Chang k c methods in nonlinear analysis (SMM 2005)(ISBN 3540241337)(447s)

The theories and methods in nonlinear analysisstem from many areas of mathematics: Ordinary differential equations, partialdifferential equations, the calculus of variations, dynamical sys

Springer Monographs in Mathematics

Kung-Ching Chang

Methods in

Nonlinear Analysis

ABC

Library of Congress Control Number: 2005931137

Mathematics Subject Classification (2000): 47H00, 47J05, 47J07, 47J25, 47J30, 58-01,58C15, 58E05, 49-01, 49J15, 49J35, 49J45, 49J53, 35-01

ISSN 1439-7382

ISBN-10 3-540-24133-7 Springer Berlin Heidelberg New York

ISBN-13 978-3-540-24133-1 Springer Berlin Heidelberg New York

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Nonlinear analysis is a new area that was born and has matured from dant research developed in studying nonlinear problems In the past thirtyyears, nonlinear analysis has undergone rapid growth; it has become part ofthe mainstream research fields in contemporary mathematical analysis.Many nonlinear analysis problems have their roots in geometry, astronomy,fluid and elastic mechanics, physics, chemistry, biology, control theory, imageprocessing and economics The theories and methods in nonlinear analysisstem from many areas of mathematics: Ordinary differential equations, partialdifferential equations, the calculus of variations, dynamical systems, differen-tial geometry, Lie groups, algebraic topology, linear and nonlinear functionalanalysis, measure theory, harmonic analysis, convex analysis, game theory,optimization theory, etc Amidst solving these problems, many branches areintertwined, thereby advancing each other.

abun-The author has been offering a course on nonlinear analysis to ate students at Peking University and other universities every two or threeyears over the past two decades Facing an enormous amount of material,vast numbers of references, diversities of disciplines, and tremendously differ-ent backgrounds of students in the audience, the author is always concernedwith how much an individual can truly learn, internalize and benefit from amere semester course in this subject

gradu-The author’s approach is to emphasize and to demonstrate the most damental principles and methods through important and interesting examplesfrom various problems in different branches of mathematics However, thereare technical difficulties: Not only do most interesting problems require back-ground knowledge in other branches of mathematics, but also, in order to solvethese problems, many details in argument and in computation should be in-cluded In this case, we have to get around the real problem, and deal with asimpler one, such that the application of the method is understandable Theauthor does not always pursue each theory in its broadest generality; instead,

fun-he stresses tfun-he motivation, tfun-he success in applications and its limitations

VI Preface

The book is the result of many years of revision of the author’s lecturenotes Some of the more involved sections were originally used in seminars asintroductory parts of some new subjects However, due to their importance,the materials have been reorganized and supplemented, so that they may bemore valuable to the readers

In addition, there are notes, remarks, and comments at the end of thisbook, where important references, recent progress and further reading arepresented

The author is indebted to Prof Wang Zhiqiang at Utah State University,Prof Zhang Kewei at Sussex University and Prof Zhou Shulin at PekingUniversity for their careful reading and valuable comments on Chaps 3, 4and 5

