Partial differential equations and systems not solvable with respect to the highest-order derivative

PARTIAL DIFFERENTIAL EQUATIONS AND SYSTEMS NOT SOLVABLE WITH RESPECT TO THE HIGHEST-ORDER DERIVATIVE GENNADI! V. DEMIDENKO Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences Novosibirsk, Russia STANISLAV V. USPENSKII Moscow State University of Nature Management Moscow, Russia

MARCEL

MARCEL DEKKER, INC. D E K K E R

NEW YORK • BASEL

Translated from the Russian by Tamara Rozhkvoskaya (Novosibirsk, Russia)

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Additional Volumes in Preparation

Dedicated to Sergei L. Sobolev

Contents

Preface

xi

Chapter 1. Preliminaries § 1. Spaces C,C\Lp,Hrp § 2. Averages § 3. Fourier Transform § 4. Multipliers § 5. Laplace Transform § 6. Integral Representation of Functions § 7. Weak Derivatives § 8. Sobolev Spaces § 9. Boundary-Value Problems for Ordinary Differential Equations on the Half-Axis § 10. Boundary-Value Problems for Systems of Differential Equations on the Half-Axis

1 1 5 7 13 18 25 29 33

Chapter 2. The Cauchy Problem for Equations not Solved Relative to the Higher-Order Derivative

38 43

47

§ 1. Problems Leading to Sobolev-Type Equations 48 § 2. Classes of Equations not Solved Relative to Higher-Order Derivative 50 § 3. Equations with Invertible Operator at the Higher-Order Derivative 54 § 4. Sobolev-Type Equations without Lower-Order Terms 66 § 5. Approximate Solutions to the Cauchy Problem for Equations without Lower-Order Terms 75 § 6. Estimates for Approximate Solutions 81 § 7. Existence and Uniqueness of a Solution to the Cauchy Problem for Equations without Lower-Order Terms 98 § 8. Equations with Variable Coefficients 108 § 9. Pseudohyperbolic Equations 119

via

Contents

§ 10. Applications of Sobolev-Type Equations to the Solution of a Hyperbolic System Chapter 3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems § § § § § § § § § § § § §

1. 2. 3. 4. 5.

Examples of non-Cauchy-Kovalevskaya Type Systems Classes of non-Cauchy-Kovalevskaya Type Systems .: The Cauchy Problem for Sobolev-Type Systems Approximate Solutions to Sobolev-Type Systems Estimates for Approximate Solutions to Sobolev-Type Systems 6. Solvability of the Cauchy Problem for Sobolev-Type Systems 7. Solvability of the Cauchy Problem for Pseudoparabolic Systems 8. Parabolic Systems 9. Approximate Solutions to Pseudoparabolic Systems 10. Estimates for Approximate Solutions to Pseudoparabolic Systems 11. Solvability of the Cauchy Problem for Pseudoparabolic Systems 12. The Cauchy Problem for Pseudoparabolic Systems with Lower-Order Terms 13. Pseudoparabolic Systems with Variable Coefficients

Chapter 4. Mixed Problems in the Quarter of the Space

136

151 152 155 159 169 172 181 193 202 207 209 219 229 233 237

§ 1. Statement of the Mixed Boundary-Value Problems for Simple Sobolev-Type Equations 237 § 2. Solvability of Mixed Problems for Simple Sobolev-Type Equations 240 § 3. Approximate Solutions to Simple Sobolev-Type Equations . . . .244 § 4. Properties of the Contour Integrals 248 § 5. Estimates for Approximate Solutions to Nonhomogeneous Simple Sobolev-Type Equations 253 § 6. Estimates for Approximate Solutions to Homogeneous Simple Sobolev-Type Equations 267 § 7. Solvability of Mixed Problems 281 § 8. Necessary Solvability Conditions for Mixed Problems 289 § 9. Mixed Boundary-Value Problems for Pseudoparabolic Equations 295 § 10. Sketch of Proof of Solvability of Mixed Problems for Pseudoparabolic Equations 299

