

(6.1)

where D is the Laplace-Beltrami operator associated with the metric go, R0(x) is the scalar curvature of go, and R(x) the scalar curvature of g.

Hence to solve the Yamabe problem, it is equivalent to find a positive function u on M such that

n-2

-emu +

"+2 R,(x)u = A u^-_

x E M,

(6.2)

4(n - 1)

for some constant A.

To show the existence of a solution for (6.2), a variational approach is usually employed. Let H'(M) be the Hilbert space of functions on M with norm IIkIIH = U

f

M

(IDuI2+u2)dVo.

Let dVo be the volume element of (M, go), and for simplicity, write 2

K0(x) = 4(n

)Ro(x)

and T=

1

Traditionally, people minimized the quotient Q(u)

_

JM

[I,7ul2 + Ko(x)u2] dVo Ilullr+1

in H1(M) \ {0}, where

Ilullp = (IM

IuIPdVo)

is the LP norm on (M, go). Or equivalently, one can minimize the functional [1,7UI2

J(u) = J

+ Ko(x)u2] dVo

M under the constraint

Gr = {u E H'(M) I IIullz+r =1}.

n±2.

6.1 Introduction

181

We will adapt the latter scheme here. It is easy to verify that, the minima, or more generally, the critical points of J() in G. are weak solutions of (6.2) with some constant A. Unfortunately, the Sobolev embedding H'(M) --+ LT+1(M) is not compact, and this causes the non-compactness of the corresponding variational scheme. This value r is called the critical exponent. Let

mo =nEG, inf J(u). It is well-known that the key ingredient in solving the Yamabe problem is to verify that (6.3) mo < So,

if M is not conformal to the unit sphere S' with standard metric, where S. is the best constant in the Sobolev embedding from Hr (Rn) to LT+1(R"), or from H1(Sn) to L7+1(Sn) as indicated by the following:

Theorem 6.1.1 For all u E Hl(Rn), we have Sju11z+r

- IIvuU

(6.4)

,

where

n S°-n(n-2) r(2)r(n+1)Wn_1 r(n+1)

-

n(n-2) a 4

Wn

(6.5)

is the largest possible constant. Here IIuIIy is the L" norm of u in Rn, w,i_1 and wn are the areas of (n - 1) and n-dimensional unit spheres, respectively, with

n7r 2

+

wn.-1 T-(22

where r(-) is the Gamma function. Inequality (6.3) is verified by Aubin [Aul] for manifolds of dimensions n > 6 that are not locally conformally flat, and by Schoen [Sc] for the remaining cases. The proofs in [Aul] and [Sc] were unified by Lee and Parker [LP] via introducing the conformal normal coordinates. Later, Wang [W] provided a simpler proof by using the positive mass theorem of Schoen and Yau [SY], but not the conformal normal coordinates. The proof of this inequality is too advanced to include in this book, and the interested reader may refer to the above mentioned articles. However, we will demonstrate, via a standard variational approach, the proof of the following theorem:

Theorem 6.1.2 Assume that (6.3) holds. Then the Yamabe problem (6.2) Possesses a positive solution.

We will prove this Theorem in Section 6.2.

6 The Yamabe Problem and Prescribing Scalar Curvature on S", for n > 3

182

6.1.2 Prescribing Scalar Curvature on S'° A natural and obvious generalization of the Yamabe problem is as follows: Given a compact, smooth, and orientable manifold M of dimension n > 3 with a Riemannian metric go, which functions R(x) can be realized as the scalar curvature of some metric g that is pointwise conformal to go? This is known as the "prescribing scalar curvature" problem.

As we mentioned in the previous subsection, if we let g = u=go, then it is equivalent to consider the existence of a positive solution to equation (6.1). When M = S^ with standard metric, this problem becomes the most difficult. In this case, equation (6.1) becomes

-Au + n(n 4

2) u

= 4(n - 1) R(x)u-2 , u> 0, x E S",

(6.6)

where A is the Laplace-Beltrami operator associated with the standard metric

on S'. The equation is "critical" in the sense that lack of compactness occurs. Besides the obvious necessary condition that R(x) be positive somewhere, there are well-known obstructions found by Kazdan and Warner [KW2] and later generalized by Bourguignon and Ezin [BE]. The conditions are these:

i X (R)dVg = 0

(6.7)

where dVg is the volume element of the conformal metric 9, and X is any conformal vector field associated with the standard metric go. We call these Kazdan-Warner type conditions. These conditions give rise to many examples of R(x) for which equation (6.6) has no solution. In particular, a monotone rotationally symmetric function R admits no solution. In the last two decades, numerous studies were dedicated to these problems and various sufficient conditions were found (please see the articles [Ba], [Ball, [BC], [BCCH], [C], [CC], [ChL], (CL3], [CL4], [CL6], [CL10], [CS], [CY1], [CGY], [ES] [Ka], [J3], [J4], [LiYl], [LiY2], [LiY3], [Linl], [Sc], (Sell, [SZ], and the references therein). However, among others, one problem of common

concern left open was this: Were the necessary conditions of the KazdanWarner type also sufficient conditions? In the case where R is rotationally symmetric, the conditions become:

R > 0 somewhere and R' changes signs.

Then (a) Is (6.8) a sufficient condition? and (b) if not, what are the necessary and sufficient conditions?

(6.8)

6.1 Introduction

183

Kazdan listed these as open problems in his CBMS Lecture Notes [Ka]. Recently, the authors answered question (a) negatively [CL3] [CL4]. We found some stronger obstructions. Our results imply that for a rotationally symmetric function R, if it is monotone in the region where it is positive, then problem (6.6) admits no solution unless R is a constant. In other words, a necessary condition to solve (6.6) is that

R'(r) changes signs in the region where R is positive.

(6.9)

However, is this a sufficient condition? For prescribing Gaussian curvature equation (5.38) on S2, Xu and Yang [XY] showed that if R is 'non-degenerate,' then (6.9) is a sufficient condition. For equation (6.6) on higher-dimensional spheres, a major difficulty is that

multiple blow-ups may occur when approaching a solution by a mini-max sequence of the functional or by solutions of subcritical equations. In dimensions higher than 3, the `non-degeneracy' condition is no longer sufficient to guarantee the existence of a solution. It was illustrated by a counter example in [Bi] constructed by Bianchi. He found some positive rotationally symmetric function R on S°, which is non-degenerate and nonmonotone, and for which the problem admits no solution. In this situation, a more proper condition is called the `flatness condition'. Roughly speaking, it requires that at every critical point of R, the derivatives of R up to order (n - 2) vanish, while some higher, but less than nth, order derivative is distinct from 0. For n = 3, the 'non-degeneracy' condition is a special case of the `flatness condition'. Although people are wondering if the `flatness condition' is necessary, it is still used widely today (see [CY2], [LiY2], and [SZ]) as a standard assumption to guarantee the existence of a solution. The above mentioned Bianchi counterexample seems to suggest that the `flatness condition' is somewhat sharp.

Now, a natural question to ask is this: Under the `flatness condition,' is (6.9) a sufficient condition? We answer the question affirmatively. In the next two sections, we will prove that, under the `flatness condition,' (6.9) is a necessary and sufficient condition for (6.6) to have a solution. This is true in all dimensions n > 3, and it applies to functions R with changing signs. Thus, we essentially answer the open question (b) posed by Kazdan. There are many versions of `flatness conditions,' and a general one was presented in [LiY2] by Y. Li. Here, to better illustrate the idea, in the statement of the following theorem, we only list a typical and easy-to-verify one.

Theorem 6.1.3 Let n > 3. Let R = R(r) be rotationally symmetric and satisfy the following flatness condition near every positive critical point ro:

6 The Yamabe Problem and Prescribing Scalar Curvature on S", for n > 3

184

R(r) = 11(r0) + alr - r" Ia + h (jr - r01), with a 00 and n-2< a< n (6.10) where h'(s) = o(sa-1). Then a necessary and sufficient condition for equation (6.6) to have a solution is that R' (r) changes signs in the regions where R > 0.

The theorem is proved by a variational approach. We blend in our new ideas with other ideas in [XY], [Ba], [LiY2], and [SZ]. We use the `center of mass' to define neighborhoods of `critical points at infinity,' obtain some quite sharp and clean estimates in those neighborhoods, and construct a max-mini variational scheme at sub-critical levels and then approach the critical level.

Outline of the proof of Theorem 6.1.3. Let 7" _ n(4 2) and r =

n±2. We

first find a positive solution up of the

subcritical equation

-Au + 7,u = R(r)u1',

(6.11)

for each p < r and close to T. Then let p-*r, and take the limit. To find the solution of equation (6.11), we construct a max-mini variational scheme. Let JJ(u):=

L Rup+rdV J

and

E(u)

f "(1vu12 +7"u2)dV.

We seek critical points of Jp(u) under the constraint

H = In G H1(S") I E(u) = 7 IS'"I, u> 0}, where IS"I is the volume of S. If R has only one positive local maximum, then by condition (6.9), on each

of the poles, R is either non-positive or has a local minimum. Then similar to the approach in [C], we seek a solution by minimizing the functional in a family of rotationally symmetric functions in H. In the following, we assume that R has at least two positive local maxima. In this case, the solutions we seek are not necessarily symmetric. Our scheme is based on the following key estimates on the values of the functional J. (u) in a neighborhood of the `critical points at infinity' associated with each local maximum of R. Let r1 be a positive local maximum of R. We prove that there is an open set G1 C H (independent of p), such that on the boundary 8G1 of G1, we have

6.1 Introduction

Jp(u) < R(rl)ISRI - a,

185

(6.12)

while there is a function 01 E G1, such that

Jp(01) > R(ri)IS"I -

(6.13)

Here 8 > 0 is independent of p. Roughly speaking, we have some kind of 'upside-down mountain pass' associated with each local maximum of R. The set Gl is defined by using the `center of mass'. Let r1 and r2 be two smallest positive local maxima of R. Let 1' and 02 be two functions defined by (6.13) associated with r1 and r2. Let ry be a path in H connecting 01 and 02, and let r be the family of all such paths. Define cp = sup min Jp(u).

(6.14)

lEr 7

For each p G r, by compactness, there is a critical point up of Jp(-), such that Jp(up) = Cp.

Intuitively, in the graph of functional Jp, the neighborhoods of (01, J(01)) and (02, J(02)) look like the two humps of a camel, and P = (u7,, J(up)) is the saddle point between the two humps, as shown in the following figure.

(''2, J('2))

(0i, J(t1))

H1(S")

Fig. 6.1. Saddle Point P

Obviously, a constant multiple of up is a solution of (6.11). Moreover, by (6.12),

Jp(up) < R(ri) I S"I - S,

for any positive local maximum ri of R.

(6.15)

6 The Yamabe Problem and Prescribing Scalar Curvature on S°, for n > 3

186

To find a solution of (6.6), we let p-+r, and take the limit. To show the convergence of a subsequence of {u,}, we established a priori bounds for the solutions in the following order. (i) In the region where R < 0: This is done by the `Kelvin Transform' and a Maximum Principle. (ii) In the region where R is small: This is mainly due to the boundedness of the energy E(up). (iii) In the region where R is positively bounded away from 0: First due to the energy bound,-{up} can only blow up at finitely many points. Using a Pohozaev type identity, we show that the sequence can only blow up at one point, and that point must be a local maximum of R. Finally, we use (6.15) to argue that even a one-point blow up is impossible, thus establishing an a priori bound for a subsequence of {up}. In Section 6.3, we carry on the max-mini variational scheme and obtain a solution up of (6.11) for each p. In Section 6.4, we establish a priori estimates on the solution sequence {up}.

The arguments in Sections 6.3 and 6.4 are rather complicated, and beginners in this area are welcome to skip them.

6.2 The Variational Approach for the Yamabe Problem In this section, we prove Theorem 6.1.2. The following special case of Aubin's theorem (See 2.21 Theorem in [Au]) plays an important role in the proof.

Theorem 6.2.1 Let M be a complete n-dimensional manifold with bounded curvature and injectivity radius b > 0. Then for any e > 0, there exists a constant A(e), such that every u e HI(M) satisfies: So11u117}I rnp.

(6.21)

On the other hand, similar to an argument made in Chapter 2, we can show that IM

Ipupl2dVo < liminf f Ipukj2dVo. k

M

Consequently,

J(up) 3

194

In particular,

E(Tqu) = E(u) and

J is.

