APPLIED FUNCTIONAL ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS

Milan Miklavcic
Department of Mathematics
Michigan State University
East Lansing, MI 48824-1027
U S A

World Scientific
ISBN: 981-02-3535-6,    Autumn 1998
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PREFACE

This book is an introduction to partial differential equations (PDEs) and the relevant functional analysis tools which PDEs require. This material is intended for second year graduate students of mathematics and is based on a course taught at Michigan State University for a number of years. The purpose of the course, and of the book, is to give students a rapid and solid research-oriented foundation in areas of PDEs, like semilinear parabolic equations, that include studies of the stability of fluid flows and, more generally, of the dynamics generated by dissipative systems, numerical PDEs, elliptic and hyperbolic PDEs, and quantum mechanics. In other words, the book gives a complete introduction to and also covers significant portions of the material presented in such classics as

Partial Differential Equations by Avner Friedman

Geometric Theory of Semilinear Parabolic Equations by Dan Henry

Semigroups of Linear Operators and Applications to PDEs by A. Pazy.

The need for such a book is due to the fact that in order to study PDEs one needs to know some functional analysis, which requires a thorough knowledge of real analysis (Lebesgue integral). Therefore, if real analysis is studied in the first year of graduate school, and functional analysis in the second year, the student only begins with PDEs in the third year - and may even have to re-learn functional analysis if the prior instructor ignored unbounded operators (which sometimes happens).

The reader is expected to be comfortable with the Lebesgue integral; more specifically, with the material presented in Examples 1.3.4 and 1.5.2. The Cauchy Theorem is also used in a couple of places, with the most difficult version used in (4.44). These are the only real prerequisites for the whole book. Above this level, all theorems used are proved in the text. One may, and perhaps should, skip over some of the proofs. However, they are included in case they are needed.

With regard to the writing style, all formal statements, like Theorems, contain all assumptions except for those declared at the beginning of the section in which the statement appears. This should make it easy, even for a casual reader, to figure out what is actually assumed in a given statement. There is, however, one exception. Throughout Chapter 3 it is assumed, unless otherwise specified, that O is an arbitrary nonempty open set in Rn.

In the first two chapters functional analysis tools are developed and differential operators are studied as examples. Sturm-Liouville operators are nice examples of self-adjoint operators with compact resolvent and are reused in Chapter 4 as generators of strongly continuous semigroups. Hormander's treatment of weak solutions of constant coefficient PDEs is also presented early on as an example. The foundation of elliptic, parabolic and wave equations, as well as of Galerkin approximations, is given in the section on Sectorial Forms. Throughout the text, completeness of a number of orthonormal systems is proven.

The Fourier transform and its applications to constant coefficient PDEs are presented in Chapter 3. We briefly touch upon distributions and fundamental solutions, and prove the Malgrange-Ehrenpreis Theorem. Most of Chapter 3 is devoted to study of Sobolev spaces. Many sharp results concerning existence and compactness of imbeddings, as well as interpolation inequalities, are proven. These results are applied to elliptic problems in the last two sections.

The study of evolution equations begins in Chapter 4 where the semigroup theory is introduced. The Hille-Yosida Theorem for strongly continuous semigroups and Hille's construction of analytic semigroups are presented. The semigroup theory and the results of the previous chapters enable us to discuss linear parabolic and wave equations. In preparation for studies of nonlinear evolution equations, the invariant subspaces associated with the semigroups and the inhomogeneous problem are also examined.

A dynamical systems approach to weakly nonlinear evolution equations is given in Chapter 5 with a nonlinear heat equation studied as an example. Trotter's approximation theory is adapted to such equations giving convergence of Galerkin and finite difference type approximations.

The chapter on semilinear parabolic equations begins with a very technical section on fractional powers of operators. Our main results contain existence, uniqueness, continuous dependence, maximal interval of existence, stability and instability results. These results are applied to the Navier-Stokes equations, to a stability problem in fluid mechanics, to showing how a classical solution can be obtained, and to the Chafee-Infante problem as an example of a gradient system.

I wish to thank S. N. Chow and D. R. Dunninger for their early encouragement and my wife Pam for checking the grammar.


