Volterra Equations and Inverse Problems 9783110943245, 9783110354881

Table of contents :
Chapter 1. Basic concepts of ill-posed problem theory
1.1. Classical well-posedness and Tikhonov’s well-posedness
1.2. Simplest properties of the well-posedness set and the continuity modulus. Connection with estimates of the interpolation type
1.3. Projection on the convex closed set. Quasisolution
1.4. The structure of correctness sets
1.5. Regularizators and regularising operator sets
1.6. Final remarks
Chapter 2. Linear Volterra operators and their properties
2.1. Abstract Volterra operators
2.2. Completely continuous Volterra operators in a Hilbert space
2.3. Operators Jα and their properties
2.4. Classical uniqueness and stability theorems
2.5. σ(Α) estimates and σ-continuity criteria
Chapter 3. Linear operator Volterra equations
3.1. Operator Volterra equations in the scales of Banach spaces
3.2. Operator Volterra equations of the first kind
3.3. Operator Volterra equation of the first kind with the nondifferentiable kernel
3.4. Examples of scales of Banach spaces
3.5. Examples of operator Volterra equations
Chapter 4. Nonlinear operator Volterra equations in the scales of Banach spaces
4.1. Formulations of the basic theorems
4.2. Definitions and auxiliary statements
4.3. Proofs of the basic theorems
4.4. Inverse kinematic problem of seismology
Chapter 5. Abstract integro-differential equations and inverse problems
5.1. Basic estimates
5.2. Uniqueness and stability of solution of integro-differential equations and inequalities
5.3. The problem of determining the right side of the evolutionary equation
5.4. Evolutionary equation of the second order
5.5. Operator Volterra equations with commuting kernels
Chapter 6. Multidimensional inverse problems
6.1. Inverse problems, commutators and a priori estimates
6.2. Theorems of decomposition
6.3. Examples of inverse problems that admit decomposition
Chapter 7. Multidimensional integro-differential equations of Volterra type
7.1. Statement of the problem
7.2. Necessary conditions
7.3. Sufficient conditions
Chapter 8. Inverse problems of wave propagation and scattering
8.1. The inverse problem of wave propagation in layered media
8.2. The problem of determining the right side of the Lameé equations
8.3. Statement of the inverse scattering on the barrier problems
8.4. Definitions and auxiliary facts
8.5. Uniqueness of the inverse scattering problem in the Kirchhoff approximation
8.6. Problems of coefficients determination
Bibliography

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INVERSE A N D ILL-POSED PROBLEMS SERIES

Volterra Equations and Inverse Problems

Also available in the Inverse and Ill-Posed Problems Series: Elements of the Theory of Inverse Problems AM. Denisov Small Parameter Method in Multidimensional Inverse Problems AS. Barashkov Regularization, Uniqueness and Existence of Volterra Equations of the First Kind A Asanov Methods for Solution of Nonlinear Operator Equations V.P. Tanana Inverse and Ill-Posed Sources Problems Yu.E. Anikonov, B.A. Bubnov and G.N. Erokhin Methods for Solving Operator Equations V.P. Tanana Nonclassical and Inverse Problems for Pseudoparabolic Equations A Asanov and E.R. Atamanov Formulas in Inverse and Ill-Posed Problems Yu.E. Anikonov Inverse Logarithmic Potential Problem V.G. Cherednichenko Multidimensional Inverse and Ill-Posed Problems for Differential Equations Yu.E Anikonov Ill-Posed Problems with A Priori Information V.V. Vasin and A.L Ageev Integral Geometry of Tensor Fields У.А. Sharafutdinov Inverse Problems for Maxwell's Equations V.G. Romanov and S.I. Kabanikhin

Related Journal: Journal of Inverse and Ill-Posed Problems Editor-in-Chief: M.M. Lavrent'ev

INVERSE A N D ILL-POSED PROBLEMS SERIES

Volterra Equations and Inverse Problems A.L Bughgeim

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Preface The basic problems of scientific researchers often involve problems of recovering the cause given the effect. In other words, these problems are inverse relative to the cause and effects. The first step in solving the inverse problem is, usually, the formulation of the laws connecting causes and effects. Since the basic laws of Nature are usually formulated using differential equations we obtain, as a result, the inverse problem for differential equations. In this case, the "causes" are expressed in terms of unknown coefficients that represent, the right side, initial conditions, or unknown domain of definition of the difierential equation. The "effects" are the functionals from solutions of the differential equation. Usually, they are the solution traces in certain manifolds or some solution averaging characteristics. The fact that the cause precedes the effect mathematically implies that many inverse problems reduce to operator Volterra equations. The other peculiarity of the inverse problems of mathematical physics is their ill-posedness in the natural (from the standpoint of appUcations) functional spaces, especially those inverse problems, the investigations of which is connected with operator Volterra equations. The first results relating to inverse problems for ordinary difí'erential equations of the second order were obtained by A. Ambartsumjan, G. Borg, A.N. Tikhonov, L.A. Chudov and N. Levinson. In some sense, the complete theory was derived by V.A Marchenko, I.M. Gelfand, B.M. Levitan and M.G. Krein (see Levitan, 1962; Marchenko, 1978; Romanov, 1973; and the bibliography cited there). Multidimensional analogues of these problems both in exact and difference statements were considered first by Berezanskii (1958, 1965). The most effective methods for solving one-dimensional inverse problems proved to be the transformation operator method and the factorization method. In both methods, Volterra operators are of the importance. Further, the factorization method was essentially developed by

ii

Α. L. Bughgeim.

Volterra equations and inverse

problems

Fadeev (1974) and Nizhnik (1973). As a result, this method was extended to certain multidimensional inverse problems. Systematic investigation of multidimensional inverse problems for hyperbolic equations then followed (see Lavrent'ev and Romanov, 1966; Lavrent'ev et ai, 1969, 1980; Romanov, 1971, 1972, 1973). The problems considered there turned out to be connected with multidimensional operator Volterra equations. At the International Congress of Mathematicians (Nice, 1970) M.M. Lavrent'ev had started the investigation of different classes of such equations. The greater part of this book deals with the theory of these equations and their applications to multidimensional inverse problems. In the first (introductory) chapter the basic concepts of the general illposed problem theory are stated. They were defined by A.N. Tikhonov, M.M. Larent'ev and V.K. Ivanov. In the second chapter, the elements of abstract operator Volterra theory are represented. The third and fourth chapters deal with linear and nonlinear operator equations in scales of Danach spaces and their applications to inverse problems. The basis of this equation theory lies in the result in the abstract Cauchy-Kovalevskii problem (see Ovsyannikov, 1965; Nirenberg, 1974) and the methods of operator Volterra theory in Hilbert spaces (see Gokhberg and Krein, 1967). In the fifth chapter, the linear operator Volterra equations and inverse problems connected with them are investigated by the method of the weight a priori estimates and by the method based on the spectral von Neumann theorem. Multidimensional inverse problems for differential equations with partial derivatives are solved by the method of weight a priori estimates in the sixth chapter. A priori estimates for Volterra operators connect these problems with the uniqueness theory of the Cauchy problem created by Carleman (1939), Calderón (1958), Hörmander (1965), and the others. The seventh chapter deals with multidimensional integro-differential Volterra equations and, in particular, with some problems of integral geometry. In the eight chapter the inverse problems of wave scattering and propagation are considered. The inverse problems investigated in this monograph include only a small amount of all the inverse problems studied nowadays. As for the other inverse and ill-posed problems and classes of operator and equations connected with them, they may be found in the bibliography. The author has not given a complete bibliography. He has given, for the most part, references to the literature which have been used and literature where the references on earlier works may be found. The author expresses his sincere gratitude to M. M. Lavrent'ev, who

Preface

iii

acquainted him with the inverse problem theory. The author also thanks V. G. Romanov, who has made useful comments. The author also is thankful to A. F. Gafarova and 0 . N. Chaschina for their help in preparing the text for publication.

Contents Chapter 1. Basic concepts of ill-posed problem theory 1.1. Classical well-posedness and Tikhonov's well-posedness 1.2. Simplest properties of the well-posedness set and the continuity modulus. Connection with estimates of the interpolation type . . . . 1.3. Projection on the convex closed set. Quasisolution 1.4. The structure of correctness sets 1.5. Regularizators and regularising operator sets 1.6. Final remarks

6 11 14 17 20

Chapter 2. Linear Volterra operators and their properties 2.1. Abstract Volterra operators 2.2. Completely continuous Volterra operators in a Hilbert space 2.3. Operators J'^ and their properties 2.4. Classical uniqueness and stability theorems 2.5. σ{Α) estimates and σ-continuity criteria

23 23 29 36 43 46

Chapter 3. Linear operator Volterra equations 3.1. Operator Volterra equations in the scales of Banach spaces 3.2. Operator Volterra equations of the first kind 3.3. Operator Volterra equation of the first kind with the nondifferentiable kernel 3.4. Examples of scales of Banach spaces 3.5. Examples of operator Volterra equations

55 55 63

Chapter 4. Nonlinear operator Volterra equations in the scales of Banach spaces 4.1. Formulations of the basic theorems 4.2. Definitions and auxiliary statements 4.3. Proofs of the basic theorems 4.4. Inverse kinematic problem of seismology

1 1

69 72 78

81 82 84 89 95

VI

Α. L. Bughgeim. Volterra equations and inverse problems

Chapter 5. Abstract integro-difFerential equations and inverse problems 5.1. Basic estimates 5.2. Uniqueness and stability of solution of integro-differential equations and inequalities

108

5.3. The problem of determining the right side of the evolutionary equation 5.4. Evolutionary equation of the second order 5.5. Operator Volterra equations with commuting kernels

112 116 118

Chapter 6. M u l t i d i m e n s i o n a l inverse problems

127

6.1. Inverse problems, commutators and a priori estimates 6.2. Theorems of decomposition

127 133

6.3. Examples of inverse problems that admit decomposition

138

Chapter 7. M u l t i d i m e n s i o n a l integro-difFerential equations of Volterra t y p e

143

99 99

7.1. Statement of the problem

144

7.2. Necessary conditions

147

7.3. Sufficient conditions

151

Chapter 8. Inverse problems of wave propagation and scattering 157 8.1. The inverse problem of wave propagation in layered media 157 8.2. The problem of determining the right side of the Lameé equations . . . . 164 8.3. Statement of the inverse scattering on the barrier problems 176 8.4. Definitions and auxifiary facts 179 8.5. Uniqueness of the inverse scattering problem in the Kirchhoff approximation 184 8.6. Problems of coefficients determination

188

Bibliography

193

Chapter 3asic Concepts of Ill-Posec ^^roblem Theory The object of this chapter is to set forth the definitions and results of illposed problem theory which will be used further. The papers of A.N. Tikhonov, M.M. Lavrent'ev and V.K. Ivanov have laid the groundwork for this theory. For those who wish to read more widely in this subject we recommend Bakushinskii (1968), Lavrent'ev (1962), Arsenin (1973), Morozov (1974), Tikhonov and Arsenin (1974), Ivanov et al, (1978), Lavrent'ev et ai, (1980).