(T9u)T+1dV =

Jsn

ur}ldV.

Proof. (i) We will use a property of the conformal Laplacian to prove the invariance of the energy

E(Tqu) = E(u). The proof for (i) is similar. On a n-dimensional Riemannian manifold (M, g),

L9 :_ %9 - c(n)R9 is called a "conformal Laplacian," where c(n) = 4(n-1) , O9 and R. are the Laplace-Beltrami operator and the scalar curvature associated with the metric g, respectively.

The conformal Laplacian has the following well-known invariance property w4/(n-2)g, w > 0, we have under the conformal change of metrics. For

L9u = w--2 L9(uw) ,

for all u E C°°(M).

(6.42)

Interested readers may find its proof in [Bes]. If M is a compact manifold without boundary, then multiplying both sides of (6.42) by u and integrating by parts, one would arrive at fM{Iobu12 +c(n)RyIuI2}dVg

=

fM{I09(uw)12 +c(n)RgIuw12}dV9. (6.43)

In our situation, we choose g to be the Euclidean metric g in R" with gig = bij and

Co, the standard metric on Sn. Then it is well-known that g° _ W "

(x)g

with

satisfying the equation We also have n(n - 2)

4 and c(n)Ry = c(n)0 = 0.

6.3 The Variational Approach for Prescribing Scalar Curvature

195

Now formula (6.43) becomes

f {IV.u12+7n.iu12}dVo= f nIo(uw)l2dx,

(6.44)

R

S

where vo and p are gradient operators associated with the metrics go and respectively.

Again, let q" be the intersection of S" with the ray passing through the center of the sphere and the point q. Make a stereo-graphic projection from Sn to Rn, where q" is placed at the origin of Rn. Under this projection, the point (r, 0) E Sn becomes x = (xi, , xn) E Rn, with 1x1

2

r

=tan g

and

(Tqu)(x) = u(Ax) I A(4+Ix12) 4 + Aixi2

I

2

It follows from this and (6.44) that E(Tqu)

=

fs {loo

z'

U('\x) 12+7n1u(Ax)

(A4+ajxj2)\

1 12J

4-2

f

R

=

f

117

Lu (Ax)

t

H

(

M(4+ x12)l I- w(x)] 4+A1x12

v-

(4Ix12)n

12dx

] 12x

l

= fRT 10 [u(y) (4+41y12 n22] I2dy

=

f

R^

117 (u(y)w(y))12dy

= E(u).

(ii) This can be derived immediately by making the change of variable y = ha(x). This completes the proof of the lemma. One can also verify that Tq¢q = 1.

196

6 The Yamabe Problem and Prescribing Scalar Curvature on S", for n > 3

The relations between q and A, 4 and Tq for 4 # 0 can be expressed in a similar way.

We now carry on the estimates near the south pole (0, 0) which we assume to be a positive local maximum. The estimates near other positive local maxima are similar. Our conditions on R imply that

R(r) = R(0) - ara, for some a > 0, n - 2 < a < n

(6.45)

in a small neighborhood of O. Define

E = {u E H I

q(u)I 0, by (6.49), one can choose A0, such that ¢ao,o E E, and

JJ(Oao,o) > R(0)IS"I - 21.

(6.50)

It is easy to see that for a fixed function 4ao,p, Jp is continuous with respect to p. Hence, by (6.50), there exists a p1, such that for all p > pl,

6.3 The Variational Approach for Prescribing Scalar Curvature

197

Jp('ao,o) > R(0)IS" I - 6r. This completes the proof of Proposition 6.3.1.

The proof of Proposition 6.3.2 is rather complex. We first estimate Jp for a family of standard functions in E.

Lemma 6.3.2 For p sufficiently close to r, and for h and Iq"I sufficiently small, we have

Jp(Oa,4) c(IxI" + 141°) for some c > 0 in one half of the ball B,(0) and the symmetry of 0a,o. Noticing that a < n and 6p->0, we conclude that (6.52), (6.53), and (6.54) imply (6.51). This completes the proof of the Lemma. To estimate Jp for all u E 8E, we also need the following two lemmas that describe some useful properties of the set E.

6 The Yamabe Problem and Prescribing Scalar Curvature on S", for n > 3

198

Lemma 6.3.3 (On the `center of mass') (i) Let q, A, and q be defined by (6.38). Then for sufficiently small q, Ig12 < C(IaJ2 + ),4).

(6.55)

(ii) Let po, q, and v be defined by (6.46). Then for po sufficiently small, (6.56)

Po si(xi). From this, we derive 2

2

ui(ai)(ri - jai - x''I)Pi > ui(xi)ri=' = (2K)P-i It follows that

ri-jai-x`j>AiK,

(6.79)

2 hence

BAiK(ai) C Bri(x`).

To see the second part, we use si(ai) > si(x) and derive from (6.79), 2

ui(x) < ui(ai)

1

r1 - jai - xil \ Pi-1

ri - Ix - x`lJ a

G ui(ai) 2Pi=1

Vx E BA:K(ai).

Now we can rescale: ui(a1

Vi(x)

=

i)ui(Aix + ai).

Then it satisfies R(Aix + ai)vri(x)

with R(Aix + ai)--30.

Obviously vi is bounded in Bx(0), and it follows by a standard argument that {vi} converges to a harmonic function vo in BK(0) C R", with vo(0) = 1. Consequently for i sufficiently large, vi(x)r+1dx > cK",

(6.80)

for some c > 0.

On the other hand, the boundedness of the energy E(ui) implies that uir}1dV 0, uq+1dV > f va+ldx. (6.82)

J S.

JBx(0)

Obviously, (6.81) and (6.82) contradict with (6.80). This completes the proof of the Proposition.

206

6 The Yamabe Problem and Prescribing Scalar Curvature on S", for n > 3

6.4.3 In the Regions Where R > 0. Proposition 6.4.3 Let {up} be solutions of equation (6.35) obtained by the variational approach in Section 6.3. Then there exists a po < r, such that for all po < p < r and for any e > 0, {up} are uniformly bounded in the regions where R(x) > e. Proof. The proof is divided into 3 steps. In Step 1, we argue that the solutions

can only blow up at finitely many points because of the energy bound. In Step 2, we show that there is no more than a one-point blow up and the point must be a local maximum of R. This is done mainly by using a Pohozaev type identity. In Step 3, we use (6.78) to conclude that even a one-point blow up is impossible.

Step 1. The argument is standard. Let {x'} be a sequence of points such that and xi-*xo with R(xo) > 0. Let ri, ai, si(x), and vi(x) be defined as in the previous subsection. Then

-Ovi(x) = R(Aix + ai)vP;(x)

(6.83)

The difference between this and the previous case is that now

R(Aix + ai)-.R(xo) > 0. Similar to the previous case, one can derive that vi(x) is uniformly bounded in BK(0) for any fixed K > 0. It follows from (6.83) that wiIIw=.9(BK(o)) < C

for any q > 1. Hence by Sobolev embedding, wiIIC ,a(BK(o)) < C.

Applying the Schauder estimate to equation (6.83), we have IViIIC3.°(BK(0)) < C.

Therefore, {vi(x)} converges in C2,°(BK(0)) to a function vo(x), and

R(xo)vo(x), Vx E BK(0).

(6.84)

Since K > 0 is arbitrary, equation (6.84) actually holds for all x in R". Then by the classification result, vo(x) must be radially symmetric about some point in R" and assume a certain form (see for example [CL1), or Chapter 8 for more details). Hence va+i(x)dx > co > 0.

R^

It follows that, for large i,

6.4 The a priori Estimates for Prescribing Scalar Curvature

f

ur+l(x)dV >

2> Co

207

0.

Because the total energy of ui is bounded, we can have only finitely many

such x0. Therefore, {ui} can only blow up at finitely many points. Step 2. We have shown that {ui} has only finitely many blow-up points. As a consequence of a result in [LiY2] (also see [CL6] or [SZ]), we have

Lemma 6.4.1 Let R satisfy the `flatness condition' in Theorem 1, Then {ui} can have at most one simple blow-up point, and this point must be a local maximum of R. Moreover, {ui} behaves almost like a family of the standard with Ai = functions and with 4 at the maximum of R. (maxui)_n?z

Step 3. Finally, we show that even a one-point blow up is impossible. For convenience, let {ui} be the sequence of critical points of Jp, we obtained in Section 6.3. From the proof of Proposition 6.3.2, one can obtain

JJ(ui) <minR(rk)IS"I -b,

(6.85)

for all positive local maxima rk of R because each ui is the minimum of Jr. on a path. Now if {ui} blows up at x0, then by Lemma 6.4.1, we have JT(u )-YR(x0)IS^I.

This is a contradiction with (6.85). Therefore, the sequence {ui} is uniformly bounded and possesses a subsequence converging to a solution of (6.6). This completes the proof of Theorem 6.1.3.

7

Maximum Principles

7.1

Introduction

7.2

Weak Maximum Principle

7.3

The Hopf Lemma and Strong Maximum Principle

7.4

Maximum Principle Based on Comparison

7.5

A Maximum Principle for Integral Equations

7.1 Introduction As a simple example of the Maximum Principle, let's consider a C2 function u(x) of one independent variable x. It is well known in calculus that at a local maximum point x° of u, we must have

-(x°) < 0. Based on this observation, then we have the following simplest version of the Maximum Principle: Assume that u"(x) > 0 in an open interval (a, b); then it cannot have any interior maximum in the interval. One can also see this geometrically. Since under the condition u"(x) > 0, the graph is concave up, it can not have any local maximum. More generally, for a C2 function u(x) of n independent variables x =

(xi,

,

at a local maximum x°, we have D2u(x°) := (us1 ; (x')) < 0,

7 Maximum Principles

210

that is, the symmetric matrix is non-positive definite at point x°. Correspondingly, the simplest version of the Maximum Principle reads as follows:

If

(aij (x)) ?0 and Eaij(x)uz;y;(x)>0

(7.1)

ij

in an open bounded domain fl, then u cannot achieve its maximum in the interior of the domain. An interesting special case is when (aij (x)) is an identity matrix, in which case condition (7.1) becomes

Au>0. Unlike its one-dimensional counterpart, condition (7.1) no longer implies that the graph of u(x) is concave up. A simple counterexample is this: z u(xl,xz) = x l -

2 x2'

One can easily see from the graph of this function that (0, 0) is a saddle point.

In this case, the validity of the Maximum Principle derives from the following simple algebraic fact:

For any two n x n matrices A and B, if A > 0 and B < 0, then AB < 0. In this chapter, we will introduce various Maximum Principles, and most of them will be used in developing the method of moving planes in the next chapter. Of course, there are numerous other applications of maximum principles, and we list some of these below. i) Providing Estimates

Consider the boundary value problem

Au = f (x) , x E Br(0) C Rn U(X) = 0,

(7.2)

x E OB1(0).

If a < f(x) < b in B1(0), then we can compare the solution u with the two functions

2n(1-x12) and (1-Ix12) which satisfy the equation with f (x) replaced by a and b, respectively, and which vanish on the boundary as it does. Now, applying the Maximum Principle for the A operator (see Theorem 7.1.1 in the following), we obtain

2n(1 -

Ix12)

< u(x) < 2n (l

ii) Proving Uniqueness of Solutions

-

Ix12)_

7.1 Introduction

211

In the above example, if f (x) _ 0, then we can choose a = b = 0, and this implies that it - 0. In other words, the solution of the boundary value problem (7.2) is unique. iii) Establishing the Existence of Solutions (a) For a linear equation such as (7.2) in any bounded open domain S2, let

u(x) = sup O(x), where the sup is taken over all functions that satisfy the corresponding differential inequality

5 p¢C f(x),xES2 1O-(x)=0,

xEOQ.

Then u is a solution of (7.2). (b) Now consider the nonlinear problem

Lu=f(u),xES2 {u(x)=0,xEOQ. Assume that f() is a smooth function with f'( ) > 0. Suppose that there exist two functions u(x) < u(x), such that -ou < f(u) < f(u) < -Au.

