Techniques of Functional Analysis for Differential and Integral Equations

By Paul Sacks

Techniques of Functional Analysis for Differential and Integral Equations describes a variety of powerful and modern tools from mathematical analysis, for graduate study and further research in ordinary differential equations, integral equations and partial differential equations. Knowledge of these techniques is particularly useful as preparation for graduate courses and PhD research in differential equations and numerical analysis, and more specialized topics such as fluid dynamics and control theory. Striking a balance between mathematical depth and accessibility, proofs involving more technical aspects of measure and integration theory are avoided, but clear statements and precise alternative references are given . The work provides many examples and exercises drawn from the literature.

• Provides an introduction to mathematical techniques widely used in applied mathematics and needed for advanced research in ordinary and partial differential equations, integral equations, numerical analysis, fluid dynamics and other areas
• Establishes the advanced background needed for sophisticated literature review and research in differential equations and integral equations
• Suitable for use as a textbook for a two semester graduate level course for M.S. and Ph.D. students in Mathematics and Applied Mathematics

• Language: English
• Publisher: Academic Press
• Release date: May 16, 2017
• ISBN: 9780128114575

Paul Sacks

Professor Paul Sacks received his B.S. degree from Syracuse University and M.S. and Ph.D. degrees from the University of Wisconsin-Madison, all in Mathematics. Since 1981 he has been in the Mathematics department at Iowa State University, as Full Professor since 1990. He is particularly interested in partial differential equations and inverse problems. He is the author or co-author of more than 60 scientific articles and conference proceedings. For thirty years he has regularly taught courses in analysis, differential equations and methods of applied mathematics for mathematics graduate students.