September, 2003

1 Linearization 1

1.1 Differential Calculus in Banach Spaces 1

1.1.1 Frechet Derivatives and Gateaux Derivatives 2

1.1.2 Nemytscki Operator 7

1.1.3 High-Order Derivatives 9

1.2 Implicit Function Theorem and Continuity Method 12

1.2.1 Inverse Function Theorem 12

1.2.2 Applications 17

1.2.3 Continuity Method 23

1.3 Lyapunov–Schmidt Reduction and Bifurcation 30

1.3.1 Bifurcation 30

1.3.2 Lyapunov–Schmidt Reduction 33

1.3.3 A Perturbation Problem 43

1.3.4 Gluing 47

1.3.5 Transversality 49

1.4 Hard Implicit Function Theorem 54

1.4.1 The Small Divisor Problem 55

1.4.2 Nash–Moser Iteration 62

2 Fixed-Point Theorems 71

2.1 Order Method 72

2.2 Convex Function and Its Subdifferentials 80

2.2.1 Convex Functions 80

2.2.2 Subdifferentials 84

2.3 Convexity and Compactness 87

2.4 Nonexpansive Maps 104

2.5 Monotone Mappings 109

2.6 Maximal Monotone Mapping 120

VIII Contents

3 Degree Theory and Applications 127

3.1 The Notion of Topological Degree 128

3.2 Fundamental Properties and Calculations of Brouwer Degrees 137 3.3 Applications of Brouwer Degree 148

3.3.1 Brouwer Fixed-Point Theorem 148

3.3.2 The Borsuk-Ulam Theorem and Its Consequences 148

3.3.3 Degrees for S1Equivariant Mappings 151

3.3.4 Intersection 153

3.4 Leray–Schauder Degrees 155

3.5 The Global Bifurcation 164

3.6 Applications 175

3.6.1 Degree Theory on Closed Convex Sets 175

3.6.2 Positive Solutions and the Scaling Method 180

3.6.3 Krein–Rutman Theory for Positive Linear Operators 185

3.6.4 Multiple Solutions 189

3.6.5 A Free Boundary Problem 192

3.6.6 Bridging 193

3.7 Extensions 195

3.7.1 Set-Valued Mappings 195

3.7.2 Strict Set Contraction Mappings and Condensing Mappings 198

3.7.3 Fredholm Mappings 200

4 Minimization Methods 205

4.1 Variational Principles 206

4.1.1 Constraint Problems 206

4.1.2 Euler–Lagrange Equation 209

4.1.3 Dual Variational Principle 212

4.2 Direct Method 216

4.2.1 Fundamental Principle 216

4.2.2 Examples 217

4.2.3 The Prescribing Gaussian Curvature Problem and the Schwarz Symmetric Rearrangement 223

4.3 Quasi-Convexity 231

4.3.1 Weak Continuity and Quasi-Convexity 232

4.3.2 Morrey Theorem 237

4.3.3 Nonlinear Elasticity 242

4.4 Relaxation and Young Measure 244

4.4.1 Relaxations 245

4.4.2 Young Measure 251

4.5 Other Function Spaces 260

4.5.1 BV Space 260

4.5.2 Hardy Space and BMO Space 266

4.5.3 Compensation Compactness 271

4.5.4 Applications to the Calculus of Variations 274

4.6 Free Discontinuous Problems 279

4.6.1 Γ-convergence 279

4.6.2 A Phase Transition Problem 280

4.6.3 Segmentation and Mumford–Shah Problem 284

4.7 Concentration Compactness 289

4.7.1 Concentration Function 289

4.7.2 The Critical Sobolev Exponent and the Best Constants 295 4.8 Minimax Methods 301

4.8.1 Ekeland Variational Principle 301

4.8.2 Minimax Principle 303

4.8.3 Applications 306

5 Topological and Variational Methods 315

5.1 Morse Theory 317

5.1.1 Introduction 317

5.1.2 Deformation Theorem 319

5.1.3 Critical Groups 327

5.1.4 Global Theory 334

5.1.5 Applications 343

5.2 Minimax Principles (Revisited) 347

5.2.1 A Minimax Principle 347

5.2.2 Category and Ljusternik–Schnirelmann Multiplicity Theorem 349

5.2.3 Cap Product 354

5.2.4 Index Theorem 358

5.2.5 Applications 363

5.3 Periodic Orbits for Hamiltonian System and Weinstein Conjecture 371

5.3.1 Hamiltonian Operator 373

5.3.2 Periodic Solutions 374

5.3.3 Weinstein Conjecture 376

5.4 Prescribing Gaussian Curvature Problem on S2 380

5.4.1 The Conformal Group and the Best Constant 380

5.4.2 The Palais–Smale Sequence 387

5.4.3 Morse Theory for the Prescribing Gaussian Curvature Equation on S2 389

5.5 Conley Index Theory 392

5.5.1 Isolated Invariant Set 393

5.5.2 Index Pair and Conley Index 397

5.5.3 Morse Decomposition on Compact Invariant Sets and Its Extension 408

Notes 419

References 425

Linearization

The first and the easiest step in studying a nonlinear problem is to linearize

it That is, to approximate the initial nonlinear problem by a linear one linear differential equations and nonlinear integral equations can be seen asnonlinear equations on certain function spaces In dealing with their lineariza-tions, we turn to the differential calculus in infinite-dimensional spaces Theimplicit function theorem for finite-dimensional space has been proved veryuseful in all differential theories: Ordinary differential equations, differentialgeometry, differential topology, Lie groups etc In this chapter we shall seethat its infinite-dimensional version will also be useful in partial differentialequations and other fields; in particular, in the local existence, in the stability,

Non-in the bifurcation, Non-in the perturbation problem, and Non-in the gluNon-ing techniqueetc This is the contents of Sects 1.2 and 1.3 Based on Newton iterationsand the smoothing operators, the Nash–Moser iteration, which is motivated

by the isometric embeddings of Riemannian manifolds into Euclidean spacesand the KAM theory, is now a very important tool in analysis Limited inspace and time, we restrict ourselves to introducing only the spirit of themethod in Sect 1.4

1.1 Differential Calculus in Banach Spaces

There are two kinds of derivatives in the differential calculus of several ables, the gradients and the directional derivatives We shall extend these two

vari-to infinite-dimensional spaces

Let X, Y and Z be Banach spaces, with norms
·
X,
·
Y,
·
Z,

respectively If there is no ambiguity, we omit the subscripts Let U ⊂ X be

an open set, and let f : U → Y be a map.

1.1.1 Frechet Derivatives and Gateaux Derivatives

Definition 1.1.1 (Fr´ echet derivative) Let x0∈ U; we say that f is Fr´echet differentiable (or F-differentiable) at x0, if ∃A ⊂ L(X, Y ) such that

f(x) − f(x0)− A(x − x0)
Y=◦(
x − x0
X )

Let f  (x0) = A, and call it the Fr´ echet (or F-) derivative of f at x0.

If f is F-differentiable at every point in U , and if x → f  (x), as a mapping from U to L(X, Y ), is continuous at x0, then we say that f is continuously differentiable at x0 If f is continuously differentiable at each point in U , then we say that f is continuously differentiable on U , and denote it by f ∈

C1(U, Y ).

Parallel to the differential calculus of several variables, by definition, wemay prove the following:

1 If f is F-differentiable at x0, then f  (x0) is uniquely determined.

2 If f is F-differentiable at x0, then f must be continuous at x0

3 (Chain rule) Assume that U ⊂ X, V ⊂ Y are open sets, and that f is F-differentiable at x0, and g is F-differentiable at f (x0), where

By definition, we have the following properties:

1 If f is G-differentiable at x0, then df (x0, h) is uniquely determined.

2 df (x0, th) = tdf (x0, h) ∀t ∈ R1

3 If f is G-differentiable at x0, then∀h ∈ X, ∀y ∗ ∈ Y ∗, the functionϕ(t) = y ∗ , f (x0+ th)  (t) = y ∗ , df (x0, h)

4 Assume that f : U → Y is G-differentiable at each point in U, and that

the segment{x0+ th | t ∈ [0, 1]} ⊂ U, then

f(x0+ h) − f(x0)
Y
sup
df(x0+ th, h)
Y

1.1 Differential Calculus in Banach Spaces 3

5 If f is F-differentiable at x0, then f is G-differentiable at x0, with

df (x0, h) = f  (x0)h ∀h ∈ X.