These two functions are called sub (or lower) and super (or upper) solutions, respectively, as we have seen in Chapter 5. To seek a solution of problem (7.3), we use successive approximations. Let

-,Lul = f(jk) and - Dui+r = f(ui). Then by the Maximum Principle, we have

v_, < ul < 1t2 < ... < ni < ... < 9r. Let u be the limit of the sequence {ui}:

u(x) = limui(x). Then u is a solution of the problem (7.3). This chapter is organized as follows. In the remainder of this section, we introduce both strong and weak maximum principles; in Section 7.2 we prove weak maximum principles, and in Section 7.3 we prove strong maximum principles using the Hopf Lemma. In Section 7.4, as a consequence of Theorem

7.1.5, we derive the `Narrow Region Principle' and the `Decay at Infinity Principle'. These principles can be applied very conveniently in the `Method of. Moving Planes' to establish the symmetry of solutions for semi-linear elliptic equations, as we will see in later sections. Finally, in Section 7.5, we establish a Maximum Principle for integral inequalities.

7 Maximum Principles

212

Theorem 7.1.1 (Weak Maximum Principle for -A.)

i)If -Au(x) > 0, x E .fl, then

min u > min u.

- as?

f2

ii) If -Au(x) < 0,

x E £7,

then

max u < max u.

an

n

This result can be extended to general uniformly elliptic operators. Let x

09

Di = a

i,

Di

Define

L = - E aii (x)Dii + E bi (x)Di + c(x).

ii

i

In the following we always assume that affi(x), bi(x), and c(x) are bounded continuous functions in £7, and we also assume that L is uniformly elliptic,

that is, aii(xgi6i ? bI6I2 for any x E .fl, any e E R" and for some 5 > 0.

Theorem 7.1.2 (Weak Maximum Principle for L) Assume that c(x) i) If Lu > 0 in .2, then

0.

min u > min u. f2

- 8111

ii) If Lu < 0 in .2, then max u < max u. Q

an

These Weak Maximum Principles infer that the minima or maxima of u are attained at some points on the boundary a.fl. However, they do not exclude the possibility that the minima or maxima may also occur in the interior of Q. Actually this cannot happen unless it is constant, as we will see in the following.

7.1 Introduction

213

Theorem 7.1.3 (Strong Maximum Principle for L with c(x) _ 0) Assume that 12 is an open, bounded, and connected domain in Rn with smooth bound-

ary 812. Let it be a function in C2(Q) n C(Q). Assume that c(x) = 0 in 12.

i) If Lu(x) > 0, x E .(2, then it attains its minimum value only on 80 unless it is constant.

ii) If

Lu(x) 0 with slight modifications.

Theorem 7.1.4 (Strong Maximum Principle for L with c(x) > 0) Assume that S? is an open, bounded, and connected domain in Rn with smooth bound-

ary 8I1. Let it be a function in C2(0) n C(0). Assume that c(x) > 0 in R.

i) If Lu(x) > 0, x E fl, then it can not attain its non-positive minimum in the interior of 0 unless it is constant.

ii) If Lu(x) < 0, x E ,fl, then u can not attain its non-negative maximum in the interior of Q unless u is constant.

Notice that all the previous theorems require that c(x) > 0. Roughly speaking, Maximum Principles hold for `positive' operators. -A is `positive',

and obviously so is -A + c(x) if c(x) > 0. However, as we will see in the next chapter, in practice the condition c(x) > 0 frequently cannot be met. But do we really need c(x) > 0? The answer is `no.' Actually, if c(x) is not "too negative", then the operator '-A+ c(x)' can still remain "positive" to ensure the Maximum Principle. As mentioned earlier, these will be studied in Section 7.4 where we prove the `Maximum Principles Based on Comparisons'. Let ¢ be a positive function on R satisfying

-AO + A(x)O > 0.

(7.4)

Let it be a function such that

Du + c(x)u ? O x E 1? u on 490. I >0

(7.5)

7 Maximum Principles

214

Theorem 7.1.5 (Maximum Principle Based on Comparison) Assume that 12 is a bounded domain. If c(x) > A(x) ,Vx E 12,

then it > 0 in ,2.

7.2 Weak Maximum Principles In this section, we prove the Weak Maximum Principles.

Theorem 7.2.1 (Weak Maximum Principle for -z.) i) If -Du(x) > 0, x E I2,

(7.6)

then

minu > minu. an B

ii) If -Au(x) < 0,

x E 12,

then

max it < max it. an Li Proof. Here we only present the proof of part i). An entirely similar proof also works for part ii). To better illustrate the idea, we treat the one-dimensional and higher-

dimensional cases separately.

First, let 12 be the interval (a, b). Then condition (7.6) becomes u"(x) < 0. This implies that the graph of u(x) on (a, b) is concave downward, and therefore one can roughly see that the values of u(x) in (a, b) are large or equal to the minimum value of it at the end points (See figure).

a

b

To prove the above observation rigorously, we first carry it out under the stronger assumption that

7.2 Weak Maximum Principles

215

-u°(x) > 0. (7.10) Let in = minan it. Suppose, contrary to (7.7), that there is a minimum x° E (a, b) of it, such that u(x°) < rn. Then by the second order Taylor expansion of it around x°, we must have -u" (x°) < 0. This contradicts with the assumption (7.10).

Now for it satisfying only the weaker condition (7.6), we consider a perturbation of u: U,(x) = u(x) - Ext.

Obviously, for each E > 0, u,(x) satisfies the stronger condition (7.10), and hence

min u, > min u,. 00

T2

Now letting E-40, we arrive at (7.7). To prove the theorem in dimensions higher than one, we need the following lemma:

Lemma 7.2.1 (Mean Value Inequality) Let x° be a point in Si. Let Br(x°) C 11 be the ball of radius r, center x°, and boundary DBr(x°)

i) If -Au(x) > ( or =) 0 for x E Bro(x°) with some r° > 0, then for

anyr°>r>0,

u(x°)

1

> ( or =) BBr(x°)I

u(x)dS.

(7.11)

JOB,(.-)

It follows that, if x° is a minimum of it in 12, then

-Lu(x°) < 0.

(7.12)

ii) If -Au(x) < 0 for x E Bro(x°) with some r° > 0, then for any r° > r > 0, 1 (7.13) u(x°) < (x°)I JaBr(so) u(x)dS. 1'9B,

It follows that, if x° is a maximum of it in fl, then

-Au(x°) > 0.

(7.14)

We postpone the proof of the lemma for a moment. This lemma tells us

that, if -Au(x) > 0, then the value of u at the center of the small ball Br(x°) is larger than its average value on the boundary 8Br(x°). Now based on the lemma, to prove the Theorem, we first consider u(x) EIxI2. Obviously,

216

7 Maximum Principles

-AuE = -Au + 2en > 0.

(7.15)

Hence we must have

nnu, > min us.

(7.16)

a

Otherwise, if there exists a minimum x° of u in !2, then by Lemma 7.2.1, we have -Auj(x°) < 0. This contradicts with (7.15). Now in (7.16), letting e-*0, we arrive at the desired conclusion (7.7). This completes the proof of the Theorem.

The Proof of Lemma 7.2.1. By the Divergence Theorem,

JB,(.,)

Au(x)dx =

au dS = r'-'

J&B,.(x") 8v

r

au (x° + rw)dS° ur

(7.17)

where dSw is the area element of the (n - 1)-dimensional unit sphere Sn-1 =

{w w1 =1}. If An < 0, then by (7.17), { Jga

1

u(x +rw)dS,} < 0.

(7.18)

Integrating both sides of (7.18) from 0 to r yields

u(x°+rw)dS - u(x)IS"-1I 0.

Theorem 7.2.2 (Weak Maximum Principle for L with c(x) - 0). Let L be the uniformly elliptic operator defined above. Assume that c(x)

0.

i) If Lu > 0 in Q, then minu > min u. an n

(7.20)

ii) If Lu < 0 in 12, then max u < max u. an Q

For c(x) > 0, the principle still applies with slight modifications.

Theorem 7.2.3 (Weak Maximum Principle for L with c(x) > 0). Let L be the uniformly elliptic operator defined above. Assume that c(x) > 0. Let u (x) = min{u(x),0}

and u+(x) = max{u(x),0}.

i) If Lu > 0 in ,fl, then

minu>minu . D

- an

ii) If Lu < 0 in 0, then maxu < maxu+. Q

- an

Interested readers may find its proof in many standard books (see [Ev], p. 327).

7 Maximum Principles

218

7.3 The Hopf Lemma and Strong Maximum Principles In the previous section, we proved a weak form of the Maximum Principle. In

the case Lu > 0, it concludes that the minimum of u attains at some point on the boundary 812. However it does not exclude the possibility that the minimum may also attain at some point in the interior of fl. In this section, we will show that this cannot actually happen; that is, the minimum value of u can only be achieved on the boundary unless u is constant. This is called the "Strong Maximum Principle," which we will prove using the following:

Lemma 7.3.1 (Hopf Lemma). Assume that d2 is an open, bounded, and connected domain in Rn with smooth boundary 811. Let u be a function in C2 (0) fl C(12) . Let

L = - L aij (x)Dij + E bi(x)Di + c(x) z7

a

be uniformly elliptic in 12 with c(x) __ 0. Assume that

Lu> 0

in Q.

(7.21)

Suppose there is a ball B contained in 12 with a point x° E 812 fl 8B and suppose

u(x) > u(x°), Vx ED.

(7.22)

Then for any outward directional derivative at x°,

8u(x°) 8v

0, if we require additionally that u(x°) < 0, then the

same conclusion of the Hopf Lemma holds.

Proof. Without loss of generality, we may assume that B is centered at the origin with radius r. Define w(x) = e

e-a

n12.

Consider v(x) = u(x) + ew(x) on the set D = Br (x°) fl B (See figure).

7.3 The Hopf Lemma and Strong Maximum Principles

219

We will choose a and e appropriately so that we can apply the Weak

Maximum Principle to v(x) and arrive at

v(x) > v(x°) Vx E D.

(7.24)

We postpone the proof of (7.24) for a moment. Now from (7.24), we have (y) < 0.

(7.25)

Noticing that C9W

8v (x°) > we arrive at the desired inequality

0,

-(x) < 0. Now to complete the proof of the lemma, we only need verify (7.24). We will carry this out in two steps. First we show that

Lv > 0.

(7.26)

Hence we can apply the Weak Maximum Principle to conclude that minv > min v. D

(7.27)

OD

We then show that the minimum of v on the boundary 8D is actually attained at x°: (7.28) v(x) > v(x°) Vx E 8D. Obviously, (7.27) and (7.28) imply (7.24). To see (7.26), we directly calculate

7 Maximum Principles

220

Lw =

(7.29) n

e °IsI2 {4a2 E aij(x)xixj - 2a E[affi(x) - bi(x)xi] - c(x)} +c(x)e'a'' i,j=1

>

i=1

e-aIx2{4a2

n

E aij(x)xixj-2aE[aii(x)-bi(x)x,j-c(x)} (7.30) i,j=1

i=1

By the ellipticity assumption, we have

1 aj(x)xixj > 61xI2

> b(2)2 > 0 in D.

(7.31)

i,j=1

Hence we can choose a sufficiently large, such that Lw > 0. This, together with the assumption Lu > 0 implies Lv > 0, and (7.27) follows from the Weak Maximum Principle. To verify (7.28), we consider two parts of the boundary 8D separately.

(i) On 8D n B, since u(x) > u(x°), there exists a c° > 0, such that u(x) > u(x°) + c°. Take e small enough such that eIwI < b on 8D n B. Hence v(x) > u(x°) = v(x°) Vx E 8D n B.

(ii) On 8D n 8B, w(x) = 0, and by the assumption u(x) > u(x°), we have v(x) > v(x°). This completes the proof of the Lemma. We are now ready to prove the following theorem:

Theorem 7.3.1 (Strong Maximum Principle for L with c(x) - 0.) Assume that 1? is an open, bounded, and connected domain in Rn with smooth bound-

ary 80. Let a be a function in C2(dl) n C(Q). Assume that c(x)

0 in

Q.

i) If Lu(x) > 0, x E ,fl, then it attains its minimum only on OR unless it is constant.

ii) If Lu(x) < 0, x E ,fl, then u attains its maximum only on 8.2 unless u is constant.

Proof. We prove part i) here. The proof of part ii) is similar. Let m be the minimum value of it in ,2. Set E _ {x E 9 1 u(x) = m}. It is relatively closed in ,2. We show that either E is empty or E = R.