Contents

ix

§11. Statements of the Mixed Boundary-Value Problems for Sobolev-Type Systems 306 § 12. Approximate Solutions to the Mixed Problems for Sobolev-Type Systems 321 § 13. Solvability of Mixed Problems for Sobolev-Type Systems . . . . 331 § 14. Mixed Boundary-Value Problems for Pseudoparabolic Systems 358 § 15. Approximate Solutions to Mixed Problems for Pseudoparabolic Systems 363 § 16. Convergence of Approximate Solutions 371 Chapter 5. Qualitative Properties of Solutions to Sobolev-Type Equations § 1. Sobolev-Wiener Spaces § 2. Mixed Problems for Sobolev-Type Equations in Cylindrical Domains § 3. Properties of Solutions to the First Boundary-Value Problem for the Sobolev Equation § 4. Algebraic Moments of Solutions to the First Boundary-Value Problem for the Sobolev Equation § 5. Asymptotic Behavior of Solutions to Some Problems in Hydrodynamics

381 381 387 394 401 412

Bibliographic Comments

421

Reference

423

Appendix S. L. Sobolev. On a New Problem in Mathematical Physics

437

Subject Index

489

This book deals with the theory of linear differential equations and systems that are not solved with respect to the higher-derivative. In operator notation, such equations and systems can be written as evolution equations

i-i AnD' — f, ftu + ^Ai-kD^u u / _J » /v i j ;

(0.1) \ I

where AQ, AI, .. . ,Ai are linear differential operators with respect to a; — ( x i , . . . ,xn). Such equations appear in various applications such as problems of hydrodynamics, atmosphere physics, and plasma physics. Classical examples of systems of the form (0.1) are the linearized NavierStokes equations vt — fAv + Vp = 0,

div v = 0

and the Sobolev system vt - [v,u] + Vp = 0, d i v w - 0 ,

w = (0,0,0;)'.

(0.2)

Example of equations of the form (0.1) are the Boussinesq equation

the internal wave equation Au t t + N 2 ( u X l X l + uX2X2) = 0, the Sobolev equation &utt+u2uX3X3 = 0 , and the Rossby wave equation Au t + /3uX2 = 0.

(0.3)

xii

Preface

Equations not solved relative to the higher-order derivative were first studied by H. Poincare [I] in 1885. Such systems arose from the study of special equations in hydrodynamics. The results of S. W. Oseen [1], F. K. G. Odqvist [1, 2], J. Leray [1, 2], J. Leray and J. Schauder [1], E. Hopf [1] devoted to the study of the Navier-Stokes equations, as well as S. L. Sobolev's works on small-amplitude oscillations of a rotating fluid in the 1940's (communications of his result in [7] were published in [46]) stimulated great interest in these systems. S. L. Sobolev [4-9] studied the Cauchy problem, the first and second boundary-value problems for the system (0.2) and equation (0.3). He also formulated some new problems in mathematical physics. This work was the first deep study of equations not solved with respect to the higher-order derivative. The system (0.2) is often referred to as the Sobolev system, and equation (0.3) is called the Sobolev equation. The work of S. L. Sobolev was continued by P. A. Aleksandryan, N. N. Vakhaniya, G. V. Virabyan, R. T. Denchev, V. I. Lebedev, T. I. Zelenyak, V. N. Maslennikova, S. G. Ovsepyan, and others. As is well known, after the publication of the works of S. L. Sobolev, I. G. Petrovsky emphasized the necessity of studying general differential equations and systems not solved with respect to the higher-order time-derivative (systems that are not Kovalevskaya-type systems) (cf. 0. A. Oleinik [1, p. 27]). Equations of the form (0.1) are often called Sobolev-type equations because the results due to S.L. Sobolev were a starting point of the systematic study of such equations. At present, there is a huge number of theoretical and applied works devoted to the study of equations and systems not solved relative to the higher-order derivative. Solutions of some problems for specific equations and systems can be found, for example, in the monographs of S. M. Belonosov and K. A. Chernous [1], S. A. Gabov and A. G. Sveshnikov [1, 2], N. D. Kopachevsky, S. G. Krein, and Ngo Zuy Can [1], and 0. A. Ladyzhenskaya [1]. These problems are studied in various directions: qualitative properties of solutions, spectral problems, various statements of boundaryvalue problems, and numerical investigations. Interest in this topic grows. For example, in the 1960's, as was noted by J. L. Lions and E. Magenes k