1.1.

CLASSICAL WELL-POSEDNESS AND TIKHONOV'S WELL-POSEDNESS

Prior to considering ill-poser problems we shall recall the basic properties of well-posed problems using linear operator equations as example. Here we shall follow, for the most part, the monograph of Krein (1971). Let A be a linear operator acting from a Banach space X into a Banach space Y with a domain of D{A) Ç X. Let R{A) be a range of values of A, i. e. R{A) = AD{A); ker A = {i¿ e D{A) | Au = 0} be a kernel of A. As it is known, A is said to be closed if from - иЦ ->• 0, \\Aun — / | | ->· О, Un e D{A) follows, that и G D{A) and Au — f . Evidently, each bounded operator acting an all the space is closed, but a closed operator may be unbounded. By the closed graph theorem, a closed operator acting in all the space is bounded. Consider the problem Au^f

(1.1.1)

2

Α. L. Bughgeim.

Volterra equations and inverse

problems

where / is a given element from У, u is a desired element from D(A), A is a closed operator. Definition 1.1.1. Problem (1.1.1) is said to be well-posed if \\u\\ < c\\Au\\,

Vu G D{A)

(1.1.2)

where с does not depend on u, || • || is a norm in X or in У respectively. Evidently, the well-posedness of a problem essentially depends on the pair X, Y. A problem Au = f , where (1.1.2) fails is said to be ill-posed. The motivation for Definition 1.1.1 is the following T h e o r e m 1.1.1. A problem Au = f with a closed operator A is wellposed if and only if ker A — {0} and R{A) is closed. Thus, problem (1.1.1) well-posed in the sense of Definition 1.1.1 with closed A in the pair of Banach spaces X, R{A) (a norm in R{A) coincides with a norm in Y) satisfies all the conditions of the Hadamard wellposedness, i. e. the solution exist, is unique, and continuously depends on the initial data / . P r o o f of T h e o r e m 1.1.1. FVom (1.1.2) immediately follows the uniqueness of the (1.1.1) solution, therefore, ker A = {0}. Let us prove the completeness of Д(А). Let fn = Aun G R{A) and fn ^ f- Then (1.1.2) yields ||u„ - Um\\ < с||А(к„ - u)\\ = c||/„ - fm\\· Therefore, by the Cauchy criterion. Un converges to an element u. The completeness of A implies и G D{A) and / = Au e R{A). Thus, R{A) is closed. Conversely, assume that R{A) = R{A) and ker A = {0}. Then in the Banach space R{A) (with the norm from Y) the inverse operator is determined. Since it is closed (due to the completeness of A), then by the closed graph theorem it is bounded. It denotes, that WA'^fW < c||/||, V/ G R{A), or ||n|| < c||Au||, Уи G D{A). • To formulate the existence theorem in terms of Y we need the concept of a conjugate operator. Assume that D{A) is dense in X. Let g be an element from Y* conjugated with Y. Then the linear functional и -> {g, Au) is determined in D{A). Here (·,·) is the duality between Y and Y*. If this functional is bounded, we shall say that g belongs to the domain of conjugated operator A* : D{A*) X*. By the definition we assume that {A*g,u) = {g,A*u),

ueD{A).

Chapter

1. Basic

concepts of ill-posed problem

theory

3

Here from the left (·, •) is the duality between X and X*. Due to the density of in X, the element A*g is uniquely determined. Operator A* is said t o b e conjugated t o A.

Consider the structure of kervl*. Let g e кегЛ*, и Ε D{A). Then = О = {g, Au), i. е. g is orthogonal to R{A). Conversely, if g belongs to the orthogonal complement to R{A): {A*g,u)

g e R{A)^

= {g EY*

\ (gj)

= 0,f e

R{A)}

then {g,Au) = 0 = {A*g,u) for all и e D{A), and therefore g G D{A*), A*g = 0. So we have proved that ker A* is an orthogonal complement to RiA): keTA*=R{A)^.

Prom here follows Theorem 1.1.2. R{A)

Let D{A)

= Y, it is necessary

= X.

For Au = f to be dense solvable, i. e.

and sufficient that ker Л* = {0}.

Now let A be a closed operator with a dense domain of definition D(A). In this case the following theorem is valid. Theorem 1.1.3. A only iff

well-posed problem

Au = f has a solution

if and

1 k e r A*, i. e. {g, / ) = 0 for all g G k e r A*.

Proof. By Theorem 1.1.1 R{A) = R{A); therefore, it is sufficient to show that {кетА*)^

=

R{A).

Since ker A* 1 R{A), then (ker Л*)^ D R{A). Let us prove that R{A) D (кегЛ*)^. Really, let / G (ker A*)^ and / i R{A). Then by the HahnBanach theorem, due to the fact that R{A) is a closed subspace, there exists such a finear functional g EY* that {g, f) = 1, {g,v) = 0, υ E R{A). Hence {g,Au) = 0 = {A*g,u), и E D{A). Since D{A) =.X, then A*g = 0 and therefore, {g, /) = 0 which is a contradiction. • In the case when a right side / of a well-posed problem (1.1.1) is known with and error, i. e. instead of / the element Д is given: ΙΙ/-ΜΙ2 + h) - A'^V2\\ = sup \\A'\v2

+ h2 + hi) - A-\v2

+ /12) + A-\v2

+ /12) - A-^V2\\

< sup \\A-^{V2 + h2 + hi) - A-'^{v2 + /12)11 + sup \\A-^{V2 + /12) - A-^V2\\ = ωΜ·(ει) +ωΜ(ε2). We have used here the triangle inequality and the fact that h = hi + h2. The sets where we take sup are evident from the context. The continuity of ^ ^ ( ε ) in the interval [0,00) follows from the continuity in zero and estimates (ii) and (1.2.1), since for ε, Δε > 0 we have 0 < шм{е + Δε) - шм{£) < ^ м ( г ) + ωΜ(Δε) - u n i e ) = ωΜ(Δε) - ^ 0 ,

Δε

0.

Chapter 1. Basic concepts of ill-posed problem theory Estimates (1.2.2), (1.2.3) follow from (1.2.1).

7

•

Theorem 1.2.1 is analogous to the corresponding of the real functions theory (see, for example Dzjadyk, 1977). Theorem 1.2.2. Let M Ç X be a well-posedness set of the problem Au = f . Then the following statement hold (i) if A : X Y is a continuous operator, then the closure M of M is also a well-posedness set; (ii) if A is a homogeneous operator, i. e. AXu = λ°Άη for all λ > 0, и e D{A), then the set XM = {u | u = λυ, υ e M} is also a wellposedness set, where ωχΜ{ε) = (Hi) if A is a linear operator, then, for a convex balanced set M to be a well-posedness set, it is necessary and sufficient that such continuous in zero nondecreazing function ω(ε), ε > 0, ω(0) = О exists, where ||n|| 0.

Proof. Let Au — f be /-well-posed. Assume ω{ε)=

inf {α + φ{α)ε), ae(0,ao)

ε > 0.

(1.2.6)

Evidently, ω(0) = 0 and ω is a nondecreasing function continuous in zero. For и E Ml, from (1.2.5) it follows that ll^ll < a + (^(a)pîi||. Minimizing this inequality by a we obtain ||u|| < a;(||yl«||). By (iii) of Theorem 1.2.2, from here follows the Tikhonov well-posedness in Μχ. As Ms = sMi, then (see (ii) of Theorem 1.2.2) Mg is a well-posedness set for each s > 0. Conversely, let Au = / be well-posed by Tikhonov in Ml. Then, in particular, ||n|| < ^(ЦАиЦ), и e M, where ω = umi is the continuity modulus of problem Au = f , и e Μχ. By (iii) of Theorem 1.2.1, ω £ C[0, oo) and, besides, there exists such с that ω{ε) 0.

Chapter

1. Basic concepts

of ill-posed

problem

theory

This estimate follows from (1.2.3). Assume ,

,

(p(a) = sup e>0

ω(ε) -

a

,

(1.2.7)

α > 0.

ε

Evidently, for each α > 0 the function (p(a) takes a finite value, (p(a') > (p(a) for a' < a. Equation (1.2.7) yields ω(ε) < a-l· φ(α)ε,

a > 0,

ε > 0.

Let и e D(l), l(u) φ 0. Then ujl(u) G Mi and, therefore. и l{u)

J^-U) \ ( l{u) /Ы

Au

+

l{u)

Multiplying both sides of this inequality by l{u) we obtain (1.2.5). If l{u) = 0, then su e Ml for each s > 0, and we again obtain s||u||
c>0,

A; = 1,2,...

(1.4.1)

holds, where с does not depend on k. On the other hand, A is completely continuous; hence, Au^ strongly converges to Au and, in particular, since M is a well-posedness set, ll^nfc - ii||
m. If M bounded, then L = Ч); but the empty set satisfy the lemma conditions. For an unbounded M, the set F^ φ ^ for all n, and, therefore, F f | as the intersection of η

nonempty compact sets is nonempty. In this case it is easy to verify that L = {u\u = \v, υ G F, A e R} is the desired maximal subspace. •

16

Α. L. Bughgeim.