7.3 The Hopf Lemma and Strong Maximum Principles

221

We argue by contradiction. Suppose E is a nonempty proper subset of Q. Then we can find an open ball B C 1? \ E with a point on its boundary

belonging to E. Actually, we can first find a point p E .(2 \ E such that d(p, E) < d(p, 017), then increase the radius of a small ball center at p until it hits E (before hitting 852). Let x° be the point at 8B n E. Obviously we have in B Lu > 0 and u(x) > u(x°).

Now we can apply the Hopf Lemma to conclude that the normal outward derivative i9u

(x) < 0.

(7.32)

8v On the other hand, x° is an interior minimum of u in S2, and we must have Du(x°) = 0. This contradicts with (7.32) and hence completes the proof of the Theorem. In the case when c(x) > 0, the strong principle still applies with slight

modifications.

Theorem 7.3.2 (Strong Maximum Principle for L with c(x) > 0.) Assume that 52 is an open, bounded, and connected domain in R with smooth bound-

ary 852. Let it be a function in C2(S2) n C(.fl). Assume that c(x) > 0 in Q.

i) If

Lu(x) ?0, x(= Sl, then u can not attain its non-positive minimum in the interior of 5l unless it is constant.

ii) If Lu(x) < 0, x E Sl, then it can not attain its non-negative maximum in the interior of S2 unless u is constant.

Remark 7.3.1 In order that the Maximum Principle hold, we assume that the domain 52 be bounded. This is essential, since it guarantees the existence of maximum and minimum of it in Q. A simple counter example is when 52 is

the half space {x E R' I x > 0}, and u(x) = xn. Obviously, Lu = 0, but u does not obey the Maximum Principle: max u < max u. an 92

222

7 Maximum Principles

Equally important is the non-negativeness of the coefficient c(x). For exam-

ple, set Q = { (x, y) E R2 I - 2 < x < 2 , - 2 < y < 2 }. Then u = cos x cos y satisfies

Du-2u=0, in S? on 80.

U = 0,

But obviously u attains its maximum at the origin: u(0, 0) = 1. However, if we impose some sign restriction on u, say u > 0, then both conditions can be relaxed. A simple version of such a result will be present in the next theorem. Also, as will be shown in the next section, c(x) is actually allowed to be negative, but not "too negative."

Theorem 7.3.3 (Maximum Principle and Hopf Lemma for not necessarily bounded domain and not necessarily non-negative c(x). ) Let .fl be a domain in Rn with smooth boundary. Assume that u E C2 (.2) n C(Q) and satisfies

Lu+E bi(x)Diu+c(x)u>0, u(x)>0,xE.2 1

U(X) = 0

x E OQ

(7.33)

with bounded functions bi(x) and c(x). Then

0 in .2; and

i) if u vanishes at some point in 0, then u

ii) if u *0 in .2, then on 80, the exterior normal derivative av < 0. To prove the Theorem, we need the following Lemma concerning eigenvalues.

Lemma 7.3.2 Let .X be the first positive eigenvalue of enso = Ai (x), x E Bl(0) ¢(x) = 0, x e 8B1(0)

(7.34)

with the corresponding eigenfunction ¢(x) > 0. Then for any p > 0, the first positive eigenvalue of the problem on Bp(0) is

a2

. More precisely, if we let

O(x) _ ¢(P), then ,n,V) =

O(x) = 0,

Pl0(x), x E

Bp(0)

x E 8Bp(0).

The proof is straightforward and is left for the reader.

(7.35)

7.3 The Hopf Lemma and Strong Maximum Principles

223

The Proof of Theorem 7.3.3. i) Suppose that u = 0 at some point in 12, but u ,E 0 on Q. Similar to the proof of the Strong Maximum Principle, let E = {x

S? I u(x) = 0}.

It is relatively closed in d2. We show that either E is empty or E = 12. We argue by contradiction. Suppose E is a nonempty proper subset of 12. Then we can find an open ball B C 12 \ E with a point on its boundary

belonging to E. Actually, we can first find a point p E 12 \ E such that d(p, E) < d(p,812), then increase the radius of a small ball center at p until it hits E (before hitting 812). Let x° be the point at 8B n E. Obviously we have in B u(x) > u(x°).

For p > 0 sufficiently small, one can choose a ball Bp 2(x) C B with x° as its boundary point. Let 0 be the positive eigenfunction of the eigenvalue Al. problem (7.35) on Bp(x°) corresponding to the eigenvalue Obviously, P

Bp(x°) completely covers B,/200. Let v = U. Then from (7.33), it is easy to

deduce that

0 0. Now we can apply the Hopf Lemma to conclude that the outward normal derivative at the boundary point x° of Bp12(a), (7.36) because v(x) > 0 Vx E Bp 2(x) and v(x°) _

0.

On the other hand, since x° is also a minimum of v in the interior of 12, we must have

7 Maximum Principles

224

'7v(x°) = 0.

This contradicts with (7.36) and hence proves part i) of the Theorem. ii) The proof is almost the same as in part i) except we consider the point

x° on ad2 and the ball B,,/2(±) is in fl with x° E 8B.12(x). Then for the outward normal derivative of u, we have an av (x)

=

av av

a v(im) av

49V

av

0.

Here we have used a well-known fact that the eigenfunction 7P on B,(x°) is ra-

dially symmetric about the center x°, and hence 7'(x°) = 0. This completes the proof of the Theorem.

7.4 Maximum Principles Based on Comparisons In the previous section, we showed that if (-L + c(x))u > 0, then the Maximum Principle, i.e., (7.20), applies. There, we required c(x) > 0. We can think

of -0 as a `positive' operator, and the Maximum Principle holds for 'positive' operators. For c(x) > 0, -L + c(x) is also `positive'. Do we really need c(x) > 0 here? To answer the question, let us consider the Dirichlet eigenvalue

problem of -&

¢ - ao(x) = 0, x E fl x E an.

{ ¢(x) = 0,

(7.37)

We notice that the eigenfunction 0 corresponding to the first positive eigenvalue al is either positive or negative in ,fl. That is, the solutions of (7.37) with A = ar obeys the Maximum Principle; that is, the maxima or minima of 0 are attained only on the boundary a0. This suggests that, to ensure the Maximum Principle, c(x) need not be nonnegative and is allowed to be as negative as -A1. More precisely, we can establish the following more general Maximum Principle based on comparison.

Theorem 7.4.1 Assume that I? is a bounded domain. Let ¢ be a positive function on fl satisfying

-AO +A(x)O > 0.

(7.38)

Assume that u is a solution of

Du+c(x)u>OxEQ

I u>O

on OR.

(7.39)

If c(x) > A(x)

then is > 0 in ,fl.

,

Vx E 0,

(7.40)

7.4 Maximum Principles Based on Comparisons

225

proof. We argue by contradiction. Suppose that u(x) < 0 somewhere in Q.

Let v(x) = -(-T) . Then since O(x) > 0, we must have v(x) < 0 somewhere in ,fl. Let x° e f2 be a minimum of v(x). By a direct calculation, it is easy to verify that -,Lv = 2pv 0 + 1(-Au + u). (7.41) On the one hand, since x° is a minimum, we have

0 and pv(x°) = 0,

(7.42)

while on the other hand, by (7.38), (7.39), and (7.40), and taking into account

that u(x°) < 0, we have at point x°,

-Au+

u(X') > -Du+A(x°)u(x°)

> -Lu+c(x°)u(X') > 0. This is an obvious contradiction with (7.41) and (7.42) and thus completes the proof of the Theorem.

Remark 7.4.1 From the proof, one can see that conditions (7.38) and (7.40) are required only at the points where v attains its minimum, or at points where u is negative. The Theorem is also valid on an unbounded domains if u is "nonnegative" at infinity:

Theorem 7.4.2 If 0 is an unbounded domain, besides condition (7.39), we assume further that liminfu(x)/fi(x) > 0, (7.43) I=I*o where 0 is the function mentioned in Theorem 7.4.1. Then u > 0 in P. Proof. Still consider the same v(x) as in the proof of Theorem 7.4.1. Now condition (7.43) guarantees that the minima of v(x) do not "leak" away to infinity. Then the rest of the arguments are exactly the same as in the proof of Theorem 7.4.1.

For convenience in applications, we provide two typical situations where there exist such functions 0 and c(x) satisfying condition (7.38) and (7.40), so that the Maximum Principle Based on Comparison applies:

226

7 Maximum Principles

i) Narrow regions, and

ii) c(x) decays fast enough at oo. i) Narrow Regions. When

f2={xl0<x1 A(x) = Hence we can directly apply Theorem 7.4.2 to conclude that u > 0 in fl, provided lim inf x _- u(x)/¢(x) > 0. l2.

ii) Decay at Infinity. In dimension n > 3, one can choose some positive number q < n - 2, and let ¢(x) = s , Then it is easy to verify that q(n Ixl2 2 - q)

In the case c(x) decays fast enough near infinity, we can derive from Theorem 7.4.2 that

Corollary 7.4.2 (Decay at Infinity) Assume there exists R > 0, such that C(X) > - q(n

Ixl2 -

q)

V ixi > R.

(7.44)

Suppose

lim u(x)

IX19 = 0.

Let .2 be a region containing in BR (0) = R" \ BR(0). If it satisfies (7.39) on .2, then

u(x) >0 for all xe!?.

7.5 A Maximum Principle for Integral Equations

227

Remark 7.4.2 F1om Remark 7.4.1, one can see that actually condition (7.44) is only required at points where u is negative.

Remark 7.4.3 Although Theorem 7.4.1 as well as its Corollaries are stated in linear forms, they can easily be applied to a nonlinear equation, for example,

-Au - IuIP-1u = 0 x c R".

(7.45)

Assume that the solution u decays near infinity at the rate of lIs with x s(p- 1) > 2. Let c(x) = -Iu(x)Ip-1. Then for R sufficiently large, and for the I

region Sl as stated in Corollary 7.4.2, c(x) satisfies (7.44) in S2. If we further assume that

UIon>0, then we can derive from Corollary 7.4.2 that u > 0 in the entire region fl.

7.5 A Maximum Principle for Integral Equations In this section, we introduce a Maximum Principle for integral equations. Let Sl be a region in Rn, which may or may not be bounded. Assume

K(x,y)?0, V(x,y)E.2xS2. Define the integral operator T by

(Tf)(x) = Let

.

Jn

K(x,y)f(y)dy

be a norm in a Banach space satisfying If II 3;

and

-Du = e°iyl, x E R2. During the moving of planes, the Maximum Principles Based on Comparison

will play a major role. In particular, the Narrow Region Principle and the Decay at Infinity Principle will be used repeatedly in dealing with the three examples.

8.1 Outline of the Method of Moving Planes

233

In Section 8.3, we will apply the Method of Moving Planes in a `local way' to obtain a priori estimates on the solutions of the prescribing scalar curvature equation on a compact Riemannian manifold M 4(n

n

I)

L0u+

R0(x)u = R(x)u-2 x E M. ,

2

We allow the function R(x) to change sign. In this situation, the traditional blowing-up analysis fails near the set where R(x) = 0. We will use the Method of Moving Planes in an innovative way to obtain a priori estimates. Since the

Method of Moving Planes cannot be directly applied to the solution it,, we introduce an auxiliary function to circumvent this difficulty. In Section 8.4, we use the Method of Moving Spheres to prove a nonexistence of solutions for the prescribing Gaussian and scalar curvature equations

-Du + 2 = R(x)e" and

-Lu + n(n4 2)u

=

4(n

- 1)R(x)u

on S2 and on S' (n > 3), respectively. We prove that if the function R(x) is rotationally symmetric and monotone in the region where it is positive, then both equations admit no solution. This provides a stronger necessary condition than the well known Kazdan-Warner condition, and it also becomes a sufficient condition for the existence of solutions in most cases. In Section 8.5, as an application of the Maximum Principle for integral equations introduced in Section 7.5, we study the integral equation in R"

u(x)='

n

yl"a

U-(y)dy,

hx for any real number a between 0 and it. It arises as an Euler-Lagrange equation for a functional in the context of the Hardy-Littlewood-Sobolev inequalities. Due to the different nature of the integral equation, the traditional Method of Moving Planes does not work. Hence we exploit its global property and develop a new idea, the Integral Form of the Method of Moving Planes to obtain the

symmetry and monotonicity of the solutions. The Maximum Principle for Integral Equations established in Chapter 7 is combined with the estimates of various integral norms to carry out the moving of planes.