Conversely it is not true, but we have:

U , ∃A(x) ∈ L(X, Y ) satisfying

df (x, h) = A(x)h ∀h ∈ X

If the mapping x → A(x) is continuous at x0, then f is F-differentiable at x0, with f  (x0) = A(x0).

Proof With no loss of generality, we assume that the segment {x0+ th |

t ∈ [0, 1]} is in U According to the Hahn–Banach theorem, ∃y ∗ ∈ Y ∗, with

Example 1 Let A ∈ L(X, Y ), f(x) = Ax Then f  (x) = A ∀x.

Example 2 Let X =Rn , Y =Rm , and let ϕ1, ϕ2 , ϕm ∈ C1(Rn , R1) Set

Example 3 Let Ω ⊂ R n be an open bounded domain Denote by C(Ω) the

continuous function space on Ω Let

ϕ : Ω × R1−→ R1,

be a C1 function Define a mapping f : C(Ω) → C(Ω) by

u(x) → ϕ(x, u(x)) Then f is F-differentiable, and ∀u0∈ C(Ω),

(f  (u

0)· v)(x) = ϕu (x, u0(x)) · v(x) ∀v ∈ C(Ω) Proof ∀h ∈ C(Ω)

t −1 [f (u

0+ th) − f(u0)](x) = ϕ u (x, u0(x) + tθ(x)h(x))h(x) , where θ(x) ∈ (0, 1) ∀ε > 0, ∀M > 0, ∃δ = δ(M, ε) > 0 such that

| ϕu (x, ξ) − ϕu (x, ξ )|< ε, ∀x ∈ Ω ,

as |ξ|, |ξ  |
M and |ξ − ξ  |
δ We choose M =
u0
+
h
, then for

|t| < δ < 1,

|ϕu (x, u0(x) + tθ(x)h(x)) − ϕu (x, u0(x)) | < ε

It follows that df (u0, h)(x) = ϕu (x, u0(x))h(x).

Noticing that the multiplication operator h → A(u)h = ϕu (x, u(x)) · h(x)

is linear and continuous, and the mapping u → A(u) from C(Ω) into L(C(Ω), C(Ω)) is continuous, from Theorem 1.1.3, f is F-differentiable, and

(f  (u

0)· v)(x) = ϕu (x, u0(x)) · v(x) ∀v ∈ C(Ω)

1.1 Differential Calculus in Banach Spaces 5

We investigate nonlinear differential operators on more general spaces.Let Ω ⊂ R n be a bounded open set, and let m be a nonnegative integer,

γ ∈ (0, 1) C m(Ω) (and the H¨older space C m,γ(Ω)) is defined to be the function

space consisting of C m functions (with γ-H¨ older continuous m-order partial

∂ α1

x1∂ α2

x2 · · · ∂ α n

x n

We always denote by m ∗the number of the index set{α = (α1, α2, , αn)|

|α|
m}, and D m u the set {∂ α u | |α|
m}.

Suppose that r is a nonnegative integer, and that ϕ ∈ C ∞(Ω×R r ∗

where ϕ α is the partial derivative of ϕ with respect to the variable index α.

The proof is similar to Example 3

In particular, the following functional occurs frequently in the calculus of

variations (r = 1, r ∗ = n + 1) Assume that ϕ(x, u, p) is a function of the form:

∀p  1, ∀ nonnegative integer m, let

W m,p(Ω) ={u ∈ L p(Ω)| ∂ α u ∈ L p(Ω)| |α|
m} ,

where ∂ α u stands for the α-order generalized derivative of u, i.e., the derivative

in the distribution sense Define the norm

The Banach space is called the Sobolev space of index {m, p}.

W m,2 (Ω) is denoted by H m (Ω), and the closure of C ∞

0 (Ω) under this norm

is denoted by H m(Ω)

1.1 Differential Calculus in Banach Spaces 7

The motivation in introducing the Caratheodory function is to make the

composition function measurable if u(x) is only measurable Indeed, there

exists a sequence of simple functions {un (x) } ∞ , such that u n (x) → u(x) a.e., ϕ(x, u n (x)) is measurable according to (2) And from (1), ϕ(x, u n (x)) → ϕ(x, u(x)) a.e., therefore ϕ(x, u(x)) is measurable.

Theorem 1.1.5 Assume p1, p2  1, a > 0 and b ∈ L p2

dµ (Ω) Suppose that ϕ

is a Caratheodory function satisfying

|ϕ(x, ξ)|
b(x) + a|ξ| p1 p2 Then f : u(x) → ϕ(x, u(x)) is a bounded and continuous mapping from

1 if u n → u in L p1, then there is a subsequence

{un i } such that f(un i) → f(u) in L p2 Indeed one can find a subsequence

{un i } of {un} which converges a.e to u, along which
un i − un i−1
p1< 1

f (un i ) = ϕ(x, u n i (x)) → ϕ(x, u(x)) a.e ,

and

|f(un i )(x) |
b(x) + a(Φ(x)) p1 p2 ∈ L p2

dµ (Ω, R N ) ,

we have
f(un i)− f(u)
p2→ 0, according to Lebesgue dominance theorem.

p1, p2 ≤ ∞ Suppose that ϕ : Ω × R m ∗

→ R is a Caratheodory function satisfying

Proof The Sobolev embedding theorem says that the injection i : H1(Ω) →

L n−2 2n (Ω) is continuous, so is the dual map i ∗ : L 2n

n+2 (Ω) → (H1(Ω))∗.According to Theorem 1.1.5, ϕ ξ(·, ·) : L n−2 2n → L n+2 2n is continuous There-fore the Gateaux derivative