Volterra equations and inverse problems

Lemma 1.4.2. Let M e M. Then M = K + L = {t¿ | u = U1+U2, щ G К, U2 G L}, where К is a convex balanced compact set; L is a fìnitedimensional subspace. Proof. Let L be the maximum finite-dimensional subspace contained in M (constructed in Lemma L4.1 ), L-*- be its orthogonal component in X. Assume that К — M Then M = К + L. Compactness of К follows from maximality of L. • Proof of Theorem 1.4.3. Let m E M. Then by Lemma 1.4.2 M = К + L, К being a compact set, L being a finite-dimensional subspace. For и e M we have a representation u = k + l,

keK,leL

(1.4.2)

where, since AM = AK + AL == AKf]{AL)-^ -b AL {{AL)-^ is a subspace orthogonal to AL), then considering for simplicity the case A* — Л, AL Ç L we may assume that Ak 1 AL (1.4.3) As к e К, then by Theorem 1.1.4' \\k\\• 0 and, therefore, Aw = Au, which contradicts ker Л = {0}. Estimate (1.5.9) implies Ци^ — ulp = {ua — u,Ua) — {u,Ua — u) < —{u,ua — u) 0. The theorem is proved. • Remark 1.5.3. This proof is contained in Lattés and Lions (1967). A regularizing set may be built on the basis of approximation of the regularizator in (1.5.2) by linear bounded operators. Really, the following evident theorem holds true. Theorem 1.5.2. Let A and кег{Е + В) = {0}. Let II(Яа for a о regularizing set for (1.5.1) in

have the left regularizator R, RA = E + В operator Ra, a > О belong to C{Y,X), and and и e χ. Then R^ = {E + B)-^Ra is the the whole space X.

Α. L. Bughgeim.

20

Volterra equations and inverse

problems

Remark 1.5.4. The regularizing set described in Theorem 1.5.2 takes into account the specific character of initial problem Au = f and, therefore, in some cases, is better than the general method of constructing the set Ra by formula (1.5.7). In the case when a solution of (1.5.1) belongs to a domain of definition of a stabilizing functional I, the direct variational method of constructing the regularizing set (see Tikhonov, 1963) is natural. It is based on minimizing the functional l{u) under the condition \\Au - /|| < ε, where ε characterizes the explicitness with which the right side is known. Similar methods were described by Tikhonov and Arsenin (1974).

1.6.

FINAL REMARKS

We have considered the basic statements of the ill-posed problem theory in Sections 1.1-1.5. In this case, we assume a priori that the solution to the problem Au = f either is unique, or, moreover, the corresponding stabihty estimate holds. The other chapters deal with the proof of these facts, i. e. we solve the theorem of uniqueness and stability for diff'erent classes of multidimensional inverse problems of mathematical physics and operator equations. In other words we shall establish the well-posedness of these problems by Tikhonov. Further we shall often use the fact that uniqueness and stability for nonlinear problems reduce to the corresponding problems for linear equations. Such an approach is well-known in nonlinear equations theory as the "lemma of Hadamard". We shall formulate it (see Theorem 1.6.1) for the nonhnear operator equation A{u) = f

(1.6.1)

where A : D{A) ^ Y, D{A) Ç X is ш operator with a convex open domain of definition D{A); X, Y are Banach spaces. Let A have the Gateaux derivative for all и e D{A). In other words, for all и e D{A), φ&Χ

θ->0 exists and A'{u) : ψ -)• A'{и,ψ) G C{X,Y). A'{u,φ) — A'{u)ψ.

We shall use the notation

Assume, that for each ui, Κ2 G D{A) the function /(0) ξ ||A'(ti2-l-ö(ui —

Chapter 1. Basic concepts of ill-posed problem theory

21

U2))|| is integrable by θ in the interval (0,1). Then the formula B(ui,u2)= i \ ' i u 2 + e { u i - u 2 ) ) d e Jo

(1.6.2)

determines the linear bounded operator acting from X into Y, depending on parameters щ, U2 G D{A). In this case, the following theorem holds. Theorem 1.6.1. Let kerß(ui,u2) = {0} for all щ, U2 G D{A). Then (1.6.1) has a unique solution. Let M be a set contained in D{A). If for all щ, u^, v\, V2 G M the estimate ¡¡VI - V2¡¡ < ш(ЦВ(ии U2)(vi - V2)ll) holds, then ¡¡UI-U2¡¡0

p — oo

Ol, 02,..., On be a sequence of numbers such that αϊ > 02 > ... > a„. The operator A we define by the formula = {aiui,a2U2,...,anUn,...). Evidently, = αϊ -02 • . . . ·α„. Let limn_>.oo(ai • ... · = 0. Then specA = {0}. Assume that Pq = 0, PnU = (no,Ki,...,u„_i,0,0,...), η ^ 1 , 2 , . . . , Poo = E . Since (P„+i P n ) A { P n + i — P n ) = 0, then V — { P n } is the proper maximal chain of A . Therefore, A is a Volterra operator. Example 2.1.3. Let χ be a n-dimensional space of vectors u = { щ , . . . , Un), and suppose A has only eigenvalues, i.e., specA = {0}. Then by the Shura theorem, in corresponding orthonormal basis {e^}, it reduces to the triangular form A e j — a i j C i + 02^62 + . . . + a j j C j , j = 1, η

26

Α. L. Bughgeim.

Volterra equations and inverse

problems

where a j j = 0, since specA = {0}. Set Po = 0, PkU =

(ui,...,uk,0,...,0),

Then Ρ = {Pa:}) к = 0, η is a proper maximal chain of A and (Pj Fj-i)A(Fj - Pj-i) = 0. If we set APj = Pj - Pj^u 3 = V " , then, by η

the formulae APjA+APj

= 0, A+ = (A + A*)/2,

obtain

η

A = 2j^Pj^iA+APj.

^ APj = E, it is easy to (2.1.4')

Otherwise, in a finite-dimensional space, a Volterra operator A is restored be the operator A+ and the chain V. As a completely continuous operator may be approximated in the norm C{X) by ñnite-dimensional ones, we may expect that each completely continuous operator with the spectrum in zero will be a Volterra operator, and Λ

= 2 J PA+

dP

if we suitably define the integral by the chain V. This important statement will be proved in Section 2.2 If α G C{X) and specA = {0}, then for each complex λ (a Banach space χ is assumed to be complex) the operator {E — XA)~^ G C{X) exists, where 00

(Ε-λΑ)-^ =

(2.1.5) n=0

and the series (2.1.5) converges by the norm C{X), and determines the entire operator function. By means of the well-known formula for convergence radius г : — lim it is easy to show that specA = {0} if and . Tl->00 only if lim ||A"||V" = 0. (2.1.6) n—>oo An operator A G C{X) satisfying (2.1.6) is called a quasinilpotent operator. If specA = {0}, then by σ(Α) and by p(A) we shall denote respectively an order and a type of the entire function 00

/л(Л) = ^ | | А " | | А " . Ti=0

(2.1.7)

Chapter 2. Linear Volterra operators and their properties

27

By the well-known formula expressing an order and a type of an entire function by means of the Taylor coefficients, we have p{A) = - T i i i r - ^ ^ σ{Α) =

(2.1.8)

i — ер{А) In^oo

(

2

J

.

1

.

9

)

Thus, О < p(A) < oo, О < σ < 00 (for p(A) — oo we set σ(Α) — oc). In some cases another definition is more convenient. Set МА{Г)=

sup

||(£;-ЛЛ)-1||

|λ|=Γ

and define ρ and σ by the formulae T-^oo

(2.1.10)

Inr

— I n Мл hm г^оо rP Lemma 2.1.1. p{A) = p, σ{Α) - σ. Proof. As Мд(г) < Mf{r), where M/(r) == sup |/л(А)|, then ρ < p(A), |A|=r and if ρ = p(A). In particular, if ρ = oo or σ = oo, then p(A) — oo от σ(Α) = oo respectively. Now we obtain the inverse inequalities assuming that ρ < oo, σ < oo. By the Cauchy formula we have 2πι Д where 7 is an arbitrary smooth contour not containing the origin. Implementing the change of variable 1/λ = μ, we obtain ^ ¿

- μ Λ ) - ' dp

where 7' is the image of 7 under the inversion 1/λ = μ. For each ε > 0, by the ρ and σ definition, such Γ(ε) exists that \\{E - μΑ)-^\\ < e^^+'^P,

\μ\ = r > r{e).

(2.1.12)

28

Α. L. Bughgeim. Volterra equations and inverse problems

Hence, choosing 7' as the circle |/i| = r, from (2.1.12), we obtain < r - 1 exp((a + e)rP).

(2.1.13)

The right side of (2.1.13) is minimal for

\{σ + ε)ρ) therefore. |μη|| < e"/''((a +

(2.1.14)

This inequaUty implies that p{A) < p. Since ρ < p(A), we have p{Ä) = ρ, σ{Α) < σ +ε. Because ε is arbitrary and σ < σ{Α), we have σ{Α) = σ. • Formulae (2.1.8)-(2.1.11) and Lemma 2.1.1 immediately yield L e m m a 2.1.2. (i) For every natural к > I p(A')=p(A)/k,

σ{Α')=σ{Α).

(ii) If F is a linear isomorphizm of a Banach space X onto a Banach space X', then p{FAF-^) = p{A), o{FAF-^) = σ{Α). (Hi) For each complex a p{aA) = p{A),

σ{αΑ) = |α|''σ(Α).

In conclusion we shall give a following definition. Definition 2,1.2. A Volterra operator A is called σ-continuous relative to a proper maximal chain V = {P} if, for each sequence P„ weakly converging to Ρ and such that P„ < P„+i < P, p{{P - Pn)A{P - P„)) = p(A), we have lim σ((Ρ - Рп)А{Р - P„)) = 0. n->oo A Volterra operator is called σ-continuous if it is σ-continuous relative to each proper maximal chain. Sufficient conditions of the σ-continuity of a Volterra operator A and mates for the numbers p{A) and σ{Α) will be obtained in Section 2.5.