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CONTENTS

Preface v
1 Linear Operators in Banach Spaces 1
1.1 Metric Spaces 1
1.2 Vector Spaces 6
1.3 Banach Spaces 8
1.4 Linear Operators 11
1.5 Duals 16
1.6 Spectrum 23
1.7 Compact Linear Operators 29
1.8 Boundary Value Problems for Linear ODEs 39
1.9 Exercises 43
2 Linear Operators in Hilbert Spaces 47
2.1 Orthonormal Sets 47
2.2 Adjoints 54
2.3 Accretive Operators 58
2.4 Weak Solutions 62
2.5 Example: Constant Coefficient PDEs 68
2.6 Self-adjoint Operators 70
2.7 Example: Sturm-Liouville Problem 76
2.8 Sectorial Forms 80
2.9 Example: Harmonic Oscillator and Hermite Functions 88
2.10 Example: Completeness of Bessel Functions 92
2.11 Example: Finite Element Method 96
2.12 Friedrichs Extension 99
2.13 Exercises 104
3 Sobolev Spaces 109
3.1 Introduction 109
3.2 Fourier Transform 114
3.3 Distributions 122
3.4 Weak Derivatives 126
3.5 Definition and Basic Properties of Sobolev Spaces 135
3.6 Imbeddings of Wm,p (O) 140
3.7 Elliptic Problems 153
3.8 Regularity of Weak Solutions 160
3.9 Exercises 163
4 Semigroups of Linear Operators 167
4.1 Introduction 167
4.2 Bochner Integral 171
4.3 Basic Properties of Semigroups 176
4.4 Example: Wave Equation 187
4.5 Sectorial Operators and Analytic Semigroups 192
4.6 Invariant Subspaces 204
4.7 The Inhomogeneous Problem - Part I 209
4.8 Exercises 212
5 Weakly Nonlinear Evolution Equations 215
5.1 Introduction 215
5.2 Basic Theory 218
5.3 Example: Nonlinear Heat Equation 222
5.4 Approximation for Evolution Equations 224
5.5 Example: Finite Difference Method 233
5.6 Example: Galerkin Method for Parabolic Equations 236
5.7 Example: Galerkin Method for the Wave Equation 239
5.8 Friedrichs Extension and Galerkin Approximations 243
5.9 Exercises 245
6 Semilinear Parabolic Equations 247
6.1 Fractional Powers of Operators 247
6.2 The Inhomogeneous Problem - Part II 254
6.3 Global Version 257
6.4 Main Results 259
6.5 Example: Navier-Stokes Equations 267
6.6 Example: A Stability Problem 271
6.7 Example: A Classical Solution 273
6.8 Dynamical Systems 276
6.9 Example: The Chafee-Infante Problem 279
6.10 Exercises 282
Bibliography 285
List of Symbols 289
Index 291



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TEXT SAMPLES

Counter example to uniqueness for semilinear parabolic equations

Friedrichs extension and Galerkin approximations

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INDEX


Abel's integral equation, 45
accretive, 28 , 58 , 73, 185
addition, 6
adjoint, 21 , 56, 63
adjoint problem, 40
adjoint space, 16
almost separably-valued, 171
analytic, 27 , 117 , 169, 193
arcwise-connected, 5
Arzela-Ascoli Theorem, 5, 17, 31, 34, 90, 94, 150


Baire Theorem, 2
Banach space, 9
Banach-Steinhaus Theorem, 14
basis, 7
Bessel function, 95
biharmonic operator, 156
Bochner integrable, 172
Bochner integral, 172
boundary value problem, 39, 120
bounded, 12
bounded linear functionals, 16 , 55
bounded set, 9
Burger's equation, 284


Cauchy sequence, 2
Cebysev polynomials, 52
closable, 15
closed, 2 , 15
Closed Graph Theorem, 16
closure, 2 , 16
compact, 2 , 31
compact imbedding, 82, 83, 152, 156
compact resolvent, 37 , 74, 78, 82, 83, 87, 90, 91, 94, 154, 155, 157
complement, 2
complete, 2 , 48 , 50, 74, 75, 78, 91, 92, 104, 163
complex vector space, 6
cone property, 141 , 155
conjugate gradient method, 105
conjugate space, 16
connected, 5 , 130, 155
continuous, 4
continuous at the point, 4
contraction, 3
Contraction Mapping Theorem, 3, 25, 216, 259
contraction semigroup, 184
converge, 2
convex, 7
convolution, 109 , 123
cycle, 206


defined in, 12
defined on, 12
delay equation, 257
delta function, 123
dense, 2
densely defined, 12
derivative of the distribution, 122
derivative of the product, 8
    weak, 131
dimension, 7
Dirichlet problem, 153
discretization, 224
dissipative, 28
distance, 1
distribution, 122 , 127
domain, 11
dual space, 16
dynamical systems, 277


eigenvalues, 23
eigenvectors, 23
elliptic equation, 70 , 96, 102, 118, 153
energy method, 271
entire, 117
equilibrium point, 277
equivalent norms, 9
Euler method, 183 , 224, 235
extension, 12 , 13, 18, 61, 99, 104


finite difference, 129, 234
finite element, 96 , 107, 236, 237, 239
fixed-point, 3
formal adjoint, 63
Fourier cosine series, 52 , 53
Fourier series, 52 , 75, 104
Fourier sine series, 52 , 79
Fourier transform, 114, 115 , 120, 121, 129, 137, 159
fractional powers of operators, 58, 249
Fredholm alternative, 38 , 42, 57, 83, 156, 157
Friedrichs extension, 99 , 243
fundamental solution, 123