8.1 Outline of the Method of Moving Planes To outline how the Method of Moving Planes works, we take the Euclidean

space R" as an example. Let it be a positive solution of a certain partial

8 Methods of Moving Planes and Moving Spheres

234

differential equation. If we want to prove that it is symmetric and monotone in a given direction, we may assign that direction the xl axis. For any real number A, let TA = {x =

(xl,x2,...

xn) E R" I xl = A}.

This is a plane perpendicular to the xl-axis and is the plane with which we shall move. Let E'a denote the region to the left of the plane, i.e.,

La= {xER"Ix1 0 and x° E Rn.

ii) For p < 2, the only nonnegative solution of (8.10) is identically zero. The proof of Theorem 8.2.2 is actually included in the more general proof of the first part of Theorem 8.2.3. However, to better illustrate the idea, we will first present the proof of Theorem 8.2.2 (mostly in our approach). And the readers will see vividly how the "Decay at Infinity" principle is applied here.

Proof of Theorem 8.2.2. Define

Ea={x=(xi,...,x,) ERnIxl 0, and therefore we must have 'wAo (x°) = 0.

It follows that

vwA0(x°)=vwA0(x°)0(x )+1VAo(x°)04(x)=0+0=0.

(8.18)

On the other hand, by (8.14), since wA0(x°) = 0, x° must be on the boundary of EAp. Then by the Hopf Lemma (see Theorem 7.3.3), we have the outward normal derivative OWA (x°) < 0.

This contradicts with (8.18). Now, to verify (8.15), we need only prove the Lemma.

Proof of Lemma 8.2.1. Assume that x° is a negative minimum of WA. Then -AwA(x°) < 0 and VWA(x°) = 0. (8.19) On the other hand, as we argued in Step 1, by the asymptotic behavior of is at infinity, if x°j is sufficiently large, c(x°)

> _ q(n - 2 - q) _ 0(x°) x012

-

O(x°)

8.2 Applications of the Maximum Principles Based on Comparisons

243

It follows from (8.12) that 'nSwa

+wa) (x°) > 0.

This, together with (8.19) contradicts with (8.16) and hence completes the proof of the Lemma. Step 3. In the previous two steps, we showed that the positive solutions of (8.10) must be radially symmetric and monotone decreasing about some point in R. Since the equation is invariant under translation, without loss of generality, we may assume that the solutions are symmetric about the origin. They then satisfy the following ordinary differential equation -u" (r) - n=1u'(r) = ,aT r

u'(0)=0 u(0)

n(n 2) (n-2)/4 Z =2

2)a ] 2 (A2 + r2) r2

for some A > 0. One can verify that u(r) = [n(n

is a solution, and

by the uniqueness of the ODE problem, this is the only solution. Therefore, we conclude that every positive solution of (8.10) must assume the form u(x) = for A > 0 and some x° (=-

[n(n-2)A2]P n2 2

(A2 + Ix - x° I2)

Rn. This completes the proof of the Theorem.

Proof of Theorem 8.2.3. i) The general idea in proving this part of the Theorem is almost the same as that given for Theorem 8.2.2. The main difference is that we have no decay assumption on the solution u at infinity, hence the Method of Moving Planes cannot be directly applied to u. So we first make a Kelvin transform to define a new function v(x) =

Then obviously, v(x) has the desired decay rate Ixln_2 at infinity, but has a

possible singularity at the origin. It is easy to verify that v satisfies the same equation as u does, except at the origin:

Av+vT(x)=0, xERn\{0},n>3.

(8.20)

We will apply the Method of Moving Planes to the function v and show that v is radially symmetric and monotone decreasing about some point. If the

244

8 Methods of Moving Planes and Moving Spheres

center point is the origin, then by the definition of v, u is also symmetric and monotone decreasing about the origin; if the center point of v is not the origin, then v has no singularity at the origin, and hence it has the desired decay at infinity. Then the same argument in the proof of Theorem 8.2.2 would imply that it is symmetric and monotone decreasing about some point. Define va(x) = v(x'), vJA(x) = va(x) - v(x).

Because v(x) may be singular at the origin, correspondingly wa may be singular at the point xa = (2A, 0, .. 0). Hence instead of on Ea, we consider wa on Ea = Ea \ {x ,}. And in our proof, we treat the singular point carefully. Each time, we show that the points of interest are away from the singularities so that we can carry on the Method of Moving Planes to the end to show the existence of a a° such that wao (x) _- 0 for x E Eaa and that v is strictly increasing in the x1 direction in Eao. As in the proof of Theorem 8.2.2, we see that v), satisfies the same equation as v does, and -Owa = To' (x)wa(x), where *a(x) is some number between vA(x) and v(x). ,

Step 1. We show that for A sufficiently negative, we have

wa(x) > 0, Vx E EA.

(8.21)

By the asymptotic behavior v(x)

1xln-2,

we derive immediately that, at a negative minimum point x° of wa, Y',T-1(x°)

the power of

of I

(1x1n-2)T-1

- IxIV

is greater than two, and we have the desired decay rate

x

for c(x) := as mentioned in Corollary 7.4.2; hence, we can apply "Decay at Infinity" to w), (x). The difference here is that wa has a singularity at xa, and hence we need to show that the minimum of wa is away from xa. Actually, we will show that

If infta w,\(x) < 0, then the infimum is achieved in Ea \ B1(xa). To see this, we first note that for x E Br (0), v(x) > min v(x) = co > 0 8B1(0)

(8.22)

8.2 Applications of the Maximum Principles Based on Comparisons

245

due to the fact that v(x) > 0 and Ov < 0. Then let A be so negative that v(x) < co for x E Bi(x,\). This is possible because v(x) -4 0 as xI -> oo. For such A, obviously wA(x) > 0 on B1(xa) \ {xA}. This implies (8.22). Now similar to Step I in the proof of Theorem 8.2.2, we can deduce that wa(x) ? 0 for A sufficiently negative. Step 2. Again, define A. = sup{A I WM(x) > 0, Vx E ti,, µ < A}.

We will show that

If ao < 0, then w ,, (x)

0, Vx E Eao.

Define IDa = T the same way as in the proof of Theorem 8.2.2. Suppose that wao (x) L0; then, by the Maximum Principle, we have wao(x) > 0

,

for x E Eao.

The rest of the proof is similar to step 2 in the proof of Theorem 8.2.2 except that now we need to take care of the singularities. Again, let Ak N ao be a sequence such that wak(x) < 0 for some x E EAk. We need to show that for each k, infE w,\, (x) can be achieved at some point xk E Eak and that the k

sequence {xk} is bounded away from the singularities xak of wa,,. This can be seen from the following facts:

a)

There exists e > 0 and S > 0 such that wao(x) > e for x E Bs(x),0) \ {xa0}.

b)

lima-.ap infXEBk(XA) wa(x) > infZEBS(%Ao) wao(x) > C.

Fact (a) can be shown by noting that 7-v,\, (x) > 0 on Eao and Away < 0, while fact (b) is obvious.

Now through a similar argument as given in the proof of Theorem 8.2.2 one can easily arrive at a contradiction. Therefore, wao (x) - 0.

If A, = 0, then we can carry out the above procedure in the opposite direction, namely, we move the plane in the negative x, direction from positive infinity toward the origin. If our planes TA stop somewhere before the origin, we derive the symmetry and monotonicity in xl direction by the above ar-

gument. If they stop at the origin again, we also obtain the symmetry and monotonicity in the xr direction by combining the two inequalities obtained

246

8 Methods of Moving Planes and Moving Spheres

in the two opposite directions. Since the x1 direction can be chosen arbitrarily, we conclude that the solution u must be radially symmetric about some point. ii) To show the non-existence of solutions in the case p < nn+2, we notice that after the Kelvin transform, v satisfies

Ov+ xn+2ly(n-2) vP(x) = 0 , x E R" \ {0}. Due to the singularity of the coefficient of vP at the origin, one can easily see that v can only be symmetric about the origin if it is not identically zero; hence, u must also be symmetric about the origin. Now given any two points x1 and x2 in Rn, since equation (8.10) is invariant under translations and rotations, we may assume that the origin is at the midpoint of the line segment x1x2. Then from the above argument, we must have u(x1) = u(x2). It follows that u is constant. Finally, from the equation (8.10), we conclude that u - 0. This completes the proof of the Theorem.

8.2.3 Symmetry of Solutions for -Au = e" in R2 When considering prescribing Gaussian curvature on two-dimensional compact manifolds: if the sequence of approximate solutions "blows up", then by re-scaling and taking the limit, one would arrive at the following equation in the entire space R2:

4u+expu=0, xER2 fR2 exp u(x)dx < +oo.

(8.23)

The classification of the solutions for this limiting equation would provide essential information on the original problems on manifolds (and it is also interesting in its own right). It is known that 32A2 OA,

(x) = In (4 + A21x _ x°12)2

for any A > 0 and any point x° E R2 is a family of explicit solutions. We will use the Method of Moving Planes to prove the following theorem:

Theorem 8.2.4 Every solution of (8.23) is radially symmetric with respect to some point in R2 and hence assumes the form of (x). To this end, we first need to obtain some decay rate of the solutions near

infinity.

8.2 Applications of the Maximum Principles Based on Comparisons

247

Some Global and Asymptotic Behavior of the Solutions The following Theorem gives the asymptotic behavior of the solutions near infinity, which is essential to the application of the Method of Moving Planes.

Theorem 8.2.5 If u(x) is a solution of (8.23), then as lxj -4 +oo, 1 (x ))

i - Zn fR, exp u(x) dx < -4

uniformly.

This Theorem is a direct consequence of the following two lemmas.

Lemma 8.2.2 (W. Ding) If it is a solution of

-Au=e", xER2, and

fR2 expu(x)dx < +oo, then

fR, exp u(x)dx > 8ir.

Proof. For -oo < t < oo, and letting Ot = {x I u(x) > t}, one can obtain f.at

u(x)dx = - 12, An = foot 1pulds

-dtIOtl =

Jan, Foul

By the Schwartz inequality and the isoperimetric inequality, ds

Jan, IvuI Jan

foul > lantl2 > 4trlQtl.

Hence

-(d IOt1) - fn, expu(x)dx > 47rIJ?tl, so

dt(fn

fn, expu(x)dx < -8trjQtlet.

expu(x)dx)2 = 2expt'

Integrating from -oo to oo gives

(J expu(x)dx) U2R

f

2

< -87r

JR'

which implies fR2 expu(x)dx > 81r as desired.

exp u(x)dx,

8 Methods of Moving Planes and Moving Spheres

248

Lemma 8.2.2 enables us to obtain the asymptotic behavior of the solutions at infinity.

Lemma 8.2.3. If u(x) is a solution of (8.23), then as IxI -r +oo,

1

IxI

-4

27r fR2 expu(x)dx uniformly.

Proof. By a result of Brezis and Merle [BM], we see that the condition fR2 exp u(x)dx < oo implies that the solution u is bounded from above. Let

w(x) =

,r

fR2(ln Ix - yI - ln(Iyl + 1)) expu(x)dy.

Then it is easy to see that ,nsw(x) = exp u(x) , x E R2, and we will show w(x) In IxI

-4

1

fR2

21

exp u(x)dx uniformly as IxI -4 +oo.

(8.24)

To see this, we need only verify that

I:= fR2

In Ix - yl - ln(Iyi + 1) - In lxl eU(y)dy->O In IxI

as IxI-ioo. Write I = I,+I2+I3i where 11, I2 and I3 are the integrals on the three regions

D,={yIIx-yIK}, respectively. We may assume that IxI > 3. a) To estimate I,, we simply notice that 11 < C

eu(y)dy

J Ix-yl0.

c)To see 13-+0, we use the fact that for Ix - yI > 1

In Ix-yl -ln(Iyi+1) -lnixiI 0, x E S"

(8.31)

for any exponent p between 1 + a and A, where A is an arbitrary positive number. For a fixed A, the bound is independent of p. Proposition 8.3.1 Assume R E C2,"(S") and IVR(x)I > /3o for any x with IR(x)I < So. Then there are positive constants e and C depending only on Qo, So, M, and IIRIIc2.n (sn), such that for any solution it of (8.31), we have it < C in the region where IR(x) I < e.