Chapter 2. Linear Volterra operators and their properties

29

Remark 2.1.1. The basic concepts of this selection which relate to projection chains for the Hilbert space case are contained in Gokhberg and Krein (1967). There, a Volterra operator is defined as each completely continuous operator with a spectrum in zero. Such definition does not cover the operators from Example 2.1.1 which are often met in practice. In Section 2.2 we show that this definition is a particular case of Definition 2.1.1. The concept of σ-continuity was introduced by Bukhgeim (1980c).

2.2.

COMPLETELY C O N T I N U O U S VOLTERRA OPERATORS IN A HILBERT SPACE

In this section we shall show that the properties of finite-dimensional Volterra operators considered in Example 2.1.3 may be propagated on the class of completely continuous operators. Let χ be a separable Hilbert space. In this case each operator A e С{Х) may be represented in the form A — A+ -b = (A + A*)/2, A- - {A- A*)/2i. Self-conjugated operators A^ and A^ are called the real and imaginary components of the operator A respectively. An operator A is called dissipative if Л+ > 0, i.e., if {A^u,u) > 0, Vu G Χ; (·, •) is a inner product in X. In a Hilbert space we shall impose on the chain V the additional self-conjugate condition: P* = Р,У Ρ e V. Thus, •ρ is a chain of orthogonal projectors (orthoprojectors) P. Hence follows its strong closedness, because, if ((P„ - P)u,u) 0, then ||(P„ - P)u\\ ->• 0. Evidently, a maximal chain V contains operators 0 and E, is closed, has no more than a countable number of breaks (due to separability of X) and each of its breaks { P ~ i s one-dimensional: dim(P''" — P~)X = 1. Here and the further we shall denote by dimL the dimension of a subspace L С X. Denote by σοο the class of completely continuous operators acting in X. To prove that completely continuous operators have proper maximal chains of orthoprojectors V = {P}, we need the following theorem: Theorem 2.2.1. Let A e σοο- Then A has a nontrivial (i.e., different from {0} and X) invariant subspace. Proof. For A Ξ 0, the theorem is trivial. Let A ^ 0 and be the set of operators В G C{X) such that AB = BA. We shall prove that there exists a nontrivial subspace invariant relative to all operators В E CA and, in particular, relative to A. If A has a nontrivial eigenvalue λ then, evidently, the proper subspace Lx corresponding to λ will be desired. Let specj4 = {0}. Assume that we have no nontrivial invariant for all В E CA

30

Α. L. Bughgeim. Volterra equations and inverse problems

subspace. Since A ^ 0, there exists such an element щ E X that for all ues = {uex \ | | u - u o | | < 1} ||Aw||>i>0.

(2.2.1)

For arbitrary и £ U, и ^ 0 consider the subspace the closure of the linear manifold L'^ = {v E X \ ν = Bu, В € CA]· Since CA is an algebra, Lu is invariant subspace for ail operators from CA- Here и E Lu because CA contains the unit operator. Hence Lu φ {0}. According to our assumption, there exists no nontrivial, invariant for all В E CA, subspace; therefore. Lu = X for each и φ 0. In particular, L'^f]S φ Therefore, for each и φ 0 such В E CA exists that Bu E S. Hence, due to (2.2.1), we obtain the open covering of the compact set К = AS KcX/{0}=

( J B-^s. B€Ca

Choose from this covering the finite one m KÇ\JB-^S. i=l

Since AUQ E к, such number ц, 1 < ii < m exists that Aq E B~ S. In other words, the element Щ = Вг^Ащ E S. Analogously, as Ащ E К then for some ¿2) 1 < ¿2 < "т·, = ВГ^АЩ = ВГ^АВЦАЩ E S. As a result we obtain the sequence n„ = ...ВГ^^АЩ E S. Then, according to (2.2.1), ||Au„|| > σ > 0. On the other hand,

due to quasinilpotentness of A (see (2.1.6)). This contradiction proves the theorem • Remark 2.2.1. Really, we have proved the existence of the general nontrivial invariant subspace of the entire operator algebra, which commutate with A. This statement was obtained by Lomonosov (1973). The mentioned variant of the proof was taken from Kantorovich and Akilov (1977), and belongs to Hil'den. For a Hilbert space. Theorem 2.2.1 was proved by Janos von Neumann in 1935. Theorem 2.2.1 yields

Chapter 2. Linear Volterra operators and their properties

31

Corollary 2.2.1. If subspaces Я and F of a Hilbert space X are invariant relative to operator A e σ^ο, A ^ 0, where V Ç Η and d i m { Я θ F ) > 1, then such invariant subspace W exists that V ÇW ÇH,

V ^W φ H.

(2.2.2)

Proof. Denote by AQ the contraction of A on H. As the subspace V is invariant relative to AQ, then its orthogonal supplement in Я : VI= ΗQV is invariant relative to A^. By the Theorem 2.2.1, the subspace V2 Ç Vi (У2 Φ VI, V2 Φ 0) invariant relative to AQ exists. The subspace W — HQV2 satisfies condition (2.2.2). • Theorem 2.2.2. Let A e σοο, specA = {0}. operator.

Then A is a Volterra

Proof. For A = 0 the statement is trivial. Let A φ 0. We shall show that A has the maximal proper chain of orthoprojectors V = {P}, where for each break (P+ _ p-)A{P+

- P-) = 0.

Let F be a set of all proper chains of the orthoprojectors of A containing 0 and E. Evidently, this set is partially ordered by the inclusion, where each linear ordered subset of F has an upper bound. Therefore, by the Zorn lemma in F there exists the maximal element V. Because it is maximal, V is a closed chain, 0, E Ç.V. Corollary 2.2.1 and the maximality condition yield that all the breaks of V are one-dimensional. Thus, V is the desired proper maximal chain of A. Let (P"",P+) be a break of V. Let us show that (P+ - P-)A{P+ - P-) ^ 0. Indeed, let AQ be a contraction of A on R{P'^) : AQ = P'^AP~. Subspace R{P~) is invariant relative to AJ; therefore, one-dimensional subspace L = R{P'^)QR{P~) is invariant relative to AQ. Since specA = {0}, then specAo = {0} and AQL = 0. Otherwise, L Ç ker A^; therefore, R{Aq) ± L and so, (P+ - P')A{P+ - P') = 0. • To obtain the analog of formulae (2.1.4), (2.1.4') we introduce the corresponding concept of a limit. Let V be an arbitrary closed chain of orthoprojectors. The chain Vn — {Pk}, Pk ^ V, к = 0, η is said to be a partition of Ρ if Po < P i . . . < Pn, Po = minP, Pn = maxP. On the set of all pev Pev partitions of V, let a function S{Vn) be given with the values in a certain Banach space Y. An element A G У is called the limit of the function S

32

Α. L. Bughgeim.

Volterra equations and inverse problems

( И т 5 ( 7 ' „ ) = A), if for each ε > 0 there exists a partition 'Ρ]ν(ε) of V such that for all partitions Vn 5 the inequality ||A - 5(·ρ„)|| < ε holds, (II • II) is a norm in У ) E x a m p l e 2.2.1.

Ρ he ά proper maximal chain of

Let Y = £(X),

A e σοο· Set η

sCPn) - ΣPj-i^+^Pj^

^Pj

= Pj -

Pj-i·

If the limit of S{Vn) exists, then, by the definition, assume YimS{Vn) = J

(2.2.3)

PA+dP.

The operator standing in the right side of (2.2.3) is called an integral by the chain V. Analogously, the limit of the function η

is, by the definition, Jj, dPA-^.P. The existence of these limits for a continuous chain is proved in the following theorem. T h e o r e m 2.2.3. Let A G σ^ο, V be a proper maximal continuous chain of orthoprojectors

of A. Then A = 2J

Proof.

PA+ dP,

Let Vn = {Pk},

^ = 2/

к = 0,η he Ά partition of V.

V, then Po = О, Pn = E,

^Pj

= E.

(2.2.4)

dPA+P.

Since 0, E e

Further, {E - Pj)AAPj

{E ~ Pj)APÁE - Pj-i) - 0~Pj^,A*APj = {APjAPj.^)* P j - i ) A P j - i ) * = 0. These identities hold true because {E-P)AP

VP e Я

= 0,

Taking these equalities into account we have η

A =

η

AAPj j=l

= ^ AP.AAPj j=l

η

+

j=l

Pj)AAPj

=

{Pj{E

= -

Chapter 2. Linear Volterra operators

= 2

+

and their

properties

33

APjAAPj.

Therefore, be the definition of the integral by the chain it suffices to show that \ìm^APjAAPj

= 0.

(2.2.5)

Operator A = A+ + iA- is a linear combination of self-conjugated operators; therefore, (2.2.5) is sufficient to prove for a self-conjugated operator A G σοο- Let A^ be an A^-dimensional self-conjugated operator such that IIA — АдгЦ < ε. Since the vectors APju, и £ X are mutually orthogonal and i A P j f = APj, ||ΔΡ,·|| 0, operator (J'^nXt)

=

operators

and

their

properties

37

naturally may be defined by the formula Щ

l \ t

-

т Г - \ { т ) dr

(2.3.2)

where Г (a) is the gamma-function. By the known formula

/

m

m

Γ(α+/3)

the operators J°' forms the semigroup =

α,β>0.

(2.3.4)

Operator J°' is called the integrating operator of the order a . To find the inverse to we shall generalize formula (2.3.4) for the case of negative a. Note, that J ^ u may be written as the convolution J ° ' u = k a * u , where ka{t) = t%-^/r{a), = t·^-^ for t > 0, = 0 for Í < 0. We shall consider the function as the generalized function acting on the basic function φ E S as follows roo Jo

í"-V(í)dí.