$

[1], for operators of the form ^ Aj-—, where Aj are unbounded operators j=o &V there were a few results of a special character. But now only abstracts of papers on this subject can form an entire volume! We cannot mention all authors working in this direction. Some works are indicated in bibliographical comments at the end of this book. Interest in equations not solved with respect to the higher-order derivative is caused by applications. During the last 30 years, some scheme for the general theory of differential equations and systems of the form (0.1) was created by efforts of many mathematicians. The further development of this theory is influenced by the development of the theory of boundaryvalue problems for elliptic, parabolic, and hyperbolic equations, as well as modern calculus.

Preface

xiii

The general theory of boundary-value problems for equations not solved relative to the higher-order derivative was constructed in works of M. I. Vishik, S. A. Galpern, A. A. Dezin, Yu. A. Dubinskii. A. G. Kostyuchenko, G. I. Eskin, J. E. Lagnese, T. W. Ting, R. E. Showalter, and others. Boundary-value problems are discussed in some chapters of the monographs of H. Gajewski, K. Groger, and K. Zacharias [11], R. W. Carroll and R. E. Showalter [1], and S. V. Uspenskii, G. V. Demidenko, and V. G. Perepelkin [1]The case where the operator AQ at the higher-order derivative is elliptic was mostly considered. It is natural that statements of problems for such equations are different from those for the classical equation. However, for some classes of equations it is possible to establish the solvability results similar to the corresponding results in the theory of parabolic and hyperbolic equations. For example, this fact holds for the Cauchy problem if the symbol of the operator AQ does not vanish anywhere (the nondegeneracy condition). If this condition fails, no analogs to classical results exist. This fact was first observed by S. A. Galpern (cf., for example, [1, 2]) in the construction of the L2-theory of the Cauchy problem. In particular, he established that for the solvability of the Cauchy problem in the Sobolev spaces W™ some additional conditions on the data of the problem are necessary. These conditions are similar to orthogonality. A similar situation for the mixed boundary-value problems in the quarter of the space was discovered by G. V. Demidenko [1,2]. In this book we deal mainly with classes of equations and systems (0.1) for which the symbols of the operators AQ do not satisfy the nondegeneracy condition. The main goal of the monograph is to study the Cauchy problem and general mixed problems in the quarter of the space for such classes of equations and systems and to study asymptotic properties as t —> oo of solutions to some boundary-value problems in hydrodynamics. In particular, we clarify the solvability conditions for the problems under consideration, obtain the L p -estimates for solutions, and prove the uniqueness theorems in the Sobolev weight spaces. The results obtained demonstrate the essential difference from the theory of boundary-value problems for parabolic and hyperbolic equations. We note that the equations and systems we study contain, in particular, the linearized Navier-Stokes equations, the Sobolev system and Sobolev equation, the system and equation of internal and gravity-gyroscopic wave equation in the Boussinesq approximation, the Barenblatt-Zheltov-Kochina equation, and the ion-sound wave. The book contains five chapters. The first chapter is auxiliary and presents some facts of mathematical analysis and the theory of differential equations. In particular, we touch the main properties of average operators, the Fourier and Laplace operators, and the operators of weak differentiation. We prove the integral representation of summable functions established in S. V. Uspenskii [1, 2]. We give some necessary facts about Sobolev spaces and multipliers. We hope that the material of Chapters 2-5 can be understood by postgraduates and students. In Chapter 2, we consider differential equations that are not solved with