(f  (u

0)h)(x) =

|α|≤m

ϕα (x, D m u(x))∂ α h(x) ∀h ∈ C l,γ (Ω)

1.1 Differential Calculus in Banach Spaces 9

1.1.3 High-Order Derivatives

The second-order derivative of f at x0 is defined to be the derivative of f  (x)

at x0 Since f  : U → L(X, Y ), f  (x0) should be in L(X, L(X, Y )) However,

if we identify the space of bounded bilinear mappings with L(X, L(X, Y )), and verify that f  (x0) as a bilinear mapping is symmetric, see Theorem 1.1.9below, then we can define equivalently the second derivative f  (x0) as follows:For f : U → Y , x0∈ U ⊂ X, if there exists a bilinear mapping f  (x0)(·, ·) of

By the same manner, one defines the mth-order derivatives at x0

succes-sively: f (m) (x0) : X × · · · × X → Y is an m-linear mapping satisfying

as
h
→ 0 Then f is called m differentiable at x0

Similar to the finite-dimensional vector functions, we have:

Theorem 1.1.9 Assume that f : U → Y is m differentiable at x0∈ U Then for any permutation π of (1, , m), we have

Theorem 1.1.10 (Taylor formula) Suppose that f : U → Y is continuously m-differentiable Assume the segment {x0+ th | t ∈ [0, 1]} ⊂ U Then

m!

1

0(1− t) m f (m+1) (x0+ th)(h, , h)dt Proof ∀y ∗ ∈ Y ∗, we consider the function:

we obtain the desired Taylor formula for mappings between B-spaces Example 1 X = Rn , Y = R1 If f : X → Y is twice continuously differen-

u (x, u(x))ϕ(x)]dx ,

and

f  (u)(ϕ, ψ) =

Ω[∇ψ(x)∇ϕ(x) + g 

uu (x, u(x))ϕ(x)ψ(x)]dx With some additional growth conditions on g 

uu:

|g 

uu (x, u) | ≤ a(1 + |u| n−24 ), a > 0 ∀u ∈ R N ,

f is twice differentiable in H1(Ω, R N ) As an operator from H1(Ω, R N) intoitself,

f  (u) = id + ( −) −1 g 

u(·, u(·))

1.1 Differential Calculus in Banach Spaces 11

is self-adjoint, or equivalently, the operator− + g 

Geometrically, let u : Ω C → R2 3 be a parametrized surface in R3; Q(U ) is

the volume of the body enclosed by the surface

As exercises, one computes the first- and second-order differentials of thefollowing functionals:

1.2 Implicit Function Theorem and Continuity Method

1.2.1 Inverse Function Theorem

It is known that the implicit function theorem for functions of several ables plays important roles in many branches of mathematics (differentialmanifold, differential geometry, differential topology, etc.) Its extension toinfinite-dimensional space is also extremely important in nonlinear analysis,

vari-as well vari-as in the study of infinite-dimensional manifolds

Theorem 1.2.1 (Implicit function theorem) Let X, Y , Z be Banach spaces,

U ⊂ X × Y be an open set Suppose that f ∈ C(U, Z) has an F-derivative w.r.t y, and that fy ∈ C(U, L(Y, Z)) For a point (x0, y0)∈ U, if we have

f (x0, y0) = θ ,

f −1

y (x0, y0)∈ L(Z, Y ) ; then ∃r, r1> 0, ∃|u ∈ C(Br (x0), B r1(y0)), such that

⎧

⎨

⎩

Br (x0)× Br1(y0)⊂ U , u(x0) = y0,

f (x, u(x)) = θ ∀x ∈ Br (x0) Furthermore, if f ∈ C1(U, Z), then u ∈ C1(B r (x0), Y ), and

u  (x) = −f −1

y (x0, u(x0))◦ fx (x, u(x)) ∀x ∈ Br (x0) (1.1)

1.2 Implicit Function Theorem and Continuity Method 13

Proof (1) After replacing f by

g(x, y) = f −1

y (x0, y0)◦ f(x + x0, y + y0) , one may assume x0= y0= θ, Z = Y and f y (θ, θ) = id Y

(2) We shall find the solution y = u(x) ∈ Br1(θ) of the equation

f (x, y) = θ ∀x ∈ Br (θ)

Setting

R(x, y) = y − f(x, y) ,

it is reduced to finding the fixed point of R(x, ·) ∀x ∈ Br (θ).

We shall apply the contraction mapping theorem to the mapping R(x, ·).

Firstly, we have a contraction mapping:

R(x, y1)− R(x, y2)
=
y1− y2− [f(x, y1)− f(x, y2)]

=
y1− y2−

1 0

f y (x, ty1+ (1− t)y2)dt · (y1− y2)

1 0

idY − fy (x, ty1+ (1− t)y2)
dt·
y1− y2
Since f y : U → L(X, Y ) is continuous, ∃r, r1> 0 such that

f(x, θ)
< 1

it follows that
R(x, y)
< r1, ∀(x, y) ∈ Br (θ) × Br1(θ) Then, ∀x ∈ Br (θ),

∃|y ∈ Br1(θ) satisfying f (x, y) = θ Denote by u(x) the solution y.