(2.3.5)

Here S = 0. By the identity roo

roo

m—о

(,3.6) the right side of which has a meaning for all α e С (С is the complex plane) such that R e a > - η , α φ О , - 1 , - 2 , . . . , - η , "integral" ,ψ) is defined for α > 0. Thus, for α e С, α 0, —1, —2,..., the expression is the right side of identity (2.3.6) for η so large that R e a > -п. Prom this definition it follows that is an analytic function for a φ 0, —1, - 2 , . . . . For α == —m it has a pole of the first order with

38

Α. L. Bughgeim. Volterra equations and inverse problems

the residue (/J™(0)/m! = (-1)'"(ό("'\φ)/τη!, where ó'' is the fc-derivative of Dirac function S(t). Since Γ(α) = and, therefore, if it has poles in the same points a = 0 , - 1 , - 2 , . . . , then the function ξ ,φ ) / , will be an entire function of a. In particular, for a = —n it is equal to the relation of corresponding residues of and i.e. \{ka,φ)\a=-n = Thus we have proved that k-n = J". Let V — r>'(R') be a space of generalized functions, dual to the space of infinite differentiable functions with a compact support. Let be the space of generalized functions with a support on the half-axis ί > 0, i.e., V'^ — {u E D' \ suppu Ç [0,oo]}, suppn be the support of u. For и G

α G С we assume

J'^u = k a * u . Theorem 2,3.1. (i) For each

(2.3.7)

element J'^u G

jßja ^ (ii) For each / G such unique и Ε k-a * / •

and besides (2.3.8)

exists that J°'u = f . Moreover,

(Hi) For each function и G nV'^the value J'^u G In particular, if = f , a > 0, then the classical inversion formula V-^dr, u{t) = [ [ t ^ i (n — α) Jo

f>0

(2.3.9)

holds, where η = [a + 1] ( [λ] is the integral part of λ). For a fìxed a this formula is extended by the continuity on functions f of the class С"[0,Г], /*=(0) = 0, = Proof, (i) Since the supports ka and и are bounded from one and the same side, the convolution ka* и exists, where supp k^* и Ç supp ka + suppu Ç [0, oo). Thus, operator (2.3.7) J^' : D'^ is defined correctly. Formula (2.3.8) follows from the associativity of the convolution and the fact that ka*kß — ka+ß- For α,β > 0 this formula follows from (2.3.3). For the order a, /3 G С it follows from the uniqueness of analytic continuation. Statement (ii) follows from (2.3.8) for β = —a and the fact j'^u = ko*u = S*u = u. Formula (2.3.9) is a more detailed notation of the

Chapter 2. Linear Volterra operators and their properties formula и = J-^^f = J " - " J - " / = jn-aj{n)^ gj^^^g j-nj: * / = /("). The infinite differentiability of и e evident. •

^

39 */ = Πΐ^ί^ is

Remark 2.3.1. Formula (2.3.9) for a = 1/2 was obtained by Abel in 1826. By Theorem 2.3.1 is the differentiation operator of order a . Further we shall denote it by Z?", D = d/ di. The majority of operator properties are expressed by corresponding a priori estimates. To reduce them, first, recall two estimates for integral operators of the form {Ku)it) ^ ΓK{t,T)u{T)dT, Jo Lemma 2.3.1. Let Κ{ί,τ) τ >0 such that

ueLp{0,T).

be a non negative function of variables t,

K{Xt, AT) = X-^K{t, T), roo di = / Jo

fOO / X(í, Jo

(2.3.10)

Í, r > 0, λ > 0

(2.3.11)

dT = k

(2.3.12)

where ρ > 1, ì/p + 1/ρ' = 1, || · ||ρ is the norm in Lp(0, T). Then \\Ku\\p < k\\u\\p

(2.3.13)

where, if in (2.3.13) is the equality sign, then и ξ 0 Proof. The lemma is proven by the Holder inequality | | ω υ | | ι < | | ω | | ρ | | υ | | ρ ' , where equality is achieved only for u^ = cv^ , с — constant. So, we have rT fT < /

fT pT di /

пТ K{t,T){TltflP'\u{T)\PàTU

¡•τ

ρ/„/ Κ{ί,τ){ίΙτγΙ^άτγ

ρΤ

dt

Κ{ί,τ){τ/ίγ^Ρ'\η{τ)\Ράτ. (2.3.14) Jo Jo To obtain the latter inequality, we have used the fact that by (2.3.11), (2.3.12) fjy I K{t,T){t/T^/Pdt^ [ K{l,T/t){t/ry^n-UT Jo Jo

40

Α. L. Bughgeim. Volterra equations and inverse problems rl/t cT/t = / K{l,x)x-^/Pdx< Jo

roo / K{l,x)x-^/Pdx Jo

=k

holds. Changing in (2.3.14) the order of integrating and using pT / K{t,T){T/t)'/P' Jo

roo dt < / Jo

dx = k

we obtain If = /ьЦ^^Цр, then in the Holder inequality the equality sign is achieved and, hence, Κ{τΙίγΙΡ'νΡ{τ)

=

cKit/τγ^Ρ

or uP{t) = ctjT. Since и G Lp(0,T), from here it follows that с = 0, u Ξ 0.

• Lemma 2.3.2. Let K{t, r) = к{1 - r) for t > τ, K{t, τ) = 0 for t < т. If к e Li{0,T), then ll^nllp < ||fc||i||u||p,

l 1, V3('=)(í) О, α,β e β < a the estimates

Γ(α-β+


т.

Hence, \w{t)\ < [\{t

-

Jo

k{t) =

τ)\υ{τ)\άτ

exp(-sai'"/m!)/r(a - β).

By Lemma 2.3.2 ||u;||p < ||A;||i||î;||p. As ili.ll /,,¿.11

llalli < II^IILICO,«.),

Hi·!!

fas^ß-o:)/rnr{{a-ß)/m)

= ( ^ j

ж Г ( а - β)

(see Gfadshtein and Ryzhik, 1971, p. 356), we obtain estimate (2.3.19). For α = 0 we have ||A;||i = (2.3.20).

•

1). Due to k{t) < k{t)

a=0

, we obtain

Remark 2.3.3. The extending of J"" to the generalized functions and the proof of Theorem 2.3.1 we have taken from Gel'fand and Shilov (1958a). This theorem will be essentially used when investigating multidimensional inverse problems in Chapters 5 and 6.

Chapter 2. Linear Volterra operators and their properties

43

Remark 2.3.4. Theorems 2.3.1-2.3.3 remain valid for functions with values in Banach space X. Otherwise, we may substitute the space Lp(0, T) and V'iR^) by Lp(0,Γ; Ji), Т>'(В^;Х) respectively. We shall use this remark later. Now we shall consider the problem of describing all invariant subspace of operator J' in the space ¿2(0, Г). Evidently, each subspace Hg = {u G 12(0, Г) I u(t) Ξ 0, for Í e [0, s], 0 < s < Τ} will be invariant. The inverse statement is nontrivial. Theorem 2.3.4. Let L be an invariant subspace of operator J in ¿2(0, T). Then L = Hs for some s e [0,Т]. Proof. Set in Theorem 2.2.4 X = 12(0,T), A = T'^J.

Then

rT

{A+u)it)=T-'^

f

Jo

u{t)dt^

{u,e)i

e{t) = Ti/2. The system of vectors = i"/n! is complete in ¿2(0,Т), ||e|| = 1. Therefore, J is a single-block operator. Following the results of Section 2.2 it denotes that J has no invariant subspace which differs from Hs = PsX, where {Psu){t) = 9{t — s)u{t), {P^} is a proper maximal chain of J. • Remark 2.3.5. Theorem 2.3.4 was almost simultaneously established by Brodskii (1957), and by Sakhonovich (1959) (see, also, Kalish (1962)). The proof based on the abstract theory of Volterra operators was suggested by Brodskii (1965).

2.4.

CLASSICAL U N I Q U E N E S S A N D STABILITY REMS

THEO-

In this section we shall deduce from Theorems 2.3.2-2.3.4 the theorems of solution uniqueness for classical integral Volterra equations of the first, second and the third kinds. We shall begin with the Titchmarsh theorem (Titchmarsh, 1951). Theorem 2.4.1. Let a, и e i i ( 0 , T ) . If for each t G [0,Г; {Au){t)^

f a{t Jo

T)U{T)

DR

=0

44

Α. L. Bughgeim. Volterra equations and inverse problems

holds, then such s € [Ο,Γ] exists that u{t) = 0 for t e [0, s], a{t) = 0 for te [ o , T - s ] . Proof. Fist, let a, и e ¿2(0,Τ). As operators J and A commutate, then J'^Au = aj^u = 0. Prom here, setting t = T, we obtain

I

τ a ( T - T ) ( J " u ) ( r ) d T = 0,

η = 0,1,2,...

(2.4.1)

The closure of the linear hull of the functions {J"'u} is the invariant subspace of J and, consequently, by the Theorem 2.3.4, it coincides with Hs for a certain 5 e [Ο,Γ]. As i¿ e Hg, then u{t) = О, t G [0,s]. Due to (2.4.1), α ( Τ - τ ) ± Hs, i.e. α ( Τ - τ) = О, τ G [s,T]. In other words, a{t) = 0, te [Ο,Τ - s]. So, in the case a, и e ¿2(0, Τ ) , the theorem is proven. For arbitrary a, и e Li (0, Г ) we set αχ = Ja, щ — Ju. As Au = О, then j'^Au

= JAui

=

/ ai{t-

Jo

T)UI(r)

d r = 0.

We have proved the theorem for αχ, щ and, hence, for a = a[, и = u[.

•

Prom Theorem 2.4.1 immediately follows Corollary 2.4.1. Let a{t) 0 in a certain neighborhood of zero of the interval [Ο,Τ]. Then solution of the equation Au = / is unique. Let, now {Au){t)=

f\{t,T)u{T)dT,

ueLp{0,T)

(2.4.2)

where A{t, r ) is a given function of the variables (i, τ ) G Δ = { i , r | 0 < r < t 0 the estimate \Α{ί,τ)\ < m(t (I, T) 6 Δ hold. Then the solution to the Volterra equation of the second kind и + Au = f is unique and ||u||p < c||/||p, where с = c(a,m,T). Proof. Let v(t) = |«(i)|. Then υ(i) < |/(i)| + т Г ( а ) ( J ° i ; ) ( i ) . Setting in estimate (2.3.19) of Theorem 2.3.3 m = 1, 0. These inequalities yield

Chapter 2. Linear Volterra operators and their properties Choose s so that mr(a)s-"

45

= 1/2. Then

i.e. ||n||p < 2e^^||/||p. Theorem is proved.