Book preview

Techniques of Functional Analysis for Differential and Integral Equations

First Edition

Paul Sacks

Department of Mathematics, Iowa State University, Ames, IA, United States

Series Editor

Goong Chen

Table of Contents

Cover image

Title page

Copyright

Preface

Chapter 1: Some Basic Discussion of Differential and Integral Equations

Abstract

1.1 Ordinary Differential Equations

1.2 Integral Equations

1.3 Partial Differential Equations

1.4 Well-Posed and Ill-Posed Problems

Chapter 2: Vector Spaces

Abstract

2.1 Axioms of a Vector Space

2.2 Linear Independence and Bases

2.3 Linear Transformations of a Vector Space

Chapter 3: Metric Spaces

Abstract

3.1 Axioms of a Metric Space

3.2 Topological Concepts

3.3 Functions on Metric Spaces and Continuity

3.4 Compactness and Optimization

3.5 Contraction Mapping Theorem

Chapter 4: Banach Spaces

Abstract

4.1 Axioms of a Normed Linear Space

4.2 Infinite Series

4.3 Linear Operators and Functionals

4.4 Contraction Mappings in a Banach Space

Chapter 5: Hilbert Spaces

Abstract

5.1 Axioms of an Inner Product Space

5.2 Norm in a Hilbert Space

5.3 Orthogonality

5.4 Projections

5.5 Gram-Schmidt Method

5.6 Bessel’s Inequality and Infinite Orthogonal Sequences

5.7 Characterization of a Basis of a Hilbert Space

5.8 Isomorphisms of a Hilbert Space

Chapter 6: Distribution Spaces

Abstract

6.1 The Space of Test Functions

6.2 The Space of Distributions

6.3 Algebra and Calculus With Distributions

6.4 Convolution and Distributions

Chapter 7: Fourier Analysis

Abstract

7.1 Fourier Series in One Space Dimension

7.2 Alternative Forms of Fourier Series

7.3 More About Convergence of Fourier Series

7.5 Further Properties of the Fourier Transform

7.6 Fourier Series of Distributions

7.7 Fourier Transforms of Distributions

Chapter 8: Distributions and Differential Equations

Abstract

8.1 Weak Derivatives and Sobolev Spaces

8.3 Fundamental Solutions

8.4 Fundamental Solutions and the Fourier Transform

8.5 Fundamental Solutions for Some Important PDEs

Chapter 9: Linear Operators

Abstract

9.1 Linear Mappings Between Banach Spaces

9.2 Examples of Linear Operators

9.3 Linear Operator Equations

9.4 The Adjoint Operator

9.5 Examples of Adjoints

9.6 Conditions for Solvability of Linear Operator Equations

9.7 Fredholm Operators and the Fredholm Alternative

9.8 Convergence of Operators

Chapter 10: Unbounded Operators

Abstract

10.1 General Aspects of Unbounded Linear Operators

10.2 The Adjoint of an Unbounded Linear Operator

10.3 Extensions of Symmetric Operators

Chapter 11: Spectrum of an Operator

Abstract

11.1 Resolvent and Spectrum of a Linear Operator

11.2 Examples of Operators and Their Spectra

11.3 Properties of Spectra

Chapter 12: Compact Operators

Abstract

12.1 Compact Operators

12.2 Riesz-Schauder Theory

12.3 The Case of Self-Adjoint Compact Operators

12.4 Some Properties of Eigenvalues

12.5 Singular Value Decomposition and Normal Operators

Chapter 13: Spectra and Green’s Functions for Differential Operators

Abstract

13.1 Green’s Functions for Second-Order ODEs

13.2 Adjoint Problems

13.3 Sturm-Liouville Theory

13.4 Laplacian With Homogeneous Dirichlet Boundary Conditions

Chapter 14: Further Study of Integral Equations

Abstract

14.1 Singular Integral Operators

14.2 Layer Potentials

14.3 Convolution Equations

14.4 Wiener-Hopf Technique

Chapter 15: Variational Methods

Abstract

15.1 The Dirichlet Quotient

15.2 Eigenvalue Approximation

15.3 The Euler-Lagrange Equation

15.4 Variational Methods for Elliptic Boundary Value Problems

15.5 Other Problems in the Calculus of Variations

15.6 The Existence of Minimizers

15.7 Calculus in Banach Spaces

Chapter 16: Weak Solutions of Partial Differential Equations

Abstract

16.1 Lax-Milgram Theorem

16.2 More Function Spaces

16.3 Galerkin’s Method

16.4 Introduction to Linear Semigroup Theory

Appendix

A.1 Lebesgue Measure and the Lebesgue Integral

A.2 Inequalities

A.3 Integration by Parts

Bibliography

Index

Copyright

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Preface

The purpose of this book is to provide a textbook option for a course in modern methods of applied mathematics suitable for first year graduate students in Mathematics and Applied Mathematics, as well as for students in other science and engineering disciplines with good undergraduate-level mathematics preparation. While the term applied mathematics has a very broad meaning, the scope of this textbook is much more limited, namely to present techniques of mathematical analysis which have been found to be particularly useful in understanding certain kinds of mathematical problems which commonly occur in the scientific and technological disciplines, especially physics and engineering. These methods, which are often regarded as belonging to the realm of functional analysis, have been motivated most specifically in connection with the study of ordinary differential equations, partial differential equations, and integral equations. The mathematical modeling of physical phenomena typically involves one or more of these types of equations, and insight into the physical phenomenon itself may result from a deep understanding of the underlying mathematical properties which the models possess. All concepts and techniques discussed in this book are ultimately of interest because of their relevance for the study of these three general types of problems. Of course there is a great deal of beautiful mathematics, which has grown out of these ideas, and so intrinsic mathematical motivation cannot be denied or ignored.

The background the student will obtain by studying this material is sufficient preparation for more advanced graduate-level courses in differential equations and numerical analysis, as well as more specialized topics such as fluid dynamics. The presentation avoids overly technical material from measure and function theory, while at the same time maintaining a high standard of mathematical rigor. This is accomplished by including careful presentation of the statements of such results whose proofs are beyond the scope of the text, and precise references to other resources where complete proofs can be found. As the book is meant as textbook for a course, which is typically the basis for a written qualifying examination for PhD students, a great many examples and exercises are given.