xiv

Preface

respect to the higher-order derivative l-i L0(x; Dx}D[u + ^2 £ oo of solutions to boundary-value problems in cylindrical domains. This method was suggested by S. V. Uspenskii (cf. S. V. Uspenskii and E. N. Vasil'eva [5]) and is based on the proof of embedding theorems for Sobolev-Wiener spaces (cf. Section 1). In this chapter, we study asymptotic properties of algebraic moments of solutions to the first boundary-value problem for the Sobolev equation and the behavior as / —> oo of the solution to the Cauchy problems for one model equation occurring in the study of small oscillations of a rotating compressible fluid. The content of Chapters 2 and 3 follows G. V. Demidenko [4-8, 11, 12]. In Chapter 4 the results of G. V. Demidenko [2, 8-12] are used. Sections 11-13 of Chapter 4 contain results due to G. V. Demidenko and I. I. Matveeva [1,2]. A number of theorems in Chapters 2-4 are published for the first time. The main results of Chapter 5 are due to S. V. Uspenskii and his student. In this chapter, results of S. V. Uspenskii and E. N. Vasil'eva [1-5], S. V. Uspenskii and G. V. Demidenko [1] are used. The main idea of our method for studying the boundary-value problems in Chapters 2-4 is to construct a sequences of approximate solutions and obtain estimates in the corresponding norms. The method for constructing approximate solutions and obtaining the Lp-estimates for solutions to the problems under consideration was suggested by G. V. Demidenko [2, 4, 5]. The scheme of constructing approximate solutions to various problems is presented in Chapter 2, Section 5, Chapter 3, Sections 4 and 9, and Chapter 4, Sections 3, 10, 12, and 15 in detail. In order to construct approximate solutions, we suggest a special modification of the Fourier-Laplace method with the help of averaging operators. For such operators we use one introduced by S. V. Uspenskii [1, 2] (cf. Chapter 1, Section 6). We note that the averaging construction presented here was first used by S. V. Uspenskii for integral representation of solutions to quasielliptic equations in the whole space. Owing to such an approach, it is possible

xvi

Preface

to obtain a number of new results about properties of solutions to these equations (cf. S. V. Uspenskh [2], S. V. Uspenskii and B. N. Chistyakov [1, 2], P. S. Filatov [1,2], G. V. Shmyrev [I]).

We warmly remember the late V. G. Perepelkin, a talent scientist. Our first book was written together with him. We would like to note that Tamara Rozhkovskaya was an initiator of the publication of this book. We thank her for moral support and literature editing. We are grateful to Professor A. V. Kazhikhov for his useful remarks. We thank I. I. Matveeva, who was the first reader of our book, for her huge one-year work with our manuscript and for many discussions. We thank E. N. Vasil'eva for her active help in the preparation of Chapter 5. We thank the Russian Foundation for Fundamental Research for support for the Russian edition of this book (grant no. 98-01-14100, no. 01-0100609).

G. V. Demidenko S. V. Uspenskii Novosibirsk, Moscow 15 May, 1998

Chapter 1 Preliminaries In this chapter, we recall some facts of calculus and the theory of ordinary differential equations. In particular, we describe properties of averaging operators, Fourier operators, and integral Laplace operators. We also discuss the notion of a weak derivative introduced by S. L. Sobolev in the 1930's and some fundamental results concerning Sobolev spaces. We use the average method developed in works of V. A. Steklov and S. L. Sobolev. We also present an integral representation of summable functions constructed by S. V. Uspenskii [1, 2]. This representation is based on the use of special averaging functions. The chapter also contains some results from the theory of multipliers. We formulate theorems due to S. G. Mikhlin [1], L. Hormander [1], P. I. Lizorkin [1, 2] and give examples of multipliers. Some of these examples are new. At the end of the chapter, we treat the general boundary-value problem on the half-axis for ordinary differential equations and systems of equations and discuss the Lopatinskii condition of unconditional solvability [1].

§ 1.