(3) We claim that u ∈ C(Br , Y ) Since

u(x) − u(x )
=
R(x, u(x)) − R(x  , u(x ))

1

2
u(x) − u(x )
+
R(x, u(x)) − R(x  , u(x))
,

we obtain

u(x) − u(x )

2
R(x, u(x)) − R(x  , u(x))
(1.4)

Noticing that R ∈ C(U, Y ), we have

u(x )→ u(x) as x  → x (4) If f ∈ C1(U, Y ), we want to prove u ∈ C1 First, by (1.2) and (1.4)

u(x) − u(x )

2
f(x, u(x)) − f(x  , u(x ))

21

u  (x) = −f −1

y (x, u(x)) ◦ fx (x, u(x))

Remark 1.2.2 In the first part of Theorem 1.2.1, the space X may be

as-sumed to be a topological space In fact, neither linear operations nor the properties of the norm were used.

g ∈ C1(V, X) Assume y0∈ V and g  (y0)∈ L(X, Y ) Then there exists δ > 0 such that B δ (y0)⊂ V and

g : B δ (y0)→ g(Bδ (y0))

is a differmorphism Furthermore

(g −1) (x0) = g −1 (y0), with x0= g(y0) (1.5)

1.2 Implicit Function Theorem and Continuity Method 15

g : Bδ (y0)→ g(Bδ (y0)) is a diffeomorphism And (1.5) follows from (1.1) 

In the spirit of the IFT, we have a nonlinear version of the Banach openmapping theorem

Theorem 1.2.4 (Open mapping) Let X, Y be Banach spaces, and let δ > 0

and y0∈ Y Suppose that g ∈ C1(B δ (y0), X) and that g  (y0) : Y → X is an open map, then g is an open map in a neighborhood of y0.

Proof We want to prove that ∃δ1∈ (0, δ) and r > 0, such that

B r (g(y0))⊂ g(Bδ1(y0)) With no loss of generality, we may assume y0= θ and g(y0) = θ Let A = g  (θ). Since A is surjective, ∃C > 0 such that

Now,∀x ∈ Br (θ), we are going to find y ∈ Bδ1(θ), satisfying g(y) = x Write

it follows that h n+1 ∈ Bδ1(θ) Then we can proceed inductively.

The sequence h n has a limit y Obviously y is the solution of (1.7)

Essentially, the implicit function theorem is a consequence of the

contrac-tion mapping theorem The continuity assumpcontrac-tion of f y in Theorem 1.2.2seems too strong in some applications We have a weakened version

Theorem 1.2.5 Let X, Y, Z be Banach spaces, and let B r (θ) ⊂ Y be a closed ball centered at θ with positive radius r Suppose that T ∈ L(Y, Z) has a bounded inverse, and that η : X × Br → Z satisfies the following Lipschitz condition:

η(x, y1)− η(x, y2)

K
y1− y2
∀y1, y2∈ Br (θ), ∀x ∈ X where K <
T −1
−1 If η(θ, θ) = θ, and

η(x, θ)

(
T −1
−1 −K)r ; then ∀x ∈ X, there exists a unique u : X → Br (θ) satisfying

T u(x) + η(x, u(x)) = θ ∀x ∈ X Furthermore, if η is continuous, then so is u.

Proof ∀x ∈ X we find the fixed point of the map −T −1 η(x, y) It is easily verified that T −1 · η(x, ·) : Br (θ) → Br (θ) is a contraction mapping

1.2 Implicit Function Theorem and Continuity Method 17

Example 1 (Structural stability for hyperbolic systems)

A matrix L ∈ GL(n, R) is called hyperbolic if the set of eigenvalues of

L, σ(L) ∩ iR1=∅ The associate differential system reads as:

˙

x = Lx, x ∈ C1(R1, R n ) The flow φ t = e Lt ∈ GL(n, R) is also linear The flow line can be seen on the left of Fig 1.1 Let ξ ∈ C 0,1 (R n , R n) be a Lipschitzian (Lip.) map weinvestigate the hyperbolic system under the nonlinear perturbation:

˙

x = Lx + ξ(x), x ∈ C1(R1, R n ) , and let ψ t be the associate flow, the flow line of which is on the right ofFig 1.1

What is the relationship between φ t and ψ t , if ξ is small? One says that the hyperbolic system is structurally stable, which means that the flow lines φ t and ψ tare topologically equivalent More precisely, there is a homeomorphism

h : R n → R n such that h ◦ ψt = φ t ◦ h.

We shall show that the hyperbolic system is structurally stable Let

A = e L , then the set of eigenvalues σ(A) of A satisfies σ(A) ∩ S1 = ∅.

We decompose ψ1 = A + f , where f ∈ C 0,1 (R n , R n), it is known that the

Lipschitzian constant of f is small if that of ξ is.

To the matrix A ∈ GL(n, R), since σ(A)∩S1=∅, provided by the Jordan

form, we have the decomposition R n = E u



Es , where E u, Esare invariant

subspaces, on which the eigenvalues of A u := A |E u lie outside the unit circle,

and those of A s := A |E s lie inside the unit circle Due to these facts, one has

As
< 1,
A −1

u
< 1 The following notations are used for any Banach spaces X, Y C0(X, Y ) stands for the space of all bounded and continuous mappings h : X → Y with

Let 0 <  <
A −1
−1 , then both (A + f ) −1 and (A + g) −1 exist Wedecompose R n = E u



Es and let P u and P s be the projections onto E uand

E s , respectively Let r = h − id, then equation (1.9) for r is equivalent to

(A + g) ◦ (id + r) = (id + r) ◦ (A + f) ,

1.2 Implicit Function Theorem and Continuity Method 19

+ P s g ◦ (id + r) ◦ (A + f) −1 .

(1.10)

We do the same for the equation of k.