The topics presented in this book have served as the basis for a two-semester course for first year graduate students at Iowa State University. A normal division is that Chapters 1–8 are covered in the first semester, and Chapters 9–16 in the second. A partial exception is Chapter 14, all of whose topics are optional in relation to later material, and so may be easily omitted.

I would like to thank past and present friends and colleagues for many helpful discussions and suggestions about the teaching of Applied Mathematics at this level and points of detail in earlier drafts of this book: Tuncay Aktosun, Jim Evans, Scott Hansen, Fritz Keinert, Howard Levine, Gary Lieberman, Hailiang Liu, Tasos Matzavinos, Hien Nguyen, John Schotland, Mike Smiley, Pablo Stinga and Jue Yan. All mistakes and shortcomings are of course my own doing.

Paul Sacks

October 28, 2016

Chapter 1

Some Basic Discussion of Differential and Integral Equations

Abstract

This chapter provides an introduction to the main types of problems which motivate the techniques developed throughout the textbook. Some general discussion of problems involving ordinary differential equations, partial differential equations, and integral equations is given, in order to establish notations, review needed background material, and explain some of the basic ways that such problems are classified. The chapter concludes with some discussion of the concepts of well-posed and ill-posed problems.

Keywords

Differential equation; Integral equation; Characteristics; Boundary conditions; Laplace equation; Heat equation; Wave equation; Well-posed problem

In this chapter we will discuss standard problems in the theory of ordinary differential equations (ODEs), integral equations, and partial differential equations (PDEs). The techniques developed in this book are all meant to have some relevance for one or more of these kinds of problems, so it seems best to start with some awareness of exactly what the problems are. In each case there are some relatively elementary methods, which the reader may well have seen before, or which rely only on simple calculus considerations, which we will review. At the same time we establish terminology and notations, and begin to get some sense of the ways in which problems are classified.

1.1 Ordinary Differential Equations

An nth order ordinary differential equation for an unknown function u = u(tis any equation of the form

   (1.1.1)

and also u(n) for derivative of order n. Unless otherwise stated, we will assume that the ODE can be solved for the highest derivative, that is, written in the form

   (1.1.2)

For the purpose of this discussion, a solution of either equation will mean a real valued function on (a, b) possessing continuous derivatives up through order n, and for which the equation is satisfied at every point of (a, b). While it is easy to write down ODEs in the form (1.1.1) without any solutions (e.g., (u′)² + u² + 1 = 0), we will see that ODEs of the type (1.1.2) essentially always have solutions, subject to some very minimal assumptions on f.

The ODE is linear if it can be written as

   (1.1.3)

, and homogeneous if also g(t) ≡ 0. It is common to use operator notation for derivatives, especially in the linear case. Set

   (1.1.4)

so that u′ = Du, u″ = D(Du) = D²u, etc., in which case Eq. (1.1.3) may be given as

   (1.1.5)

By standard calculus properties L is a linear operator, meaning that

   (1.1.6)

for any scalars c1, c2 and any n times differentiable functions u1, u2.

An ODE normally has infinitely many solutions—the collection of all solutions is called the general solution of the given ODE.

Example 1.1

By elementary calculus considerations, the simple ODE u′ = 0 has general solution u(t) = c, where c is an arbitrary constant. Likewise u′ = u has the general solution u(t) = cet and u″ = 2 has the general solution u(t) = t² + c1t + c2, where c1, c2 are arbitrary constants.

1.1.1 Initial Value Problems

The general solution of an nth order ODE typically contains exactly n arbitrary constants, whose values may be then chosen so that the solution satisfies n additional, or side, conditions. The most common kind of side conditions of interest for an ODE are initial conditions,

   (1.1.7)

where t0 is a given point in (a, bare given constants. Thus we are prescribing the value of the solution and its derivatives up through order n − 1 at the point t0. The problem of solving Eq. (1.1.2) together with the initial conditions (1.1.7) is called an initial value problem (IVP). It is a very important fact that under fairly unrestrictive hypotheses a unique solution exists. In stating conditions on f.

Theorem 1.1

Assume that

   (1.1.8)

. Then there exists ϵ > 0 such that the IVP (1.1.2), (1.1.7) has a unique solution on the interval (t0 − ϵ, t0 + ϵ).