•

Theorem 2.4.3. Let a function τ) be continuous by (t, r) £ Δ and diiferentiable by τ in a domain A, where A{t,t) - l , t e [Ο,Τ], < m{t — for certain m,a > 0. Then solutions to the Volterra equation of the fìrst kind Au — f is unique and ||i7w||p < c||/||p, с = c{a,m,T). Proof. obtain

As u = DtJu, Au = Ju+

then integrating in Au — ADrJu ^ [

Jo

by parts we

AUt,r){Ju){r)dT^f{t).

Prom here, \{Ju){t)\ < \f{t)\ + mr{a){J°'\Ju\){t)·, hence, as when proving Theorem 2.4.2 we have | | < 2e^^||/||p, where s is such that τηΓ(α)δ~" = 1/2.

•

Solution to the Volterra equation of the third kind a{t)u{t) + {Au){t) = fit),

a(0) = 0 , α

0

where A is defined by (2.4.2), is, generally speaking, nonunique. Really, let A = J , и = Ì. Then tu — Ju = 0. Show, now, that if solution u{t) is sufficiently small, then the uniqueness holds. Theorem 2.4.4. (i) Let |A(i,r)| < m, (ί,τ) e Δ and t^-'u G Lp for a certain s such that s > {l+pm)/p, ρ > 1. Then the solution to the equation tu + Au — / is unique. For s > (1 + mp)/p, ρ > 1 the stability estimates < c\\t-'f\\p, с - c{m,p,s) holds. (ii)Let \A{t,T)\ 0, (ί,τ) G Δ, m < Γ(1 - 1/ρ)/Γ(α + 1 — 1/ρ), ρ > 1. Then the solution to equation t^u + Au = / is unique in Lp(0,T). If m < Γ(1 - 1/ρ)/Γ(α + 1 - 1/ρ), then the stability estimate \\u\\p s„+i(A), η = 1,2,... Note, that si = ЦАЦ, Sn(^) = Sn(A*) and for each a the equality Sn{otA) = | (A) holds. Theorem 2.5.1. Let operators A, В € σοο- Then (i)

Sn+m-liA

+ В)




Volterra

0 the

lim

+

operators

Итп''5п(Л)

limits

B )

=

and

their

=

a,

properties

47

limn''sji(ß) = 0

hold,

a.

Denote by σρ the class of all completely continuous operators for which l/p

n=l hold. Evidently, for p' < ρ the class σρ' Ç σρ. For any operators В e C{X), A e σοο the inequalities Sn{BA) < ||5||s„(A), Sn{AB) < ||ß||s„(A) hold. Prom here follows that if α G σρ, then AB, BA e σρ. Note also that if A e σρ, ρ > 1, В e σρΐ, l/p + l/p' = 1, then AB, BA e σι. Operator Л e σι is called kernel operator; operator G σ2 is called G Uberi-Schmidt operator.

Theorem 2.5.2.

Let

X

=

L2{0,T)

{Au){t)^

I f A - A * number

and

\ A{t,

τι)

с is independent

derivative

by

τ

A{t, of t,

Theorem 2.5.3. ρ

-

Let

is taken

A in

be

the

[

Jo

and

Τ2)| < τι,

Τ2,

c\ti then

A

determined sense

(2.5.1)

A{t,T)u{T)dT.

Г2|"

for

Ε σρ,

by

a certain

ρ >

(2.5.1),

of generalized

2/{2a

Α(ί,τ)

functions);

a

G (0,1],

the

+ 1).

G ¿2(Ω) then

A

(the €

σρ,

>2/3.

Let A be a Volterra operator, A^ e σι, V = { Ρ } be a proper maximal chain of opthoprojectors of A. Set η

j=i

where Vn ^ {Pj}, j = 0,n is a partition of V, APj = Pj function h{Vn) has the hmit h{A+), where tr^+


0,

c{t) > 0,

^

a{t)c{t) - b^{t) > 0 ,

o^dí^

Vi G [О, Τ]

Mí)d0.

(2.5.7)

(2.5.8)

Denote by п+(г), п_(г) the distribution functions of the positive and negative parts of spectrum { ß j } of boundary-value problem (2.5.5), (2.5.6).

Chapter 2. Linear Volterra operators and their properties

49

Theorem 2.5.5. Under the given assumptions, lim {n±{r)/r)

=

rT cT / ^{a{t)c{t)

- b'^{t)) dt

Jo /0

holds true. Theorem 2.5.6. Let A be a Volterra operator. Then (i) if A+ E σρ, p> (a) if

1, then A, A- e σ^;

g σι, then A, A- Ε σρ, ρ > 1;

(Hi) if A^ Εσχ, A is a dissipative single-block operator, кет A — {0}, then

(iv) if A e σρ for a certain ρ > 1, then p(A) p{A) - p > 1.

< p, where σ(Α)

= 0 for

Theorem 2.5.7. Let A be a Volterra operator, A e σι. Then for each complex number λ we have 00

| | ( £ ; - а л ) - 1 | | < ^ ( 1 + |Л|5„(Л)).

(2.5.9)

Now we shall come to the main subject of this section: establishing the estimates for σ{Α). Theorem 2.5.8. Let A be a Volterra operator and A^ 6 σρ for a certain ρ > 1. Then either p{A) < p, or p{A) = ρ and A is σ-continuous, where σ(Α) for ρ >1, σ(Α) < 2h(A+) < 2\A+\i (2.5.10) for ρ = 1. To prove this theorem we need. Lemma 2.5.1. For some ρ G (0,1], let Sn(A) ~ cn'^/'P for η - A*) = o{n-'^lT^)(for example. A-A* Ε σρ). Then

oo and

= (s„(A))^

(2.5.11)

A; = 1,2,...

50

Α. L. Bughgeim. Volterra equations and inverse problems

Proof. Statement (2.5.11) may be established for к ~ 2. From the identity {A*fA^

= {A*Af+B,

В = А*{А*-А)А^+А*А{А-А*)А

(2.5.12)

and the definition of the s-number of A we have sl{A')

= Sn{iA*fA')

= Sr,{{A*Af + B)

(2.5.13)

5„((AM)2) = siiA) ~ (cn-i/P)4.

(2.5.14)

By the lemma conditions, Sn(A - A*) = whence, using Theorem 2.5.1 inequaUties, it is easy to show that for В from (2.5.12) Sn{B) -

(2.5.15)

holds. Equations (2.5.13)-(2.5.15) and (ii) of Theorem 2.5.1 yield (2.5.11) for к = 2. • Proof of Theorem 2.5.8. First, let ρ = 1, i.e. Theorem 2.5.4 limns„(A) = 2h{A+)/w

€ σχ. Then, by (2.5.16)

holds. Since, by Lemma 2.1.2, p{A) = 2p{Ä^), σ(Α) = σ(Α^), it is sufíicient to show that either 2p{A'^) < 1, or 2p{A'^) = 1 and σ{Α'^) < 2h{A+). By the condition A+ e σχ and, consequently, by Theorem 2.5.6 (ii), Ae Oq for each q > I. Therefore, A^ G σχ and by Theorem 2.5.7 00

IKE - λΛ2)-ΐ|| > Π ( 1 + |λμη(Α2)).

(2.5.17)

Setting, in Lemma 2.5.1, ρ = 1 and taking instead of A operator lA, from (2.5.16), (2.5.17) we obtain l i m ( n V s ^ A ^ ) = 2/ι(Α+)/7Γ.

(2.5.18)

Let f{z) = + and nj{r) be the number of zeros of f{z) (taking into account their multiplicity) in the disk \z\ < r. According to (2.5.18), n/(r) ~ Γ^/^2/ι(Α+)/π. From here, using the theorem on connection of zero density of entire function / with its growth (see Evgrafov, 1979, p. 200) we obtain In

~

cos((/j/2)

(2.5.19)

Chapter 2. Linear Volterra operators and their properties

51

for r oo. Prom (2.5.17)-(2.5.19) it follows that either p{A) = 2p{Ä^) < 1, or p{A) = 1, σ{Α) Ξ σ{Α^) < 2h{A+). (2.5.20) Or, more precisely, since {E - λ ^ {E + XA){E - λΜ^)"!, then ||(E asymptotically for λ ->· oo is not greater then (1 + |Л|||А||)ехр(2|Л|МА+)). Thus, taking into account (2.5.2), we see that (2.5.10) is established. Now, we show that A is σ-continuous. Let V be a proper maximal chain of orthoprojectors οΐ A; P, Pn E V, Pn < Pn+i < P,

p{{P - Pn)A{P - Pn)) = p{A) = 1

and Pn be strongly converging to P. As A+ G σι, then the sequence An — {P — Pn)A^{P — Pn) converges to zero in the kernel norm: \An\i —^ 0. On the other hand, taking operator {P—Pn)A{P—Pn) in (2.5.20) instead of A we obtain σ ( ( Ρ - Рп)А{Р - Pn)) < 2h{An) < 2|A„|i ^ 0.