Spaces C, C A , Lp, Hrp

Let G be a domain in the Euclidean space M n . We denote by G the closure and by dG the boundary of G. If the boundary is smooth, we denote by vx the outward unit normal vector at a point x = ( x ± , . . . , xn) 6 dG. If G and G' are domains in M n such that G' C G, then G' is called a subdomain of G. If, in addition, G' C G, then G' is referred to as an interior subdomain of G. If G' is a bounded interior subdomain of G, then G' is called a compact set relative to G. By the distance p(x,G) between a point x G M n and a set G C Mn we mean the lower bound of the distances between the point x and points y E G. By the distance p(Gi,Gi) between sets G\,Gi> C M n we mean inf p(x,Gi}. x£Gi

If fi = (/?!,... ,/? n ) is a multi-index with integer components, we set n

\a\ - V* /?• x? ~- TPl \P\ — 2^,Pl> 1

•'

nP> r*"9" ' D?' - — *< ~~ a ft-

'

3 D/ - D^1 D-3" x — Ux-i • • -ux -

u

n

2

1. Preliminaries

The closure of the set of points at which a function u(x) differs from zero is called the support of u and is denoted by supp u(x). We say that a function u(x) is compactly supported or has compact support in G if the support of u is a compact set relative to G. Function Classes The set of continuous functions in a domain G C Rn is denoted by C(G). The set of continuous functions in G possessing the derivatives of order up to / is denoted by Cl(G). It is obvious that the sets C(G) and Cl (G) are linear. We set

\\u(x),cl(G)\\=

sup IDX*) , / : > o .

We denote by C'(G) the set of functions u G Cl(G] whose derivatives D®u(x), \a\ 0 such that I u(x + y) — u(x)\pdx ^ e G

for \y\ <J 6e, where u(z) = u(z] for z £ G and u(z) — 0 for z £ G. Theorem 1.4. Let u ( x , y ) be a Lebesgue measurable function in a domain GI x G-2 C ffin x M m . Then the generalized Minkowski inequality holds: r

r

u ( x , y ) d y , Lp(Gi) ^ I \\u(x, y), L p ( G i ) \ \ dy,

1 ^ p < oo.

Theorem 1.5. Let p, r, q be real numbers such that I ^ p ^ q < oo, 1 + \/q - l/r+ l / p , and let u(x) e Lp(Rn), K(x] e Lr(Rn). Then the convolution (K * u)(x) satisfies the Young inequality \\(K * u ) ( x ) ,

K ( x ) , L r (M n )|| \u(x),

Theorem 1.6. Let x l ~ f ) u ( x ) e L p ( R f ) , (3 ^ l / p , 1 ^ p < oo. T/ien Hardy inequality holds: (y)dy,

u(y)dy,

1

-^^), Lp(Mf)||,

/ ? < l/p.

The proof of these theorems and relative information about the Lebesgue integral can be found, for example, in O. V. Besov, V. P. Il'in , and S. M. Nikol'skii [1], A. N. Kolmogorov and S. V. Fomin [1], S. M. Nikol'skii [1], S. L. Sobolev [3, 10]. A function u(x) is said to be locally summable in a domain G if it belongs to the space L\(G'} for any bounded interior subdomain G' C G. The set of all locally summable functions in G is denoted by L\OC(G). The convergence in L\OC(G] is defined as follows: a sequence {um(x)} £ L\QC(G] converges to a function u(x) in L\OC(G) if ||if m (x) — w(x), LI(G')|| —> 0 as 77i —> oo for any bounded interior subdomain G' C G. The spaces H^(G] were introduced and studied by S. M. Nikol'skii. We will consider a special case of such spaces below. Now, we introduce the norm Hrp(G)\\ = IK*), LP(G)\\ + £ sup h-r'\\At(h)u(x), LP(G)\\,

§ 2. Averages

5

where Ai(h)u(x) is defined in (1.1), r = ( T I , . . . ,r n ), 0 < r,- < 1, i = 1 , . . . , n, 1 ^ p < oo. The NikoVskii space H^(G} is the set of functions u(x) G LP(G) equipped with the above norm. In what follows, we identify equivalent functions, i.e., functions coinciding almost everywhere.