A proof of this theorem may be found in standard ODE textbooks (see, e.g., [4] or [7]). A slightly weaker version of this theorem will be proved in Section 3.5. As will be discussed there, the condition of continuity of the partial derivatives of f with respect to each of the variables yi can actually be replaced by the weaker assumption that f is Lipschitz continuous with respect to each of these variables. If we assume only that f then it can be proved that at least one solution exists, but it may not be unique, see Exercise 1.3. Similar results are valid for systems of ODEs.

It should also be emphasized that the theorem asserts a local existence property, that is, only in some sufficiently small interval centered at t0. It has to be this way, first of all, since the assumptions on f . But even if the continuity properties of f , then as the following example shows, it would still only be possible to prove that a solution exists for points t close enough to t0.

Example 1.2

Consider the first order IVP

   (1.1.9)

for which the assumptions of Theorem 1.1 hold for any γ. It may be checked that the solution of this problem is

   (1.1.10)

, which can be arbitrarily small.

With more restrictions on f it may be possible to show that the solution exists on any interval containing t0, in which case we would say that the solution exists globally. This is the case, for example, for the linear ODE (1.1.3).

Whenever the conditions of Theorem 1.1 hold, the set of all possible solutions may be regarded as being parametrized by the n , so that as mentioned above, the general solution will contain exactly n arbitrary parameters. In the special case of the linear equation (1.1.3) it can be shown that the general solution may be given as

   (1.1.11)

where up is any particular solution of Eq. (are any n linearly independent solutions of the corresponding homogeneous equation Lu is also called a fundamental set for Lu = 0.

Example 1.3

If Lu = u″ + u is a fundamental set for Lu is the general solution of Lu = 0. For the inhomogeneous ODE u″ + u = et .

1.1.2 Boundary Value Problems

For an ODE of degree n ≥ 2 it may be of interest to impose side conditions at more than one point, typically the endpoints of the interval of interest. We will then refer to the side conditions as boundary conditions and the problem of solving the ODE subject to the given boundary conditions as a boundary value problem (BVP). Since the general solution still contains n parameters, we still expect to be able to impose a total of n side conditions. However we can see from simple examples that the situation with regard to existence and uniqueness in such BVPs is much less clear than for IVPs.

Example 1.4

Consider the BVP

  

(1.1.12)

, the two boundary conditions lead to u(0) = c2 = 0 and u(π) = c2 = 1. Since these are inconsistent, the BVP has no solution.

Example 1.5

For the BVP

  

(1.1.13)

for any constant C, that is, the BVP has infinitely many solutions.

The topic of BVPs will be studied in much more detail in Chapter 13.

1.1.3 Some Exactly Solvable Cases

Let us finally review explicit solution methods for some commonly occurring types of ODEs.

• For the first order linear ODE

   (1.1.14)

define the so-called integrating factor ρ(t) = eP(t) where P is any function satisfying P′ = p. Multiplying the equation through by ρ we then get the equivalent equation

   (1.1.15)

so if we pick Q such that Q′ = ρq, the general solution may be given as

   (1.1.16)

• For the linear homogeneous constant coefficient ODE

   (1.1.17)

if we look for solutions in the form u(t) = eλt then by direct substitution we find that u is a solution provided λ is a root of the corresponding characteristic polynomial

   (1.1.18)

We therefore obtain as many linearly independent solutions as there are distinct roots of P. If this number is less than n, until a total of n linearly independent solutions have been found. In the case of complex roots, equivalent expressions in terms of trigonometric functions are often used in place of complex exponentials.

• Finally, closely related to the previous case, is the so-called Cauchy-Euler type equation

   (1.1.19)

. In this case we look for solutions in the form u(t) = (t−t0)λ with λ to be found. Substituting into Eq. (1.1.19) we will find again an nth order polynomial whose roots determine the possible values of λ. The interested reader may refer to any standard undergraduate level ODE book for the additional considerations which arise in the case of complex or repeated roots.