So, for ρ = 1, the theorem is proved. If ρ > 1 and A^ e Op then, by Theorem 2.5.6 (i), A G Op. Applying the statement (iv) of Theorem 2.5.6 we obtain that either p{A) < p, or p{A) — p, σ{Α) = 0. For σ{Α) = 0, operator A, evidently, is σ-continuous. The theorem is proved. • Prom this proof immediately follows Theorem 2.5.9. Let A be a Volterra operator, A 6 σοο, lim ns„(A) = 2/ι/π, s„(A+) = o(n-i). Then, either p{A) < 1, or p(A) = 1, σ{Α) < 2h and A is a σ-continuous operator. Remark 2.5.1. Prom statement (iii) of Theorem 2.5.6, it follows that estimate (2.5.10) is exact. Theorem 2.5.10. Let X = //2(0,T), {Au){t)=

Ρ

Jo

A{t,T)u{T)dT

and kernel A{t, τ) satisfy one of the following conditions:

52

Α. L. Bughgeim. (i)

Volterra equations and inverse problems

r) G for a certain a > 1/2, is the space of functions satisfying the Holder condition of the order a; Ω, = {t,T \ 0 < τ < t < T};

(ii) A{t,T) = αο(τ) + Ai{t,T), where A{t,t) = ao e ί-2(0,Τ), operator with the kernel Θ(ί - τ)Αι{ί,τ) (Θ{ί) is the Heaviside function) is a kernel operator; (Hi) A{t,T) = αο+Αι{ί,τ), r)Ai{í,r))

where A{t,t) ξ ao e ¿2(0,Τ), ^ ι ( ί , τ ) , Α ( Θ ( ΐ -

Then, either p(A) < 1, or p{A) — I, A being a σ-continuous where ^ σ(Α) 1 and the scale X we naturally construct the scale У — Lp (Ω; Χ) which consist of such functions u : Ω -> A" that y = \jYs, sei

Ys =

Lp{n-Xs)

1 < ρ < 00 ||г«||оо,5 = vrai sup ||u(í) Ili. ten Similarly, we define the scale

sei ll^llc(ñ^) =sup||u(í)|| ^ ' ten where (7(Ω; Xg) is a Banach space of functions η : Ω —>·

continuous and

bounded in the closure Ω of the set Ω. Let Л be a linear bounded operator acting in Lp(Ω) such that spec(A) = { 0 } and {Au){t)

> 0, iiu{t)

> 0.

Definition 3.1.1. Operator V from C{Lp{ü-,X)) i'p{X, A,a), a>0 if for each í G Ω the estimate

belongs to the class

(3.1.5) Consider the equation (3.1.6)

u = Vu + f. Theorem 3.1.1. Let V e Up{X,A,a), αρ(Α)

= 1,

f G Lp{n-,Xs).

ер{А)а(А)

Ifap{A)

< 1, or

< (s - s')

then equation (3.1.6) has a solution и G Lp(Q;

Xg'),

ll^llp,.' < c||/||p,. where с is independent of u, f . Lp{n;Xg).

Solution

to this equation is unique in

58

Α. L. Bughgeim. Volterra equations and inverse problems

Proof. For a sequence Sj e I such that sq = s' < s, Sj+i = sj + {s—s')/n, J = 0,n — 1, from (3.1.5) by the induction method we obtain

Hence, using the definitions of r(s,s', A), p{A), σ{Α) we have

f 0,

if αρ{Α) < 1

Using Lemma 3.1.1 completes the proof

•

If we additionally suppose that A is a Volterra σ-continuous operator, then uniqueness of solution to (3.1.6) holds for ap{A) < 1. More precisely, the following theorem holds. Theorem 3.1.2. Let A be a Volterra operator, σ-continuous relative to a proper maximal chain V = {P}, where ψ > Ρψ > О for all φ G Lp{Cl), φ > О and Ρ e V. If V e v>p{A,X,a), the scale X is continuous and ap(A) < 1, then solution to equation (3.1.6) is unique in Lp(Q;Xs) for each s G I. Proof. The case a:p(A) < 1 is considered in Theorem 3.1.1. Let ap(A) = 1 and и e Lp(^;Xg) satisfy the homogeneous equation и — Vu. Show that u Ξ 0. Let P' be a chain consisted of projectors P' of the form P' — E — P., PeV. Since PAP = AP, then P'A = P'AP'

VP' e V

(3.1.7)

where, as > Ρφ > 0 for cp > 0, then φ > Ρ'ψ > 0. Applying Ρ ' to estimate (3.1.5) and taking into account identity (3.1.7) we obtain P'\\Vu\\s' PQ estimate (3.1.8), taking into account (3.1.10) and the fact that P' - PL^ = E - Ρ - {E - PQ) ^ PQ - P, Ρ >PQ, takes a form

Prom here, by the induction method we obtain ΙΙ^ΊΙ4Ι.ΊΙμω) < ( η / ( 5 - 5 ' ) ) " Ί Ι [ ( ί ' ο - Ρ ) Α ( Ρ ο - Ρ ) ] Ί Ι Ι Η Ι ρ , . ·

(зл-п)

If P'Q enters into the break {PQ,P[), P{ > PQ of the chain V , then {P{ PQ)A{P[ - P¿) = (Po - PI)Ä{PO - Pi) and, consequently, from (3.1.11) follows Pi'||u||ä' = 0. Tending s' to s, by the continuity of X, we obtain Pillulli = 0, P{ > P¿ which contradicts the maximality of P¿. Now, let V have no break of the form {P¿,P{), i. е. it is continuous from the right on projector P¿. Then, by σ-continuity of A relative to V, given ε > 0, we may find such P¡ > P¿, that either p{{PO ~ Ρε)Α{Ρο - Ρε)) < ρ{Α), or σ((Ρο-Ρ,)Λ(Ρο-Ρ,)) P¿, which again contradicts the maximality of PQ. SO, PQ = E and the theorem is proved. • For certain applications we need to have estimates of the solution u norm by way of the right side / norm in the same space. We shall formulate now the corresponding theorem confining ourselves, for simplicity, to the Hilbert scale X = UX^, s e I = ( - α , a) of the form ||и||, = ||ехр(5Л^'^)и||,

sel.

Here X = Jío is a separable Hilbert space with a norm || · || = || · ||o, Л is a self-conjugated nonnegative operator in X. Thus, the Hilbert space Xg, s e I is defined as the closure in the norm ||n||s of the set of those elements u G ЛГ for which ЦиЦ^ < oo. Theorem 3.1.3. Let V e Vp{X,A^a). Assume, that either ap(A) < 1, or ap(A) = 1,

ea{A)p{A) < s.

(3.1.12)

Α. L. Bughgeim.

60

Then solution to equation in it.

Volterra equations

problems

(3.1.6) is unique in Lp(Q; X) and densely

If и E M — {u I ||Ли||р,о < "г},

stability estimate

and inverse

/ e Lp{ü-,X),

then the

solvable following

holds:

MU^'^.dl/M ω,(ε) Äim(s/In 1/ε)°,

(3.1.13) ε 0 .

(3.1.14)

Remark 3.1.1. We shall write uJs(e) ^ φ(ε,3ο}, ε 0, if for each s > So the hmit = lime_+o (^(ε, s) holds true. As for ар{А) — 1, number s in conditions of Theorem 3.1.3 may be taken arbitrarily close to ea{A)p{A); then ω{ε) ^ m{{ea{A))/{a

In 1 / ε ) ) " ,

ε

0.

Proof of Theorem 3.1.3. Uniqueness of (3.1.6) solution follows from Theorem 3.1.1. To prove the dense solvability, note, that by Theorem 3.1.1 for each / G Lp{Q,;Xs), where s > 0 for αρ{Α) < 1 and s > eo{A)p{A) for ap{A) = 1, there exists a solution и G Lp(íi;Xo) = Lp{íi;X). Since Lp{Ct;Xs) is dense in Lp{Cl;X) (due to the density of Xg in X), the dense solvability is proved. Now, solve estimate (3.1.13). By Theorem 3.1.1 for / G Lp{Çl;X) there exists a solution и G Lp{ü;X-s), where ll^llp,-. < c||/IU.

(3.1.15)

Since for each e > 0 the interpolation estimate libilo < ε||Λη|| + exp(sε-l/")||u||_,

(3.1.16)

holds, then, by the Minkovskii inequality, ||u||p,o < ε||Λω||ρ,ο + exp(s6"^/")||n||p_s. Prom here, taking into account (3.1.15), we obtain ΙΙ^ΙΙρ,ο < ε||Λη||ρ,ο + c e M s e - ' h \ \ ñ p , o ·

(3.1.17)

Thus, if we introduce the functional l{u) = ||Au||p,o, then problem (3.1.16), due to (3.1.17), is i-well-posed. By Corollary 1.2.1 we obtain (3.1.13), (3.1.14). It remains to prove estimate (3.1.16) used above. For this purpose use the spectral theorem according to which there exists an isometric map F : u -> û of X onto the direct integral X=

roo

/

Jo

X(λ)dμ(λ)

Chapter 3. Linear operator Volterra equations

61

of Hilbert spaces, diagonalizing operator Λ: (Λί/)(λ) = λΰ(λ). Here Û = { ΰ ( λ ) } is the image of и under map F, fOO

= l

(3.1.18)

The measure άμ{λ) is concerned in [Ο,οο) since Λ > 0. (The proof of this theorem is, for example, in Moren, 1965). Equahty (3.1.18) we rewrite in the form 1Ы|2

+ Í

\>1/ε

||û(A)||J(,)dM(À)=/i+/2

(3.1.19)

and evaluate the integrals Д, l2' h =

exp(2sAi/")exp(-25Ai/'^)||û(A)||J^^^ d/i(A)

J λ^Ι/ε Substituting these estimates in (3.1.19) and taking the square root we obtain (3.1.16). The theorem is proved. • In applications, often, Ω = (Ο,Τ), α = 1, iAu){t)=

Jo

[\{ί,τ)η{τ)άτ

where A{t, τ) is continuous for (ΐ, τ ) 6 Δ = {ί, r | 0 < τ < ί < Τ } a real function, Α{ί,τ) > 0. In this case, (3.1.5) takes the form < Theorems 3.1.1-3.1.3 yield

f Jo

A{t,r)\\u{T)\UdT.

(3.1.20)

62

Α. L. Bughgeim. Volterra equations and inverse problems

Theorem 3.1.4. Let V G £(Ιρ(0,Γ; Χ)) and for each s, s' e I, s' < s condition (3.1.20) holds. Then (i) solution to the equation u = Vu + f is unique in Lp(0,

(3.1.21)

s e I;

(ii) if for certain s,s' E I the function f G Lp{0, T; Xg) and rT J Ait,t) dt Jo

e ¡^ A{t,t) di, s G / = ( - α , α) ||n|Uo 0, such a function As{t,T) G C2(A) exists that ¿ > Λ ( ί , τ ) - α ( ί , τ ) >0,

0 - 1 , 1 - 2 a > - 1 . Prom definition (3.2.6) of this function h{t,T), it follows that h{t,t) = 0, N{t,t) =0, Dg{t)=

Jo

í\l{t,T)dT.