§ 2. Averages We introduce averaging operators in order to approximate summable functions by infinitely differentiable functions. Let K(x) be a compactly supported infinitely differentiable function in M n such that K ( x ) d x = l.

(2.1)

For an example we consider the function /*exp(l/(|z|2-l)) \0

A V( x I =
0 such that \\ueh(x) - u £ ( x ) , LP(G)\\ ^ e/2 for 0 < h < HI. Therefore,

§ 3. Fourier Transform

7

||u(o;) — u£h(x), Lp(G}\\ ^ £ for 0 < h < min{/io, hi}. Since e is arbitrary, we obtain the required assertion. D To conclude the section, we recall the following important result. Lemma 2.1 (DuBois-Reymond). Let u(x) G L\OC(G). Then I u(x}(f(x}dx

- 0

(2.5)

G

for any function y>(x] £ C*o°(^) if and only ifu(x) = 0 almost everywhere in G. PROOF. If u(x) = 0 almost everywhere in G, then the equality (2.5) is obvious. To prove the converse assertion, we consider an arbitrary compact set G7 C G. Let 0 < 2/i0 ^ p(dG,dG'}. For any fixed z G G' we introduce the function (x) = h~nK((z — x } / h ) , 0 < h (x) = 0 for \z - x\ ;> /i. Therefore, y?(:r) 6 Co°(G). By the assumptions of the lemma, = /Tn f u(x) J

Since ||«/j(2) - u(z}} Lp(G'}\\ -)• 0 as h -> 0 in view of Theorem 2.2, we have u(z] — 0 for almost all z 6 G' . Since the compact set G' was taken arbitrarily, the equality u(z] — 0 holds almost everywhere in G. D

§ 3. Fourier Transform In this section, we discuss some facts concerning the Fourier transform and recall the definition of the space 5(M n ) of test functions that rapidly decay at infinity. Definition 3.1. The space 5(M n ) is the set of all functions of class C^Rn) that, together with all the derivatives, decay faster than \x\~k for any k > 0 as \x\ —>• oo. The convergence in S'(IRn) is defined as follows: a sequence of functions {(pm(x}} e 5(M n ) converges to x^D^ip(x) for any a,/3 as ra —>• oo uniformly with respect to x 6 M n . Definition 3.2. The function n

f

u(x)dx,

^GMn,

(3.1)

is called the Fourier transform of u(x) G 5(M n ) and the operator associating with a function u(x] its Fourier transform w(£) is called the Fourier operator and is denoted by F, i.e., u(£) = F[tz](^).

8

1. Preliminaries

Theorem 3.1. Let u ( x ) e 5(M n ). Then u(f) E 5(M n ) and f/ze Fourier formula holds: (3.2)

PROOF. By the definition (3.1), the Fourier transform u(£) belongs to n ). To prove the Fourier formula, we consider the function

(X) = (V^rnf By the Lebesgue theorem, the limit u°(x) = lim ue(x) exists. We show that £—>-0

it°(x) = u ( x ) . Taking into account the formula

we write uex as follows:

Since

we have / [u(x — \/£j/J — u(x)J£

dy

\y\r It is obvious that for any 8 > 0 there exists r$ > 0 such that l-/!^)! ^ ^ for r ^ rg. Choosing r JJ> rg, we have J|(z) —> 0 as £ —>• 0 uniformly with respect to x E K n . Consequently, lim|u £ (x) — u ( x ) \ ^ (^. Since 0 is f-*0

arbitrary, w°(x) = u(a;), i.e., the Fourier formula (3.2) is proved.

D

Corollary 1. If u ( x ) , v(x) E 5(M n ), i/ien dx =

u(t)W)d£-

(3-3)

§ 3. Fourier Transform

9

In particular, the Parseval identity holds: \\ = \\u(^,L2(Wn)\\.