In this proposition, we essentially required R(x) to grow linearly near its zeros, which seems a little restrictive. Recently, in the case p = nn+2, Lin [Linl] weakens the condition by assuming only polynomial growth of R near its zeros. To prove this result, he first established a Lionville type theorem in R" and then used a `blowing up' argument. Later, by introducing a new auxiliary function, we sharpened our Method of Moving Planes to further weaken the condition and to obtain a more general result:

Theorem 8.3.1 Let M be a locally conformally flat manifold. Let

P= {xeMIR(x)=0}. OR

Assume that r E C2," and R E C2,°(M) satisfying < 0, where v is the outward normal (pointing to the region where R is negative) of P, and R(x)

IVR(x)I

1

continuous near r,

= 0 at P.

(8.32)

8.3 Method of Moving Planes in a Local Way

253

Let D be any compact subset of M and let p be any number greater than 1. Then the positive solutions of the equation

-Lou + R0(x)u = R(x)up, X E M

(8.33)

are uniformly bounded near r n D. Here, Lo is the Laplace-Beltrami operator associated with a base metric g, and R4(x) the corresponding scalar curvature.

One can see that our condition (8.32) is weaker than polynomial growth restriction. To illustrate, one may simply look at functions R(x) that grow like exp{-dx) }, where d(x) is the distance from the point x to P. Obviously, these kinds of functions satisfy our condition but are not polynomial growth. Moreover, our restriction on the exponent is much weaker. Besides being

n+2

n-2' p can be any number greater than 1. 8.3.2 The A Priori Estimates Now, we estimate the solutions of equation (8.33) and prove Theorem 8.3.1. Since M is a locally conformally flat manifold, a local flat metric can be chosen so that equation (8.33) in compact set D C M can be reduced to

-emu = R(x)up, p > 1

(8.34)

in compact set K in R". This is the equation we will consider throughout the section.

The proof of the Theorem is divided into two parts. We first derive the bound in the region(s) where R is negatively bounded away from zero. Then based on this bound, we use the `Method of Moving Planes' to estimate the solutions in the region(s) surrounding the set P.

Part I. In the region(s) where R is negative In this part, we derive estimates in the region(s) where R is negatively bounded away from zero. It comes from a standard elliptic estimate for subharmonic functions and an integral bound on the solutions. We prove Theorem 8.3.2 The solutions of (8.34) are uniformly bounded in the region {x E K : R(x) C 0 and dist(x, P) > b}. The bound depends only on b, the lower bound of R.

To prove Theorem 8.3.2, we introduce the following lemma of Gilbarg and Trudinger [GT].

254

8 Methods of Moving Planes and Moving Spheres

Lemma 8.3.1 (See Theorem 9.20 in /GTJ). Let u E W2,"(D) and suppose au > 0. Then for any ball B2r(x) C D and any q > 0, we have sup u
0 if R(x) = 0

Let

sgnR =

0

-1 if R(x) < 0.

Let a = p 21 Multiply both sides of equation (8.34) by p°IRJ'sgnR and .

then integrate. Taking into account that a > 2 and o and R are bounded in fl, through a straightforward calculation, we have In

wvaI RI'+updx < Ci

fo wa-2iR1a-2udx < C2 J

2iRI°-2udx.

Applying the Holder inequality to the right-hand-side, we arrive at

'paIRl1+°updx 0

Vx E D.

We consider the following two cases.

Case 1) 0(y) + s < x1 < 0(y) + E. In this region, R(x) is negative and bounded away from 0. By Theorem 8.3.2 and the standard elliptic estimate, we have

8.3 Method of Moving Planes in a Local Way

au 8x1

257

-w(x) +Com2 > -2m+Com2. Again, choosing Co sufficiently large (and depending only on e), we arrive at v(x) > 0. Step 3. Applying the Method of Moving Planes to v in the x1 direction

In the previous step, we showed that v(x) > 0 in D and v(x) = 0 on 01D. This lays a foundation for the moving of planes. Now we can start from the rightmost tip of region D and move the plane perpendicular to the x1 axis toward the left. More precisely, let

Ea={xEDIxl>A), TA = {xER"I x1=A}. Let xA = (2A - x1, y) be the reflection point of x with respect to the plane

TA. Let va(x) = v(xa)

and wa(x) = va(x) - v(x).

We are going to show that W'\ (X) > 0 V x E Ea.

(8.39)

We want to show that the above inequality holds for -cl < A < e for some 0 < cl < e. This choice of el is to guarantee that when the plane Tl, moves to A = -el, the reflection of Ea about the plane Ta still lies in region D. Now, (8.39) is true when A is less but close to e, because Ea is a narrow region and f (x, v) is Lipschitz continuous in v. For detailed arguments, the readers may see the first example in Subsection 8.2.1.

8 Methods of Moving Planes and Moving Spheres

258

We now we decrease A; that is, move the plane TA toward the left as long as the inequality (8.39) remains true. We show that the moving of planes can be carried on provided f(x,v(x)) < f(x),v(x)) for x = (xl,y) E D with xl > A >

(8.40)

In fact, it is easy to see that vA satisfies the equation ,Lva + z/i(y) + f (xa, va) = 0,

and hence wa satisfies -L WA = f (X A, vA) - f (X A, v) + Ax A, v) - .f (x, v)

= R(xa)C), (x) (v), - v) + f (xa, v) - f (x, v) = R(XA)CA(x)wa + f(xa,v) - f(x,v)

Here we have applied the Mean Value Theorem, CA(x) = XP-1(x), and eA(x) is valued between

{v(x) + w(x) - Come + 0(y) - xl] - m[e + 0(y)

- xl]2}

and

{v,\(x) + w(x) - Corm.[e + O(y) - xl] - rn[e + O(y) - xi]2}. If (8.40) holds, then we have

-Ow;, - R(x')CA(x)wa(x) > 0.

(8.41)

This enables us to apply the Maximum Principle to wA. Let ao = inf{A

j

wµ(x) > 0,Vx E E1,P > A}.

If ao > -el, then since v(x) > 0 in D, by virtue of (8.41) and the Maximum Principle, we have

wa° (x) > 0, Vx E Ea° and

,< 0, Vx E

Then, similar to the argument given in the examples in Subsection 8.2.2, one can derive a contradiction.

It is elementary to see that (8.40) holds if Of(Xv) < 0 in the set

{x E D I R(x) < 0 or xl > -2ci}.

To estimate a , we use 1

8.3 Method of Moving Planes in a Local Way

aX

259

+Com+2m(e+QS(y)-xr)1}.

First, we notice that the uniform convexity of the image of P near the origin implies

,Lo(y) < -ao < 0

(8.42)

for some constant ao. We consider the following two possibilities. a) R(x) < 0. Choose Co sufficiently large, so that aw

+Com>0.

ax,

Noticing that a R < 0 and (e + (y) - x1) > 0 in D, we have Of 0, x = (x1, y) with x1 > -2e1. In the part where u > 1, we use it to control

/

aR

aw

1R(x)/axl p[axl +Com+2m(e+0(y)-x1)1. More precisely, we write

of _ axl

p-1 OR 6x1

u

=

(

x { u+

OR aw R(x)/ax) p[axl +Com+2m(e+O(y) - xl )1 }

Choose el sufficiently small, such that I

-RIHIVRI.

Then by condition (8.32) on R, we can make R(x)/9' arbitrarily small by a small choice of e1, and therefore arrive at a CO. In the part where it < 1, by virtue of (8.42), we can use 2mLO(y) to control

Rp[ axl On'

-x1)1.

Here again we use the smallness of R. So far, our conclusion is this: The Method of Moving Planes can be carried on up to A = -El. More precisely, for any A between -el and c, the inequality (8.39) is true.

8 Methods of Moving Planes and Moving Spheres

260

Step 4. Deriving the a priori bound Inequality (8.39) implies that, in a small neighborhood of the origin, the function v(x) is monotone decreasing in the xr direction. A similar argument

will show that this remains true if we rotate the xl-axis by a small angle. Therefore, for any point xo E P, one can find a cone 4zp with xo as its vertex and staying to the left of xo, such that v(x) > v(xo)

(8.43)

dx E 4yp.

Noticing that w(x) is bounded in D, (8.43) leads immediately to

u(x) + C > u(xo)

Vx E 4so.

(8.44)

More generally, by a similar argument, one can show that (8.44) is true for any point xo in a small neighborhood of P. Furthermore, the intersection of the cone 4.a with the set {xIR(x) > 2 } has a positive measure, and the lower bound of the measure depends only on bo and the C' norm of R. Now the a priori bound of the solutions is a consequence of (8.44) and an integral bound on u, which we have derived from Lemma 8.3.2. This completes the proof of Theorem 8.3.1.

8.4 Method of Moving Spheres 8.4.1 The Background In this section, we revisit the prescribing Gaussian and scalar curvature equations

-Au + 2 = R(x)e°lsl, x E Sz.

n(n4 2)u(x)

=

R(x)un2(x),

(8.45)

n > 3, x E S", u > 0.

4(n - 1)

(8.46)

We will use the Method of Moving Spheres to establish some stronger necessary conditions. Both equations are so-called `critical' in the sense that lack of compactness occurs. Besides the obvious necessary condition that R(x) be positive somewhere, there are other well-known obstructions found by Kazdan and Warner [KW1] and later generalized by Bourguignon and Ezin [BE]. The conditions are:

Jsf X (R) dVg = 0,

(8.47)

8.4 Method of Moving Spheres

261

where dVy is the volume element of the metric e" go and u^=ago for n = 2 and for n > 3, respectively. The vector field X in (8.47) is a conformal Killing field associated to the standard metric 9o on Sh. In the following, we call these necessary conditions "Kazdan-Warner type" conditions.

These conditions give rise to many examples of R(x) for which (8.46) and

(8.45) have no solution. In particular, a monotone rotationally symmetric function R admits no solution. Then for which R can one solve the equations? What are the necessary and sufficient conditions for the equations to have a solution? These are very interesting problems in geometry.

In recent years, various sufficient conditions were obtained by many authors (For example, see [BC] [CY1] [CGY] [CY2] [CY3] [CL1O] [CD1] [CD2] [CC] [CS] [ES] [Ha] [Ho] [KW1] [KW2] (LiY1) [LiY2] [Mo] and the references therein.) However, there are still gaps between those sufficient conditions and

the necessary ones. Then one may naturally ask: "Are the necessary conditions of Kazdan-Warner type also sufficient?" This question has been open for many years. In [CL3], we gave a negative answer to this question. To find necessary and sufficient conditions, it is natural to begin with functions having certain symmetries. A pioneer work in this direction is [Me] by Moser. He showed that, for even functions R on S2, the necessary and sufficient condition for (8.45) to have a solution is that R be positive somewhere. Note that, in the class of even functions, (8.47) is satisfied automatically. Thus one can interpret Moser's result as follows: In the class of even functions, the Kazdan-Warner type conditions are the necessary and sufficient conditions for (8.45) to have a solution. An interesting question was asked by Kazdan [Ka]: What are the necessary and sufficient conditions for rotationally symmetric functions?

To be more precise, for rotationally symmetric functions R = R(0), where 0 is the geodesic distance from the north pole, the Kazdan-Warner type conditions take the following form: R > 0 somewhere and R' changes signs.

(8.48)

Now, is this a sufficient condition? In [XY], Xu and Yang proved:

Proposition 8.4.1 Let R be rotationally symmetric. Assume that i) R is non-degenerate ii)

R' changes signs in the region where R > 0. Then problem (8.45) has a solution.

(8.49)

262

8 Methods of Moving Planes and Moving Spheres

The above results and many other existence results tend to lead people to believe that what really counts is whether R' changes signs in the region where R > 0. And we conjectured in [CL3] that for rotationally symmetric R, condition (8.49), instead of (8.48), would be the necessary and sufficient condition for (8.45) or (8.46) to have a solution. The main objective of this section is to present the proof of the `necessary' part of the conjecture, which appeared in [CL4]. We use the Method of Moving Spheres. Instead of showing the symmetry of the solutions, we obtain a comparison formula which leads to a direct contradiction. In an earlier work [P]., a similar method was used by P. Padilla to show the radial symmetry of the solutions for some nonlinear Dirichlet problems on annuli. The following are our main results.