(2) From the decomposition C0(R n , R n ) = C0(R n , Eu)

C0(R n , Es), we

reduce (1.10) to the form of Theorem 1.2.5 Set Sr = A −1

u ◦Pur ◦(A+f)As◦ Psr ◦ (A + f) −1 Then S ∈ L(C0(R n , R n ), C0(R n , R n)), and
S
< 1 Thus

η(f, g; θ)
< 2 For sufficiently small  > 0, all conditions of Theorem 1.2.5 are met There exists a unique continuous map r in a neighborhood

of θ in C0(R n , R n ), which is continuously dependent on f and g Similarly, let

q = k − id, one proves the existence of a unique continuous map q = q(f, g)

in the same neighborhood

(3) Setting h = id + r, k = id + q, we prove that h ◦ k = k ◦ h = id In fact

h ◦ k and k ◦ h satisfy the equations:

h ◦ k = (A + g) −1 ◦ (h ◦ k) ◦ (A + g) ,

and

k ◦ h = (A + f) −1 ◦ (k ◦ h) ◦ (A + f) ,

respectively Both the equations have the unique solution id in a neighborhood

of θ, and the conclusion is proved.

(4) We now prove the conclusion for arbitrary t We have

φ1◦ (φt ◦ h ◦ ψ −t ) = (φ t ◦ φ1)◦ (h ◦ ψ −t ) = (φ t ◦ h ◦ ψ −t)◦ ψ1.

Also as |t − 1| < δ for small δ > 0, φt ◦ h ◦ ψ −t − id ∈ C0(R n , R n) is in

a small neighborhood of θ, and the above equation has a unique solution h

there; therefore

φt ◦ h ◦ ψ −t = h Then we can extend the procedure step by step to all t, i.e we have

Example 2 (Local existence of isothermal coordinates)

Given a surface M2 with a Riemannian metric g, i.e., in local coordinates

x = (x1, x2),

g = Edx2+ 2F dx1dx2+ Gdx2, where E, F and G are functions of local coordinates x = (x1, x2), and

∀(x1, x2), Eξ2+2F ξη +Gη2is a positive definite quadratic form Our problem

is to find a local coordinate u(x) = (u1(x1, x2), u2(x1, x2)) in a neighborhood

of x0= (x01, x02) such that there exists a function λ = λ(x1, x2) > 0 satisfying

g = λ(x1, x2)(du21+ du22) The local coordinate u = (u1, u2) is called an isothermal coordinate

(1.11)

We may assume (x0, x0) = (0, 0) and F (0, 0) = 0 (by translation and rotation

of the local coordinates) After eliminating λ, (1.11) is equivalent to:

F (x) · |∂x1u(x) |2− E(x) · ∂x1u(x) · ∂x2u(x)

F (x) · |∂x2u(x) |2− G(x) · ∂x1u(x) · ∂x2u(x) ,

Let x = εy, where y ∈ D, the unit disk on the coordinate plane, and ε > 0 is

a parameter, and let

where u = (p1x1+ p2x2, q1x1+ q2x2), and (p1, p2, q1, q2) is chosen such thatthe following conditions are satisfied:

∂(u1, u2)

∂(x , x ) = 0 ,

1.2 Implicit Function Theorem and Continuity Method 21

ϕ(εy, ∇y v(y) + ∇x u(εy)) = 0

Let ˙C 1,α( ¯D) denote the space C 1,α( ¯D) modulo a constant Set F :R1× (C 1,α ∩ C0(D) ˙

C 1,α (D)) → C α (D)2, for some α ∈ (0, 1), where

F (ε, v) = ϕ(εy, ∇yv(y) + ∇xu(εy))

Thus

F (0, θ) = θ

It remains to find a pair (ε, v) for small ε > 0, satisfying

F (ε, v) = θ Then by the transform(1.14), the solution u in a small neighborhood of θ εD

is obtained Note that (p1, q1, p2, q2) = (u 1,x1, u 2,x1, u 1,x2, u 2,x2) Let us

intro-duce the notation (E0, F0, G0) = (E, F, G) | (0,0); we have

F v (0, θ)h = −(A1∇h1+ A2∇h2) , where h = (h1, h2)∈ C 1,α ∩ C0(D) ˙

Since

det L = −2F0(E0ξ2− 2F0ξη + G0η2)∂(u, v)

∂(x, y) = 0

∀(ξ, η) ∈ R2\{θ}, L is elliptic.

According to the elliptic theory, F v (0, ·) has a bounded inverse The IFT

is applied to conclude the existence of ε0> 0 such that the equation

F (ε, v) = θ has a unique solution v ε,∀ε ∈ (−ε0, ε0)

Remark 1.2.7 The boundary condition for the first-order system can be

de-duced from the second-order elliptic theory In fact, h = (h1, h2) satisfies

is elliptic

Thus the Dirichlet boundary condition for h1 is well posed, and then h2follows from (1.16) modulo a constant

Remark 1.2.8 There are many applications of the IFT similar to the above

examples, e.g a necessary and sufficient condition for an almost complex structure being a complex structure (Newlander–Nirenberg theorem, see L Nirenberg [NN]), prescribing Ricci curvature problem (see DeTurck [DT] etc.) For applications to boundary value problems in ordinary and partial differen- tial equations see S.N Chow, J Hale [CH] and J Mawhin [Maw 3]

1.2 Implicit Function Theorem and Continuity Method 23

1.2.3 Continuity Method

We have shown the usefulness of the IFT in the existence of solutions forsmall perturbations of a given equation which has a known solution As tolarge perturbations, the IFT is not enough, we have to add new ingredients.The continuity method is a general principle, which can be applied to provethe existence of solutions for a variety of nonlinear equations

Let X and Y be Banach spaces, and f : X → Y be C1 Find the solution

of the equation:

f (x) = θ Let us introduce a parameter t ∈ [0, 1] and a map

F : [0, 1] × X → Y such that both F and F x are continuous; in addition,

F (1, x) = f (x) Assume that there exists x0 ∈ X satisfying F (0, x0) = θ; we want to extend the solution x0 of the equation

F (0, x) = θ

to a solution of

For this purpose, we define a set

S = {t ∈ [0, 1] | such that F (t, x) = θ is solvable}

What we want to do is to prove:

(1) S is an open set (relative to [0, 1]) For this purpose, it is sufficient to

prove that ∀t0 ∈ S, ∃xt0 ∈ X, which solves F (t0, x t0) = θ such that

F −1

x (t0, x t0)∈ L(Y, X), provided by the IFT.