(3.4)

Corollary 2. If u ( x ) , v ( x ) G 5(K n ), then (tT*u)(f) = (2Tr}n/2v(£)u(£), where v * u denotes the convolution of the functions u(x) and v(x): (v * u)(x) =

v(x - y}u(y] dy.

Corollary 3. Ifu(x) £ S(Rn), then (£>£«)(£) = ( i £ f u ( £ ) and D%u(£) = £p«)(0. Definition 3.3. The function

w(x) = (V2^)-n

ei-*u;(0 df,

x GKn>

is called the inverse Fourier transform of w(£) G S'(Mn). The operator u>(£) —> w(x) is called the inverse Fourier operator and is denoted by F~l . By Theorem 3.1, we have F~l : S(Rn] -> 5(K n ). Moreover, F^oF = /. Similarly, F o F"1 = /, i.e., F"1 is the inverse operator. We expand the notion of the Fourier transform to functions in the spaces Li(M n ) and I^PM- We start with the case u(x) e Li(R n )- Then the function e*x^u(x}, £ € M n , belongs to the space Li(K n ). Consequently, for u(x) G Li(R n ) the Fourier transform is well denned by formula (3.1). The following theorem describes the images of the Fourier operator in Theorem 3.2 (Riemann — Lebesgue). Let u(x) G Z«i(M n ). Then the Fourier transform u(£) is a continuous function on M.n; moreover,

|u(OI < ( \ 2 ) - n | | w ( x ) , Li(R n )||, «(0->0, |€|->oo.

^ G Mn,

(3.5) (3.6)

PROOF. Since for almost all x G Mn the function e ix ^u(:c) is continuous with respect to £ G M n and has the majorant |w(ar)|, the continuity of the function w(^). follows from the Lebesgue theorem. From the definition (3.1) we immediately obtain the estimate (3-5). Let us prove (3.6). Since e l7r = —1, the Fourier transform u(£) for £ ^ 0

10

1. Preliminaries

can be written in the form

[replacement

yk = xk +

Using (3.1), we find 22(0 - (x/2^)-"

e-^(u(y) - u(y - ^|^|- 2 )) dy.

Consequently,

which implies (3.6), since, in view of Theorem 1.3, the integral tends to zero as |£| -> oo. D Let us discuss some obstacles that arise if we try to use the Fourier transform of functions in Li(R n ). Although, according to the Riemann — Lebesgue theorem, u(£) decreases at infinity, u(£) does not necessarily belong to the space Z/i(M n ). For example, in the one-dimensional case, the Fourier transform of the function 1 for - 1 < x < 1, 0 for |z| > 1

is the function u(£) = \l -- -— which does not belong to the space Li(M n ). V TT £ Furthermore, for an arbitrary function u(x) E Li(M n ) the Fourier formula is not valid. To restore a function u(x) from its Fourier transform u(£) by acting the inverse operator F~l on u(^), it is necessary to require some conditions on u(x). Therefore it is not convenient to use the Fourier transform in the space L\(Rn) if we mean to apply this theory to the study of partial differential equations. For this purposes, the most suitable tool is the Lz-theory of Fourier operators. We generalize the notion of the Fourier operator to functions in I/ 2 (K n )In this case, some difficulties appear since for u(x] E L^^n] the function e t x ^ u ( x ) , £ E Mr,, is not necessarily summable on R n . Therefore, it is necessary to "correct" the classical definition of the Fourier transform (3.1) for functions u(x) E L2(K n )DLi(]R n ). For this purpose, we use the Parseval