1.2 Integral Equations

In this section we discuss the basic set-up for the study of linear integral equations. See, for example, [be an open set, K a given function on Ω ×Ω and set

   (1.2.20)

Here the function K is called the kernel of the integral operator T, which is linear since Eq. (1.1.6) obviously holds.

A class of associated integral equations is then

  

(1.2.21)

for some scalar λ and given function f in some appropriate class. If λ = 0 then Eq. (1.2.21) is said to be a first kind integral equation , otherwise it is second kind . Let us consider some simple examples which may be studied by elementary means.

Example 1.6

and K(x, y) ≡ 1. The corresponding first kind integral equation is therefore

   (1.2.22)

For simplicity here we will assume that f is a continuous function. The left-hand side is independent of x, thus a solution can exist only if f(x) is a constant function. When f is constant, on the other hand, infinitely many solutions will exist, since we just need to find any u with the given definite integral.

For the corresponding second kind equation,

   (1.2.23)

a solution, if one exists, must have the specific form u(x) = (f(x) + C)/λ for some constant C. Substituting into the equation then gives, after obvious algebra, that

   (1.2.24)

Thus, for any continuous function f and λ≠0, 1, there exists a unique solution of the integral equation, namely

   (1.2.25)

In the remaining case that λ = 1, it is immediate from Eq. (, in which case u(x) = f(x) + C is a solution for any choice of C.

This very simple example already exhibits features which turn out to be common to a much larger class of integral equations of this general type. These are

• The first kind integral equation will require much more restrictive conditions on f in order for a solution to exist.

• For most λ≠0 the second kind integral equation has a unique solution for any f.

• There may exist a few exceptional values of λ for which either existence or uniqueness fails in the corresponding second kind equation.

All of these points will be elaborated and made precise in Chapter 12.

Example 1.7

Let Ω = (0, 1) and

   (1.2.26)

corresponding to the kernel

   (1.2.27)

The corresponding integral equation may then be written as

   (1.2.28)

This is the prototype of an integral operator of so-called Volterra type, see Definition 1.1 below.

In the first kind case, λ = 0, we see that f(0) = 0 is a necessary condition for solvability, in which case the solution is u(x) = −f′(x), provided that f is differentiable in some suitable sense. For λ≠0 we note that differentiation of Eq. (1.2.28) with respect to x gives

   (1.2.29)

This is an ODE of the type (1.1.14), and so may be solved by the method given there. The result, after some obvious algebraic manipulation, is

  

(1.2.30)

Note, however, that by an integration by parts, this formula is seen to be equivalent to

  

(1.2.31)

Observe that Eq. (1.2.30) seems to require differentiability of f even though Eq. (1.2.31) does not, thus Eq. (1.2.31) would be the preferred solution formula. It may be verified directly by substitution that Eq. (1.2.31) is a valid solution of Eq. (1.2.28) for all λ≠0, assuming only that f is continuous on [0, 1].

Concerning the two simple integral equations just discussed, there are again some features which will turn out to be generally true.

• For the first kind equation, there are fewer restrictions on f needed for solvability in the Volterra case (1.2.28) than in the non-Volterra case (1.2.23).

• There are no exceptional values λ≠0 in the Volterra case, that is, a unique solution exists for every λ≠0 and every continuous f.

Finally let us mention some of the more important ways in which integral operators, or the corresponding integral equations, are classified:

Definition 1.1

The kernel K(x, y) is called

• symmetric

• Volterra type if N = 1 and K(x, y) = 0 for x > y or x < y

• convolution type if K(x, y) = F(x − y) for some function F

• Hilbert-Schmidt type

• singular if K(x, y) is unbounded on Ω ×Ω

Important examples of integral operators, some of which will receive much more attention later in the book, are the Fourier transform

   (1.2.32)

the Laplace transform

   (1.2.33)

the Hilbert transform

   (1.2.34)

and the Abel operator

   (1.2.35)

1.3 Partial Differential Equations

An mth order partial differential equation (PDE) for an unknown function u = u(xis any equation of the form

   (1.3.36)

Here we are using the so-called multiindex notation for partial derivatives which works as follows. A multiindex is vector of nonnegative integers

  

(1.3.37)

In terms of α we define

   (1.3.38)

the order of α, and

   (1.3.39)

the corresponding α derivative of u. For later use it is also convenient to define the factorial of a multiindex

   (1.3.40)

The PDE (1.3.36) is linear if it can be written as

   (1.3.41)

for some coefficient functions aα.