(2) S is a closed set Usually it depends on the a priori estimates for the

solu-tion set {x ∈ X | ∃t ∈ S, such that F (t, x) = θ} For most PDE problems

it requires special knowledge and features of the equations and techniques

in hard analysis

We present here two major ideas:

(a) If there exist a Banach space X1, which is compactly embedded in X, and

a constant C > 0 such that

xt
X1
C ∀t ∈ S , where x t is a solution of F (t, x) = θ, then S is closed.

In fact, we have{tn} ∞ ⊂ S, which implies tn → t ∗, and

xt n
X1
C Since the embedding X1 → X is compact, xt n subconverges to some

point x ∗ ∈ X in X From the continuity, it follows that F (t ∗ , x ∗ ) = θ. This proves t ∗ ∈ S, i.e., S is closed.

(b) If∀t ∈ S, there exists a unique local solution xt of the equation F (t, ·) = θ, and if there exists C > 0 such that

as n  m → ∞ Let x ∗ be the limit If the IFT is applicable to (t ∗ , x ∗),

Once (1) and (2) are proved, S is a nonempty (0 ∈ S) open and closed set Therefore S = [0, 1], and then the equation F (1, ·) = θ is solvable.

As an application of the continuity method, we have:

Theorem 1.2.9 (Global implicit function theorem) Let X, Y be Banach

spaces, and let f ∈ C1(X, Y ) with f  (x) −1 ∈ L(Y, X) ∀x ∈ X If ∃ constants

A, B > 0 such that

f  (x) −1

A
x
+B ∀x ∈ X , then f is a diffeomorphism.

Proof (1) Surjective We want to prove that ∀y ∈ Y, ∃x ∈ X satisfying

f (x) = y

∀x0∈ X, define F : [0, 1] × X → Y as follows:

F (t, x) = f (x) − [(1 − t)f(x0) + ty] Set S = {t ∈ [0, 1]| F (t, ·) = θ is solvable} Obviously, 0 ∈ S, and since

F −1

x (t, x) = f  (x) −1 ∈ L(Y, X), S is open, from the IFT.

1.2 Implicit Function Theorem and Continuity Method 25

It remains to prove the closeness of S Indeed, in a component (a, b) of S, there is a branch of solutions x tsatisfying

(2) Injective We argue by contradiction If∃y ∈ Y and x0, x1∈ X, fying f (x i ) = y, i = 0, 1 Let γ : [0, 1] → X be the segment connecting these

satis-two points:

γ(s) = (1 − s)x0+ sx1 s ∈ [0, 1] Thus f ◦γ is a loop passing through y If we could find x : [0, 1] → X satisfying

x(i) = xi i = 0, 1, and

f ◦ x(s) = y ∀s ∈ [0, 1] , then this would contradict with the locally homeomorphism of f

Define I = [0, 1] and T : I × C0(I, X) → C0(I, Y ) as follows:

r x

Tu (t, u) = f  (γ( ·) + u(·)) ∈ L(C0(I, X), C0(I, Y )) ,

which has a bounded inverse Therefore S = {t ∈ [0, 1]| T (t, u) = θ

is solvable} is open, from the IFT.

2 Let u t (s) be a solution at t ∈ S Then

f  (γ(s) + u

t (s)) · ˙ut (s) = y − f ◦ γ(s) ,

where ˙ut denotes the derivative with respect to t Again we obtain

˙ut
C0(I,X)
(A
u t
C0(I,X) +B1)
y − f ◦ γ
C0(I,Y ) , where B1 > 0 is another constant depending on B and x0, x1 only As inparagraph (1),∃C > 0 such that

+→ R1 + such that

|f(x, η, ξ)| ≤ c(|η|)(1 + |ξ|2), ∀ (x, η, ξ) ∈ Ω × R1× R n

(2)

∂f

∂η (x, η, ξ)
0

1.2 Implicit Function Theorem and Continuity Method 27

(3) Assume ∃M > 0 such that

f (x, η, θ) =



< 0, if η > M

> 0, if η < −M Assume φ ∈ C 2,γ , for some γ ∈ (0, 1) Then the equation



−u = f(x, u(x), ∇u(x)) x ∈ Ω

possesses a unique solution in C 2,γ

(1.21), then

u
C(Ω)
maxmax

∂Ω |φ(x)|, M Proof Assume that |u(x)| attains its maximum at x0 ∈ Ω We divide our

discussion into two cases

(1) x0∈ ∂Ω; the proof is done.

(2) x0∈Ω, then◦ ∇u(x0) = 0 and−u(x0) = f (x0, u(x0), θ).

If u(x0) > M , then LHS  0, but RHS < 0 It is impossible.

Similarly, if u(x0) < −M, then LHS
0, but RHS > 0 Again, it is

(u, τ ) → (−u + u − a(τ + |∇u|2), u |∂Ω − τφ) ,

where X = C 2,γ (Ω), Y = C 0,γ(Ω)× C 2,γ (∂Ω) and γ ∈ (0, γ).