Galerkin approximation, 98 , 99, 103, 236, 239, 243
Garding's inequality, 103, 156, 157
generator, 168
gradient system, 279
Gram-Schmidt orthogonalization, 48
Gronwall Theorem, 216, 258


Hahn-Banach Theorem, 18
Hamiltonian, 88, 187
harmonic functions, 120
hat function, 98
Hausdorff Maximality Theorem, 18
heat equation, 222, 260, 267
Heisenberg inequality, 117
Hermite functions, 91 , 163
Hilbert space, 48
Hilbert space adjoint, 56
Hilbert-Schmidt Theorem, 74
Hille-Yosida Theorem, 180
Holder continuous, 4
Hopf-Cole substitution, 284
Hormander Theorem, 70
hyperbolic equation, 66, 187


imbedding compact, 82, 83, 152, 156
infinitesimal generator, 168
initial value problem, 120, 121
inner product, 47
inner product space, 47
instability, 201, 265
intermediate derivatives, 132
interpolation inequalities, 132, 135
invariant, 277
invariant subspace, 204
inverse Fourier transform, 115


Laplace operator, 70
Laplace transform, 116
Laplacian, 70 , 92, 118, 153, 155, 163, 185, 193
Legendre polynomials, 48 , 104
Liapunov function, 278 , 280
linear operator from X to Y, 11
linearly independent, 7
Lipschitz continuous, 4


m-accretive, 58 , 61, 73, 87, 186, 283
Malgrange-Ehrenpreis Theorem, 125
maximal totally ordered, 18
maximum principle, 275
Maxwell's equations, 164
metric, 1
metric space, 1
mild solution, 210 , 234, 242
mild solutions, 218
mollifier, 110


Neumann problem, 155 , 165
norm, 8
normalized, 47
normalized tangent functional, 20 , 172
normed space, 8
null space, 12
numerical range, 28 , 80, 99, 192


omega-limit set, 277
open, 2
open ball, 1
Open Mapping Theorem, 14
orbit, 277
origin, 6
orthogonal, 47
orthogonal projection, 55
orthonormal, 48
orthonormal basis, 48


parabolic equation, 66, 120, 169, 185, 192 , 200 , 222, 233, 236, 237, 243
Parseval equality, 50, 52
partially ordered, 18
Partition of Unity Theorem, 110
perturbation of a generator, 186
Pettis Theorem, 171
point spectrum, 23
Ponicare inequality, 138 , 154, 165
positive definite, 67
positively invariant, 277
projection, 55, 205
projection associated with the spectral set, 206


range, 12
rapidly decreasing functions, 114
real subspace, 7 , 174, 205, 220, 263
real vector space, 6
reflexive, 20
regularity of weak solutions, 64, 160
relatively compact, 2
resolvent, 23
resolvent identity, 23
resolvent set, 23
restriction, 12 , 61
restriction of A to Y, 204
retarded functional differential equation, 257
retraction map, 217
Riesz Lemma, 55


scalar, 6
scalar field, 6
scalar multiplication, 6
scalar product, 47
Schrodinger equation, 187
Schwartz space, 114
second dual space, 20
sectorial operator, 193 , 201, 253, 254, 258, 259, 283
sectorial sesquilinear forms, 80 , 89, 93, 97, 99, 153, 155--157, 186, 187, 193, 236, 239, 253, 273
self-adjoint, 57 , 71, 73--75, 78, 83, 84, 88, 91, 93, 94, 99, 105, 107, 154--156
semigroup, 167
    analytic, 192
semilinear parabolic equations, 260
seminorm, 9
separable, 2 , 5, 11, 17
simple functions, 173
singular differential equation, 103
smoothing operator, 170, 201
span, 7
spectral representation, 74
spectral set, 206 , 213
spectrum, 23
square root, 58 , 87
stability, 201, 264, 271
stability condition, 225
star-shaped, 138 , 165
strong ellipticity condition, 156 , 157, 200
strongly continuous semigroup, 167
strongly elliptic operator, 157
strongly measurable, 171
Sturm-Liouville problem, 76
subspace, 7
support, 7
symmetric, 57 , 70
symmetric hyperbolic system, 66


Taylor formula, 8
totally ordered, 18
Trotter Theorem, 226


uncertainty principle, 118
uniform boundedness principle, 14 , 17
uniformly continuous, 4
uniqueness, 260, 283


variational formulation, 85
vector space over , 6
Volterra integral equation, 26


wave equation, 68 , 121, 189 , 239, 243, 246, 279
    generalized, 170, 187 , 190
weak derivative, 127
weak limit, 18
weak solution, 63
weakly bounded, 44
weakly convergent, 18
weakly measurable, 171
weakly nonlinear evolution equations, 215


Young Lemma, 112




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