1.3.1 First Order PDEs and the Method of Characteristics

Let us start with the simplest possible example.

Example 1.8

When N = 2 and m = 1 consider

   (1.3.42)

By elementary calculus considerations it is clear that u is a solution if and only if u is independent of x1, that is,

   (1.3.43)

for some function f. This is then the general solution of the given PDE, which we note contains an arbitrary function f.

Example 1.9

Next consider, again for N = 2, m = 1, the PDE

   (1.3.44)

where a, b are fixed constants, at least one of which is not zero. The equation amounts precisely to the condition that u has directional derivative 0 in the direction θ = 〈a, b〉, so u is constant along any line parallel to θ. This in turn leads to the conclusion that u(x1, x2) = f(ax2 − bx1) for some arbitrary function f, which at least for the moment would seem to need to be differentiable.

The collection of lines parallel to θ, that is, lines ax2 − bx1 = C obviously play a special role in the previous example, they are the so-called characteristics, or characteristic curves associated to this particular PDE. The general concept of characteristic curve will now be described for the case of a first order linear PDE in two independent variables (with a temporary change of notation)

   (1.3.45)

Consider the associated ODE system

   (1.3.46)

and suppose we have some solution pair x = x(t), y = y(t) which we regard as a parametrically given curve in the (x, y) plane. Any such curve is then defined to be a characteristic curve for Eq. (1.3.45). The key observation now is that if u(x, y) is a differentiable solution of Eq. (1.3.45) then

  

(1.3.47)

so that u satisfies a certain first order ODE along any characteristic curve. For example, if c(x, y) ≡ 0 then, as in the previous example, any solution of the PDE is constant along any characteristic curve.

We now use this property to construct solutions of Eq. (be some curve, which we assume can be parametrized as

  

(1.3.48)

The Cauchy problem for Eq. (1.3.45) consists in finding a solution of Eq. (1.3.45) with values prescribed on Γ, that is,

   (1.3.49)

for some given function h. Assuming for the moment that such a solution u exists, let x(t, s), y(t, s) be the characteristic curve passing through (f(s), g(s)) ∈Γ when t = 0, that is,

   (1.3.50)

We must then have

  

(1.3.51)

This is a first order IVP in t, depending on s as a parameter, which is guaranteed to have a solution at least for |t| < ϵ for some ϵ > 0, provided that c is continuously differentiable. The three relations x = x(t, s), y = y(t, s), z = u(x(t, s), y(t, scontaining Γ. If we can eliminate the parameters s, t to obtain the surface in nonparametric form z = u(x, y) then u is the sought after solution of the Cauchy problem.

Example 1.10

Let Γ denote the x axis and let us solve

   (1.3.52)

with u = h on Γ. Introducing f(s) = s, g(s) = 0 as the parametrization of Γ, we must then solve

  

(1.3.53)

We then easily obtain

  

(1.3.54)

and eliminating t, s yields the solution formula

   (1.3.55)

The characteristics in this case are the curves x = set, y = t for fixed s, or x = sey in nonparametric form. Note here that the solution is defined throughout the x, y plane even though nothing in the preceding discussion guarantees that. Since h has not been otherwise prescribed we may also regard Eq. (1.3.55) as the general solution of Eq. (1.3.52), again containing one arbitrary function.

The attentive reader may already realize that this procedure cannot work in all cases, as is made clear by the following consideration: if c ≡ 0 and Γ is itself a characteristic curve, then the solution on Γ would have to simultaneously be equal to the given function h and to be constant, so that no solution can exist except possibly in the case that h is a constant function. From another, more general, point of view we must eliminate the parameters s,