Theorem 8.4.1 Let R be continuous and rotationally symmetric. If R is monotone in the region where R > 0, and R 0 C,

(8.50)

Then problems (8.45) and (8.46) have no solution at all. This result can be generalized to a family of non-symmetric functions.

Theorem 8.4.2 Let R be a continuous function. Let B be a geodesic ball 1. Assume that centered at the North Pole i) R(x) > 0, R is non-decreasing in xn}1 direction, ii) R(x) < 0, for x E Sn \ B; Then problems (8.45) and (8.46) have no solution at all.

for x E B

Combining our Theorem 8.4.1 with Xu and Yang's Proposition 8.4.1, one can obtain a necessary and sufficient condition in the non-degenerate case:

Corollary 8.4.1 Let R be rotationally symmetric and non-degenerate in the sense that R" y6 0 whenever IT = 0. Then the necessary and sufficient condition for (8.45) to have a solution is R > 0 somewhere and R' changes signs in the region where R > 0.

As we proved in Chapter 6, in higher dimensions, condition (8.49) is also sufficient for equation (8.46) to have a solution under an additional flatness assumption on R(x).

8.4.2 Necessary Conditions In this subsection, we prove Theorem 8.4.1 and Theorem 8.4.2.

8.4 Method of Moving Spheres

263

For convenience, we make a stereo-graphic projection from S" to R". Then it is equivalent to consider the following equation in Euclidean space R" :

-Lu = R(r)e", x E R2;

(8.51)

-,n u=R(r)uT, u>0, xER", n>3;

(8.52)

with appropriate asymptotic decay of the solutions at infinity

u--4lnIxI forn=2; u_CIxl2-nforn>3

(8.53)

for some C > 0. Here 4 is the Euclidean Laplacian operator, r = IxI and r = '±2 is the so-called critical Sobolev exponent. The function R(r) is the projection of the original R in equations (8.45) and (8.46); and this projection

does not change the monotonicity of the function. The function R is also bounded and continuous, and we assume these throughout the subsection. For a radial function R satisfying (8.50), there are three possibilities: i) R is non-positive everywhere, ii) R 0 C is nonnegative everywhere and monotone, iii) R > 0 and non-increasing for r < ro, R < 0 for r > ro; or vice versa. The first two cases violate the Kazdan-Warner type conditions, hence there is no solution. Thus, we only need to show that in the remaining case, there is also no solution. Without loss of generality, we may assume that:

R(r) > 0, R'(r) < 0 for r < 1; R(r) < 0 for r > 1.

(8.54)

The proof of Theorem 8.4.1 is based on the following comparison formulas.

Lemma 8.4.1 Let R be a continuous function satisfying (8.54). Let it be a solution of (8.51) and (8.53) for n = 2, then u(Ax) > u(IAI2) - 41n IxI

Vx E Br(0), 0 < A < 1.

(8.55)

Lemma 8.4.2 Let R be a continuous function satisfying (8.54). Let u be a solution of (8.52) and (8.53) for n > 2, then u(Ax) >

IxI2_"u(IA X12

)

Vx E Br (0), 0 < A < 1.

(8.56)

We first prove Lemma 8.4.2. A similar idea will work for Lemma 8.4.1.

264

8 Methods of Moving Planes and Moving Spheres

Proof of Lemma 8.4.2. We use a new idea called the `method of moving spheres.' Let x E Br(0), then Ax E Ba(0). The reflection point of Ax about the sphere 8Ba(0) is ,2. We compare the values of u at those pairs of points Ax and yi2. In step 1, we start with the unit sphere and show that (8.56) is true for A = 1; this is done by applying a Kelvin transform and the strong Maximum Principle. In step 2, we move (shrink) the sphere 8Ba (0) towards the origin. We show that one can always shrink 8Ba (0) a little before reaching the origin. By doing so, we establish the inequality for all x E Bl (0) and A E (0,1]. Step 1. In this step, we establish the inequality for x E Br(0), A = 1. We show that

U(X) > Ixl2-nu(II2) b'x E Bi(0).

(8.57)

Let v(x) = IxI2-su(m) be the Kelvin transform of it. Then it is easy to verify that v satisfies the equation

v=R(1)v' r

(8.58)

and v is regular. It follows from (8.54) that Lu < 0, and nsv > 0 in Br(0). Thus

-0(u-v)>0. Applying the Maximum Principle, we obtain u > v in B1(0)1

(8.59)

which is equivalent to (8.57).

Step 2. In this step, we move the sphere 8Ba(0) towards A = 0. We show that the moving cannot be stopped until A reaches the origin. Again this is done by applying the Kelvin transform and the strong Maximum Principle. Let u), (x) = a%-ru(Ax). Then

-qua = R(Ax)u')(x).

(8.60)

Let vA(x) = IxI2-'u,\(' ) be the Kelvin transform of ua. Then it is easy to verify that -Ova = R(A)va(x). (8.61) Let wa = ua - va. Then by (8.60) and (8.61),

Lwa + R(A) {Gawa = [R( ) - R(Ar)]ui r

r

,

(8.62)

8.4 Method of Moving Spheres

where V)a is some function with values between rua 1 and account assumption (8.54), we have

R(A) - R(AT) < 0 T

265

rva-1. Taking

into

for r < 1, A < 1.

It follows from (8.62) that Owa +CA(x)WA < 0

(8.63)

where CA(x) is a bounded function if A is bounded away from 0. It is easy to see that for any A, strict inequality holds somewhere for (8.63). Thus, by applying the strong Maximum Principle, we know that the inequality (8.56) is equivalent to wa > 0. (8.64) From step 1, we know (8.64) is true for (x, A) E Bi(0) x {1}. Now we decrease A. Suppose (8.64) does not hold for all A E (0, 1] and let Ao > 0 be the smallest number such that the inequality is true for (x, A) E B1 (0) x ]A,,1]. We will

derive a contradiction by showing that for A close to and less than A0, the inequality is still true. In fact, we can apply the strong Maximum Principle and then the Hopf lemma to (8.63) for A = A. to get: wao > 0 in B1 and

OWA °

< 0 on 8B1.

(8.65)

These combined with the fact that wa - 0 on .B1 imply that (8.64) holds for A close to and less than A0.

Proof of Lemma 8.4.1. In step 1, we let v(x) = u( p) - 4ln x1.

In step 2, let

ua(x)=u(Ax)+2lnA, va(x)=uA( -[ )-41njxj. Then arguing as in the proof of Lemma 8.4.2 we derive the conclusion of Lemma 8.4.1.

We are now ready to prove Theorem 8.4.1.

Proof of Theorem 8.4.1. Taking A to 0 in (8.55), we get In xl > 0 for xj < 1, which is impossible. Letting A-+0 in (8.56) and using the fact that u(0) > 0, we obtain again a contradiction. These complete the proof of Theorem 8.4.1.

Proof of Theorem 8.4.2. Noting that in the proof of Theorem 8.4.1, we only compare the values of the solutions along the same radial direction, one can easily adapt the argument in the proof of Theorem 8.4.1 to derive the conclusion of Theorem 8.4.2.

8 Methods of Moving Planes and Moving Spheres

266

8.5 Method of Moving Planes in Integral Forms Let R" be the n-dimensional Euclidean space, and let a be a real number satisfying 0 < a < it. Consider the integral equation u(x) =

,a(y)(n+a)l(n-a)dy.

1

R^

(8.66)

Ix-7/I"-a

It arises as an Euler-Lagrange equation for a functional under a constraint in the context of the Hardy-Littlewood-Sobolev inequalities. In [L], Lieb classified the maximizers of the functional and thus obtained the best constant in the H-L-S inequalities. He then posed as an open problem the classification

of all critical points of the functional-the solutions of the integral equation (8.66).

This integral equation is also closely related to the following family of semi-linear partial differential equations (8.67)

(-d)°1271 =

In the special case n > 3 and a = 2, it becomes -,AU = n(n+2)/(n-2).

(8.68)

As we mentioned in Subsection 8.2.2, solutions to this equation were studied by Gidas, Ni, and Nirenberg [CNN]; Caffarelli, Cidas, and Spruck [CGS]; Chen and Li [CL1]; and Li [Li]. Recently, Wei and Xu [WX] generalized this result to the solutions of (8.67), with a being any even number between 0 and n.

Apparently, for other real values of a between 0 and n, equation (8.67) is also of practical interest and importance. For instance, it arises as the EulerLagrange equation of the functional

I(u)=J nI(-A)aul2dx/(J nluluadx)-. R

R

The classification of the solutions would provide the best constant in the

inequality of the critical Sobolev embedding from H 2' (R") to L- (R"):

(

R^

IuI

adx 0 on Ea for all A in [A0, A. + e).

From the first inequality of (8.73), we have IIWAIIcq(Ea)

C {JE ur+i(y) dy1 T

(8.75)

We will show that condition u E LT+r(Rn) ensures that one choose e sufficiently small so that for all A in [A0, A0 + c),

(8.76)

C{fL-uT+i(y)dy{,. a

J

Now, by (8.75) and (8.76), we have IIWAIIL,iEa) = 0, and therefore Ej must be measure zero. Hence, for these values of A > A0, we have W.\ (X) > 0,

a.e. Vx E Ea.

This contradicts with the definition of A0. Therefore, (8.74) must hold. To complete the proof, we now need only prove inequality (8.76). For any small ri > 0, we can choose R sufficiently large, so that /fR^\Bx(o) nr+i (y)dy) J

< t).

(8.77)

We fix this R and then show that the measure of Ea f1BR(0) is sufficiently small for A close to A0. By Lemma 8.5.1, we have in fact wA0(x) > 0 in the interior of Eao. For any 6 > 0, let

8.5 Method of Moving Planes in Integral Forms

271

E6 = {x E EAa n BR(0) I WA. (x) > S} and Fa = (EA. n BR(O)) \ Ea.

Then obviously limoµ(Fs) = 0.

6-4

For A > A,, let DA = (EA \ EA.) n BR(0).

Then it is easy to see that

(Ea n BR(O)) c (Ea n E5) U Fa U D.

(8.78)

Apparently, the measure of DA is small for A close to A,. We show that the measure of Ea n Ea can also be sufficiently small as A close to A0. In fact, for any x E Ea n E6, we have W'\ (X) = u(xA) - u(xAo) + u(x"°) - u(x) < 0.

Hence u(xAo)

- u(xA)

> wAj(x) > J.

It follows that (Ea n E5) C Ga - {x c BR(0)I u(xAo) - u(x') > b}.

(8.79)

By the well-known Chebyshev inequality, we have

µ(Cs) < g +r

ca

lu(xA°)-u(xA)IT+rdx
0 such that for A < -N, we have v(x) < VA (X), Vx E ZA \ {x,\}.

(8.81)

Thus we can start moving the plane continuously from A < -N to the right as long as (8.81) holds. If the plane stops at xl = a° for some a° < 0, then v(x) must be symmetric and monotone about the plane xr = a°. This implies that v(x) has no singularity at the origin and u(x) has no singularity at infinity.

In this case, we can carry on the moving planes on u(x) directly to obtain the radial symmetry and monotonicity. Otherwise, we can move the plane all

the way to xr = 0, which is shown in step 2. Since the direction of xl can be chosen arbitrarily, we deduce that v(x) must be radially symmetric and decreasing about the origin. We will show in step 3 that, in any case, u(x) can not have a singularity at infinity, and hence both u and v are in LT+r (R" ).

Step 1 and Step 2 are entirely similar to the proof under the stronger assumption: Simply replace it by v and La by LA \ {xA}. We therefore only show Step 3 here. Step 3. Finally, we show that it has the desired asymptotic behavior at

infinity, i.e., it satisfies

u(x)=0(Ix1

).

Suppose, on the contrary, that xr and x2 are any two points in R", and let x° be the midpoint of the line segment xlx2. Consider the Kelvin type transform centered at x°: 1 x-x° v(x) Ix - x°Ina u(Ix-x°I2

Then v(x) has a singularity at x°. Carrying on the arguments as in Steps 1 and 2, we conclude that v(x) must be radially symmetric about x°, and

8.5 Method of Moving Planes in Integral Forms

273

in particular, u(x') = u(x2). Since xl and x2 are any two points in R", it must be constant, which is impossible. Similarly, the continuity and higher regularity of it follows from standard theory on singular integral operators. This completes the proof of the Theorem.