Noticing that

Fu (u, τ )v = (( −v + v − a∇u · ∇v), v|∂Ω ) ,

and from the Schauder estimates,∀(u, τ) ∈ X × [0, 1], Fu (u, τ ) has a bounded

inverse Define the set:

Since ˙uτ satisfies (1.24), from Lemma 1.2.11,
˙uτ
C(Ω) is bounded by aconstant depending on
a
C(Ω) and
φ
, and from the Gagliardo–Nirenberg

Again, by the Gagliardo–Nirenberg inequality, we have

∇uτ
L 2p
C p
∇2u τ
L1p
uτ
L1p +C
uτ
L ∞

Repeating the use of Lemma 1.2.11,
uτ
C is bounded by a constant pending on
φ
C, and we obtain

1.2 Implicit Function Theorem and Continuity Method 29

As a consequence of the Sobolev embedding theorem, for p > 1−γ n , we have

uτ
C 1,γ
C(
φ
C 2,γ ,
a
C , γ) (1.25)

Substituting into the equation F (u τ , τ ) = θ, we apply the Schauder

esti-mate:

uτ
C 2,γ
C(
φ
C 2,γ ,
a
C 0,γ , γ) The continuity method is applicable; we have a solution u of (1.22) Indeed, the solution is unique Let u1, u2 be two solutions and set ω =

u1− u2, then



−ω + ω = a∇(u1+ u2)· ∇ω in Ω ,

ω |∂Ω = 0

If max ω > 0, then ∃x0 ∈ Ω such that max◦ Ωω = ω(x0) Thus, ∇ω(x0) =

0, −ω(x0) 0 and ω(x0) > 0 This is impossible Similarly, one proves that

Now we come back to the proof of the theorem

Proof Applying the continuity method, we study the equation:

We want to solve F (1, u) = θ However, ∀u ∈ C 2,σ(Ω),

Fu (t, u)v = ( −v − tfη (x, u(x), ∇u(x))v − tfξ (x, u(x), ∇u(x))∇v, v|∂Ω )

By assumption (2) and the maximum principle for linear elliptic equations,

∀g ∈ C σ(Ω)× C 2,σ (∂Ω)

Fu (t, u)v = g has a unique solution, i.e., F u (t, u) : C 2,σ ∩ C0(Ω)→ C σ(Ω)× C 2,σ (∂Ω) has a

bounded inverse Thus the set

S = {t ∈ I| F (t, u) = θ is solvable}

is open, from the IFT

We prove that S is closed Noting that if u t satisfies

F (t, ut ) = θ ,

C 2,σ (∂Ω) is compact, we conclude that S is closed, as we have seen previously The existence of the solution for F (1, u) = θ follows from the continuity

method The uniqueness is a consequence of the maximum principle

Once we obtain a solution u in C 2,σ, it follows directly by Schauder

attains some critical values This kind of phenomenon is called bifurcation.For example, a simple algebraic equation:

x3− λx = 0 λ ∈ R1,

1.3 Lyapunov–Schmidt Reduction and Bifurcation 31

x

λ0

Fig 1.3.

has a solution x = 0 ∀λ ∈ R1 As λ
0, this is the unique solution; but as

λ > 0, we have two more branches of solutions

x = ± √ λ

See Figure 1.3

Bifurcation phenomena occur extensively in nature Early in 1744, Euler

observed the bending of a rod pressed along the direction of its axis Let θ

be the angle between the real axis and the tangent of the central line of the

rod, and let λ be the pressure The length of the rod is normalized to be

π We obtain the following differential equation with the two free end point

θ + λ sin θ = 0 ,

˙

θ(0) = ˙ θ(π) = 0 Obviously, θ ≡ 0 is always a solution of the ODE Actually the solution is unique, if λ is not large As λ increasingly passes through a certain value λ0,

it is shown by experiment that there exists a bending solution θ = 0.

The same phenomenon occurs in the bending of plates, shells etc In tion, bifurcation occurs in the study of thermodynamics (B´ernard problem),rotation of fluids, solitary waves, superconductivity and lasers

addi-Mathematically, we describe the bifurcation by the following:

Definition 1.3.1 Let X, Y be Banach spaces, and let ∧ be a topological space Suppose that F : X × ∧ → Y is a continuous map ∀λ ∈ ∧, let

S λ={x ∈ X| F (x, λ) = θ}

be the solution set of the equation F (x, λ) = θ, where λ is a parameter (Fig 1.4) Assume θ ∈ Sλ , ∀λ ∈ ∧ We call (θ, λ0) a bifurcation point, if for any neighborhood U of (θ, λ0), there exists (x, λ) ∈ U with x ∈ S λ \{θ}.

λ λ

θ

Fig 1.4.

The following problems are of primary concern:

(1) What is the necessary and sufficient condition for a bifurcation point

(θ, λ0)?

(2) What is the structure of S λ near λ = λ0?

(3) How do we compute the solutions near the bifurcation points?

(4) How about the global structure of∪λ ∈∧ S λ?

(5) Let F (x, λ) = θ be the steady equation of the evolution equation:

˙

x = F (x, λ) ,

we study the stability of solutions in S λ as λ approaches λ0

In this section, we focus our discussions on problems (1) and (2) (4) will

(1) Assume that F x (x, λ) is continuous If (θ, λ0) is a bifurcation point, then

Fx (θ, λ0) does not have a bounded inverse

(2) Assume

F (x, λ) = Lx − λx + N(x, λ) , where L ∈ L(X, Y ), λ ∈ R1, and that N : U × R1→ X is continuous with

N(x, λ)
= ◦(
x
) as
x
→ θ uniformly for λ in a neighborhood of λ0 If (θ, λ0) is a bifurcation point,

then λ0∈ σ(L), i.e., λ0 is a spectrum of L.

Proof If not, λ0 ∈ ρ(L), the resolvent set of L Since ρ(L) is open, ∃ε > 0 and C ε > 0 such that

(L − λI) −1

Cε as|λ − λ0| < ε

It follows that