§ 3. Fourier Transform

11

identity (3.4) and the fact that the set of infinitely differentiable compactly supported functions is dense in the space L 2 (]R n ). Then the Fourier operator can be regarded as a linear continuous operator F : 1/2 (K n ) —>• L 2 (IR n ) with the everywhere dense domain D(F) = 5"(M n ). Since the space Z/ 2 (]R n ) is complete, in accordance with the extension theorem, we can uniquely extend the operator F to the entire space Z-2(M n ) with the same norm. The extended Fourier operator is also denoted by F and the Fourier transform of a function u(x) £ L 2 (R n ) is denoted by u(£). The inverse Fourier transform of functions in Z/ 2 (R n ) and the inverse Fourier operator F"1 : L? (M n ) —>• L2(M n ) are defined in a similar way. By definition, to find the Fourier transform u(£) of a function u(x) £ Li(S&n), one can take any approximate sequence { u k ( x } } £ 5(M n ), i-e., \\uk(x) - u(x), L 2 (M n )|| -» 0 as k -> oo. Then u(£) = F[u](t) is the limit of the sequence {«*(£)} m Theorem 3.3 (Plancherel). (a) Ifu(x),v(x] fu(x}^(x]dx= In

£ £ 2 (R n ), then

/"t^O^Odf-

(3-7)

Kn

/n particular, the Parseval identity holds: |Kx),L 2 (M n )|| = ||w(0,L 2 (Mn)||.

(3.8)

(b) For any functions u(x),v(x) £ L2(M n ) f/ie following equalities hold: F-^F^x) = u(x) andF[F-l[v]](£) = v(t). (c) For any function u(x) £ L2(M n ) ^Ae following convergence takes place in the L2(M n )-norm: (3.9)

/

PROOF, (a) We choose sequences {wj'(z)}, {i)fc(a;)} £ 5(M n ) such that \\u (x) - u(x), L 2 (M n )|| -> 0 as j -> oo, and \\vk(x) - v(x), L 2 (M n )|| ->• 0 as /; —> oo. By Corollary 1 to Theorem 3.1, for any k and j we have j

f uj(x}vk(x)dx=

t &(£)

Passing to the limit, we obtain (3.7) and (3.8). By the Parseval identity (3.8), the norm of the Fourier operator F : Z/2(M n ) —> ./^(ffin) is equal to 1. The same assertion is valid for the operator F-1. (b) Let { u k ( x ) } £ 5(M n ), and let \\uk(x)-u(x), L 2 (M n )|| -> 0 as k -> oo. By definition, ||F[ii*](f) - F[u](£}, L 2 (R n )|| -»• 0 as k -> oo. Since F"1 is continuous, we have \\F-l[F(uk]](x) - F-l(F(u]](x), L 2 (K n )|| -> 0

12

1. Preliminaries

as k —> oo. By Theorem 3.1, we have F~l[F[uk]](x) = u k ( x ) . Hence HK^X) — F~l[F[u]](x), Z/2(IRn)|| —>• 0 as k —>• oo. By the uniqueness of limit, the equality u(x) — ^~ 1 [F[-u]](,r) holds almost everywhere. The second equality in (b) is proved in a similar way. (c) Let u(x] G /^(Kn)- Consider the cut-off function

ur(x] =

u(x] 0

for x\ < r, for x > r.

It is obvious that ur (x) G L')(lRn) O L i ( M n ) ; moreover, L 2 (M n )|| 2

as r —> oo. Since the Fourier operator is continuous in the space Z/2(M n ) we have p r (£) - £(£), L 2 (M n )| -> 0 as r ->• oo. However, u r ( and, consequently,

dx, |r| 0 and u(x] defined in (3.10) we introduce the function r

(pr(x) = I v(x - y}ur(y}dy = (v * u r ) ( x ) .

§ 4. Multipliers

13

Since v ( x ) G Li(M n ) and ur(x} G L2(lStn) n Li(M n ), we have v? r (z) € 1,2 (Kn) n Z-i(M n ). We compute the Fourier transform r(x}. It is obvious that !?(£) = (y2~7f) n tz r (£)£(£)• Let us show that ||^(0 - ( V ^ r f i C O ^ O . M K r O I I ^ O ,

r-4oo.

(3.12)

Indeed, as was proved above, ||u r (£) — u(£), £20& n )|| —> 0 as r —)• oo. By the Riemann—Lebesgue theorem, we have \v(£)\ ^ ||f(a:), Li(M n )|| < oo. Therefore, using the explicit expression for the function