A

Appendices

A.1 Notation A.1.1 Algebraic and Geometric Notation 1. A = (aid) is a an m. x n matrix with ijth entry aij. 2. det(ai5) is the determinant of the matrix (aij). 3. AT = transpose of the matrix A.

4. R" = n-dimensional real Euclidean space with a typical point x

(xi,...

X")-

5. Q usually denotes an open set in Rn 6. S2 is the closure of Si.

7. 8dl is the boundary of Si.

in R,

8. n

x'y=Exiyi, ixI=

l

4=r

1x - yI is the distance of the two points x and y in R". 10. B,(x°) = {x E R" Ix - x°I < r} is the ball of radius r centered at the point x° in R". 11. S" is the sphere of radius one centered at the origin in Rn+l, i.e., the boundary of the ball of radius one centered at the origin: 9.

B1(0)_{xER1+lI1xI 1, it has been shown (See Adams [Ad]) that Hk.P(Q) = Wk'P(,(2).

19. For p = 2, we usually write Hk(D) = Hk,2(.fl).

A.2 Notation and Basic Facts from Riemannian Geometry

279

A.1.4 Notation for Estimates 1. In the process of deriving estimates, we use C to denote various constants that can be explicitly computed in terms of known quantities. The value of C may change from line to line in a given computation. 2. We write

u = O(v) as x->x , provided there exists a constant C such that 1u(x)1 x°.

A.2 Notation and Basic Facts from Riemannian Geometry 1. M denotes a differentiable manifold. 2. TPM is the tangent space of M at point p. 3. TyM is the co-tangent space of M at point p. 4. A Riemannian metric g can be expressed in a matrix 911 .. 91. .

g=

.

9n1

gnn

)

I

where in local coordinates,

9j(P)=p.

5. (M, g) denotes an n-dimensional manifold M with Riemannian metric g. In local coordinates, the length element is 1/2

ds :_

I/

\

E gigdxidxj =1

J

280

A Appendices

and the volume element is

det(gjj)dxi ... dx .

dV :=

6. K(x) denotes the Gaussian curvature, and R(x) the scalar curvature of a manifold M at point x. 7. For two differentiable vector fields X and Y on M, the Lie Bracket is

[X, Y] := XY - YX. 8. D denotes an affine connection on a differentiable manifold M.

9. A vector field V(t) along a curve c(t) E M is parallel if

= 0.

10. Given a Riemannian manifold M with metric , theredC exists a unique connection D on M that is compatible with the metric , i.e.,

< X, Y >=(t)= constant for any pair of parallel vector fields X and Y along any smooth curve c(t). 11. In local coordinates (xi, - - , x.), this unique Riemannian connection can be expressed as

Doaa I] aa

k

k

where

are the Christoffel symbols.

12. We denote

17i=D T.a7

.

13. The summation convention. We write, for instance,

xi a

Xiaxia

axi

r.kaxk

r,.kdx" k

and so on.

14. A smooth function f (x) on M is a (0, 0)-tensor. defined by

of = vifdxi =

of axi

7f is a (1, 0)-tensor

dxi.

v2 f is a (2, 0)-tensor:

02f = o, (ax dxi) ®dxj

=(aaj -r,'ak /ll dxi®dxj. a

A.2 Notation and Basic Facts from Riemannian Geometry

281

In the Riemannian context, V2f is called the Hessian of f and denoted by Hess(f). Its ijrh component is

(02f)=; =

a2f

ak.

r"a

15. The trace of the Hessian matrix ((p2f)i;) is defined to be the LaplaceBeltrami operator A.

0=

1 a( Isl

giia)I ax;

i,;-1 axi

where gi is the determinant of (gi). 16. We abbreviate:

pupvdV9 := .1M

JM

< Vu, Vv > dV9,

where

< VU, VV >= gijl7iuvjv = 923

On av

is the scalar product associated with g for 1-forms. 17. The norm of the kth covariant derivatives of u, Ipkul is defined in a local coordinates chart by IvkuI2 _ gill ... gikik(,71a)i1..1k(Ok1l)Jl..jk

In particular, we have

to ul2 = Iv 1

2 I

=

ij

i;

ij

On On

s (ou)=(ow); = s oiwo;u = s axi ax;,

18. Ck(M) is the space of CO° functions on M such that Jr

IplulpdVg 0, we have 0 < Ix ± Ay12 = IxI2 ± 2Ax y + A2IyI2.

It follows that

fx y
inf J, G

as k-*oo.

Then again by (A.15), {uk} is bounded in H' and hence possesses a subsequence (still denoted by {uk}) that converges weakly to some uo in H'(M). From compact Sobolev embedding,

H' '->y L2, we see that uk->uo strongly in L2,

and hence so in V.

Therefore fm

uo(x)dA=lira fm uk(x)dA=0,

and

f

(A.16)

rM

f (x)uo(x)d A = lim

f(x)uk(x)d A.

(A.17)

fm Now (A.16) implies that uo E G, and hence

J(uo) > in J(u),

(A.18)

whereas (A.17) and the weak lower semi-continuity of the Dirichlet integral,

liminf J Jpuk12dA>J Ipuol2dA, M M leads to

J(uo) < inf J(u). From this and (A18) we conclude that uo is the minimum of J in the set G, and therefore there exists constant A, such that uo is a weak solution of

-Auo - f(x) = A. Integrating both sides, and by (5.4), we find that A = 0. Therefore up is a solution of equation (5.3). This completes the proof of the Lemma.

References

[Ad] [AR] [Au]

[Aul] [Ba]

[Bal]

R. Adams, Sobolev Spaces, Academic Press, San Diego, 1978. A. Ambrosetti and P. Rabinowitz, Dual variational method in critical point theory and applications, J. Funct. Anal. 14(1973), 349-381. T. Aubin, Nonlinear Analysis on Manifolds, Monge-Ampere Equations, Springer-Verlag, 1982. T. Aubin, Equations differentielles non lineaires at problem de Yamabe la courbure scalaire, J. Math. Pures Appl. (9) 55(1976), 269-296. A. Bahri, Critical points at infinity in some variational problems, Pitman, 1989, Research Notes in Mathematics Series, Vol. 182, 1988. A. Bahri, An invariant for Yamabe-type flows with applications to scalar curvature problems in high dimension, Duke Math. J. 81(1996), 323-466.

[BC]

A. Bahri and J. Coron,

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Index

(0,0)-tensor, 280 (1,0)-tensor, 280 (2.0)-tensor, 280

Ck(n), 277 CO0(!), 277 C°°(S2), 278

Hk(Q),

278

HkP(S2), 278

L°°(Q),

278 LIP, (S2), 278

W2' Regularity, 91 W,P(Q), 278 f M pupvdVP, 281 it = O(v), 279 it = o(v), 279

(PS) condition, 146

a priori estimates for prescribing scalar curvature, 203 "center of mass" analysis, 169

a Maximum Principle for integral equations, 227 a priori estimates, 253 affine connection, 135

almost a local maximum, 204 an equivalent HLS inequality, 41 an integral bound for solutions, 254 approximation by smooth functions, 10 at most one simple blow up point, 207

boost the regularity, 103 bootstrap method, 101

Calder6n-Zygmund's Decomposition, 82, 284 coerciveness, 55 compact embedding, 33 complete Banach space, 47 conformal Laplacian, 194 conformal transforms, 193 contracting map, 97 converges weakly, 56 cotangent space, 127 covariant derivatives, 134 critical case, 146 critical points, 53, 55

critical points at infinity, 192 curvature of plane curves, 131 curvature on Riemannian manifolds, 133

Decay at Infinity, 226, 239 Deformation Theorem, 65 differentiable manifolds, 124 Dirichlet boundary condition, 86 distribution of mass principle, 168 distributions, 3, 7 dual space, 48

elliptic operator in divergence form, 44 Euler-Lagrange equation, 55 existence of a mini-max, 68 existence of lower solutions, 152 existence of upper solutions, 151 existence of weak solutions, 51 Extension Theorem, 20

Index

298

first spherical harmonic functions, 198 flatness condition, 183 Frechet differentiable, 55

F}edholm Alternative, 98 fully nonlinear Wolff systems, 119 function spaces., 277 functional, 53 Gagliardo-Nirenberg-Sobolev inequality, 24

Gaussian curvature, 132 general Sobolev inequalities, 23 gradient vector, 276 Holder continuous, 277 Holder norm, 277 Holder semi-norm, 277 Holder space, 278 Hardy-Littlewood-Sobolev inequality, 38, 110 Hessian matrix, 276 Hessian of f, 139, 281

higher covariant derivatives, 137 Hilbert spaces, 47, 141

inner product, 51 integral equation, 266 integrals on Riemannian manifolds, 139 interpolation inequality for LP norms, 283

k-th differentiable functions with compact support, 278 Kazdan-Warner condition, 163 Kelvin transform, 264 Lagrange multiplier, 60 Laplace-Beltrami operator, 137, 281 Laplacian, 276 Lax-Milgram Theorem, 49 length element, 279 Levi and Civita Theorem, 135 Lie bracket, 133 linear operator, 47

Maximum Principle and Hopf Lemma for not necessarily boundeddomain and not necessarily non-negative c(a), 222 Maximum Principle Based on Comparison, 214, 224 Maximum Principle on unbounded domain, 225 Mean Value Inequality, 215 method of "frozen" coefficients, 87 method of lower and upper solutions, 150

Method of Moving Planes, 233 Method of Moving Planes in a local way, 251

Method of Moving Planes in integral forms, 268 Method of Moving Spheres, 262 Mini-max critical points, 62 minimizer, 54 minimizers under constraints, 58 minimizing sequence, 59 mollifiers, 10

Morrey's inequality, 25 Moser-Trudinger inequality, 164 Mountain Pass Theorem, 63 move the plane to the limiting position, 241

Narrow Region Principle, 226, 236 narrow regions, 226 necessary conditions, 262 Newtonian Potentials, 79 norm, 51 notation and basic facts from Riemannian geometry, 279 notation for estimates, 279

Onofri inequality, 174 Operator Completion Theorem, 19

Palais-Smale condition, 63 partition of unity, 139 Poincare's inequality, 36 pointwise conformal, 140 prepare to move the plane from near

-oo, 240 Marcinkiewicz interpolation, 85 max-mini variational scheme, 192

prescribing Gaussian curvature, 148 prescribing scalar curvature, 182

Index

principal curvature, 132 re-scaling argument, 204 recovering coerciveness, 167 reflexive, 56 Regularity Lifting by Combinations of Contracting and Shrinking Operators, 111 Regularity Lifting by Contracting Operators, 102 Regularity lifting for integral equations, 109

Regularity Lifting Theorem I, 102 Regularity Lifting Theorem II, 112 Ricci curvature, 136 Riemannian connection, 280 Riemannian connections, 134 Riemannian manifold, 128 Riemannian metrics, 128

Riesz Representation Theorem, 48 scalar curvature, 136 second order elliptic operators, 44 sectional curvature, 136 semi-linear equations, 53 shrinking operator, 112 Sobolev embeddings, 22 Sobolev embeddings on Riemannian manifolds, 141 Sobolev spaces, 8 Strong Maximum Principle, 213 Strong Maximum Principle for L with c(x) __ 0, 220

Strong Maximum Principle for L with c(x) > 0, 221 strong solution, 86

strong type operators, 80 subcritical case, 146 successive approximations, 211 super critical case, 146

symmetry of solutions for integral equations under stronger assumptions, 268

299

symmetry of solutions for integral equations under weaker assumptions, 272

tangent spaces, 125 tensor, 138 The Arzela-Ascoli Theorem, 288 the Cauchy-Schwarz inequality, 282 The Contracting Mapping Principle, 286

the differential of a differentiable mapping, 127 the Holder's inequality, 282 The Hopf Lemma, 218 the Minkowski's inequality, 283 the norm of the krh covariant derivatives, 281 the Sobolev space HkiP(M), 281 the Young's inequality, 282

uniform elliptic operators, 44 uniformly elliptic, 212 uniqueness of weak solutions, 99 variational approaches, 162 variational methods, 53, 145 vector field, 133 volume element, 280

weak compactness, 57 weak derivative, 4, 5 weak lower semi-continuity, 55 weak Maximum Principles, 211 weak solution, 45 weak solutions, 92 weak type operators, 80 XY-pair, 112

Yamabe problem, 180