Substituting this formula into (3.2.8), taking {·,ν) out from the integral sign and using formulae (3.2.5)-(3.2.8) we obtain =

+

( f Jo

Ai{t,T)u{T)dT,

ν)

66

Α. L. Bughgeim. Volterra equations and inverse problems

where Αι{ί,τ) is defined by formula (3.2.2). Because of the arbitrariness of V e Y* and the density of C{[0,T];X) in Lp{0,T-X), ρ < oo, from here follows that {Au){t) has a derivative Ό°Άη in the weak sense, determined by (3.2.3). So, for α > 1, ρ < 00 the Lemma is proved. The case α = 1 is trivial, the proof of the case ρ = oo is completely the same as the above proof. • Remark 3.2.2. Formula (3.2.2) is established for α G (0,1]. The case of arbitrary α > 0 is reduced to the considered case by the differentiating of the integer order. Lemma 3.2.2. Let A{t,T) be a real function of the class C°°{ü), A{t,t) > 0; t e [0,Т]. Then, for operator A from Ip(0,T), determined by (3.2.1), we have rT

p(A) = l/a,

σ{Α)=

Jo

(^(ί,ί))^/" di,

α > 0.

The proof of these formulae follows from Lemma 2.L2 and the fact that A is similar to operator с J : FAF~^ = J"",

F, 6 £(Lp(0,T)). (For proof of the similarity between A and J " see, for example, Dunford and Schwartz (1974) and the bibliography cited there). Further, we shall need only the inequalities p{A){X,Q,ß), we may apply Theorem 3.1.1. As when proving Theorem 3.1.4 we may consider without loss of generality, that Q(i, r ) G (7°°(Ω), Q{t,t) > 0, Í € [0,Г]. By Lemma 3.2.1 we have p{Q) = υ β гТ m

^

í

Jo

[QitM'^àt.

Chapter 3. Linear operator Volterra equations

69

Choose s" > s' so close to s' that (3.2.11) holds for s-s" instead s - s ' . Then, by Theorem 3.1.1, equation (3.2.12) has a solution υ e Lp{0,T-,Xs") and \\u\\p^s> = 1|A~^(í,í)í;||p,s' < c||î;||p,s" < c||í)"/||p,s· Uniqueness is established similarly to Theorem 3.1.4. • Corollary 3.2.1. Let under Theorem 3.2.1 conditions. Λ" be a scale of Hilbert spaces XS : ЦпЦ, = || exp(sA^/'^)K||o, X Ξ XQ» S E I = {—A,A), a> 0. Here X is a separable Hilbert space, Λ is a self-conjugate nonnegative operator in X. Further, let (3.2.11) hold for s' = -a, s — 0. Then Au — f is dense solvable in X, ker Л = {0} (in X). If u solves Au = / so that ||Λη||ρ,ο < m, D " / G Lp(0,T;X), where \\A-^t,t)\\^a,~a < c, с is independent of ί e [0, Г], 7 = then IM|p,o 0, β e [О, α] exist that for each s, s' e I, s' < s, (i, r), {ξ, τ) G Ω the estimate \\A{t, τ) - Α(ξ, T)\U,S' < u{\t - ξ\){ί -

- s')-^

(3.3.2)

holds, where u{h) is a nondecreasing function h such that I = limω{Η){\In/il'^-^) < 00. h-^O

(3.3.3)

70

Α. L. Bughgeim.

Volterra equations and inverse problems

Then solution to (3.3.1) is unique in the class Lp(0,T; Xs) П Ci'([0,T]; Xs), s > s, 7 > 0. Here, s, s Ε I, C'''([0,r];Xs) is a Banach space of functions u(t) with values in Xs, satisfying Holder condition with exponent η G (0,1]; 1к11сП[0,т];Х,)

= sup ||u(í)||, + ÍG[0,T]

sup ( | | u ( í ) - w ( t ) | | , / | í ί,τ€[0,τ]

г^).

P r o o f . Because (3.3.1) is a Volterra equation, it is sufficient to prove the theorem for small T. Let h < Τ be an arbitrary number. Apply operator Ahf{t) = (/(Í + h)- f{t))/h to equation (3.3.1). Since A{t,t) = E, then uit)+

Г

AhA{t,T)uÌT)dT

= gh{t),

t e [ 0 , T - h ]

Jo

(3.3.4)

holds, where t+n rt+h

9h{t)

/

[A{t,T)-Ait+

h,T)]u{T)dT

t+h

/

iu{t)-u{T))dT = g i + g 2 + g 3 .

(3.3.5)

From (3.3.2), it follows that \ \ A , A i t , T ) \ U , s ' < c { t - T r - H s - s r ^

with constant с = u{h)/h. Since σ(ν7°) = Τ, then for sufficiently small T, namely, for (сГ(а))^/"еТ < a(s - s'), equation (3.3.4) by Theorem 3.1.1 for Qh. 6 Lp{0, Τ - h; Xs) has a unique solution и e Lp{0, Τ - h] Xg'), where \H\Lp{0,T-h-,X,,)

< Ci||9/i||LP(0,T-/i;X,)·

Here, by Lemma 3.1.1, CX)

Ci < 1 + ^

On

n=l

an = {cr(a)(s Prom the Stirling formula it follows that Γ(ηα + 1) > where К depends only on a. Prom here ci = ci{h) 00

0 it is necessary and sufficient that the following conditions be fulfìlled: (i) Function u{x) may be analytically continued in the domain fig = {z — х + гу\х,уе E",|y| < s}; (ii) for each уо e for у -> уо, |у| < |уо| = s, the function и{х + гу) takes in the sense of the limit in L2 the boundary value u(x + гуо) G ¿2(RS), where \H-+iy)\\L, 0, and -X ^ K. Introduce in Y the relation of semiordering. By the definition xu > θ. Example 4.1.1. Let Y = С (Ω) or Lp (Ω), К be the cone of nonnegative functions and FsU = where Xs be the auxiliary scale of Banach spaces, Ω be a domain in W. In other words, Yg = C{Cl]Xs) or У^ = Lp{iì-,X) correspondingly. Let N G >C(y) be a quasinilpotent operator. As in Chapter 2, by p{N) and σ{Ν) we shall denote the order and the type of entire operator function {E correspondingly. The number a = l/p{N) will be called the degree of operator N. Definition 4.1.1. Let for each pair of numbers s, s' G /, s' < s a map V be determined in the ball Ys(r,uq) = {u G Ys I — uo||s < r}, щ G Yi and V maps it into Y^/, where scale {Υ^} is isometrically positively imbedded in Y. We shall say that Y belong to the class N{a, β), a, β >0, if in the space Y there exists a quasinilpotent positive (i. e. mapping a cone К into itself) operator N of degree a that for each u, ν £ Ys{r,uo) the inequality Fs'iViu) - V{v)) < (s - s')-^N{Fs{u

- v))

(4.1.1)

holds. Consider the equation и = V{u). Theorem 4.1.1. Let V e Ν{α,β).

(4.L2)

Then

1) if a > β > 0 or a = β, but σ{Ν) < as/e, then solution to equation (4.1.2) is unique in ball ys(r,uo), s > 0;

Chapter

4. Nonlinear

equations

2) if a > β, У К ) e У.С^'.^о),

in Banach

spaces

83

г' = r/b,

к=1 -1

00

_

dk = i s - s ' ) [ k ì n ^ { k + l ) ] - ^ п=1 [ ^ n - h i i - \ n + l)

then equation

(4.1.2)

has a solution

и

\

А>1,

s'< s

G Yg'(r,uo).

E x a m p l e 4.1.2. Let Y , K , Ys, Fg be the same as in Example 4.1.1, Ω = (Ο,Γ) and N = c j ' ^ , c > 0, J " be the operator of integrating J of degree a:

{Γψ){ί)=Τ~\α) [ \ t - T r - M r ) d T , Jo

ψΕΐ,{0,Τ)

(4.1.3)

Γ ( α ) is the Gamma-function. Then a { N ) - a, σ { Ν ) = Naturally the problem arises: will the equation (4.1.2) have in the corresponding space a solution for a — ß1 Under the condition of Example 4.1.2 this problem is solved in this chapter. In this connection it proves to be that the method of theorem proof (more precisely, the space, where V will be a compression operator) essentially depends on the fact of which inequality, ap > 1 or ap < 1, holds. In other words, the structure of space Y turns out to be connected with the order of operator N . To formulate Theorem 4.1.2 we introduce Definition 4.1.2. Let for each pair of numbers s, s' e I , s' < s a map V be defined in the ball ¿ р ' " ' ' ( 0 , Г ; Х « ) = {ω | - ио||ьр{о,Т;Х,) < щ e L p { 0 , T ; X i ) , and maps it into space L p { 0 , T \ X s ' ) . We shall call V a Volterra operator of the class J { a , ß , L p ) , α > 0, /3 > 0, if such constant с > 0 exists that for each u, ν ζ L p " ° ( 0 , T ; X j ) , s' < s and for almost all t e (0, Τ ) we have ||K(u)(i) - F(w)(i)||y < c(s Here

s r ^ J - { \ \ u { T ) - v{r)\U)it).

(4.1.4)

is a norm in X , .

Evidently, for

Y = Lp{0,T)

class

J{a,ß,Lp)

С

N{a,ß).

In the degen-

erate case β = 0, X s = 'M} the operator V of class J { a , 0, L p ) will be the

84

Α. L. Bughgeim.

Volterra equations and inverse

problems

functional Volterra operator V in the sense of the Tikhonov definition (see Tikhonov, 1938). Consider the equation u{t) = iVu){t), Let q = ρ i(ap>

ÍG[0,T].

(4.1.5)

1 and p = a — 1] q> p/a for ap < 1, a < 1.

T h e o r e m 4.1.2. Let V G J{a,a,Lp).

Then

1) solution to equation (4.1.5) is unique in Lp"°(0,T;

Xs), s > 0;

2) ifV(uo) e Lg'''°(0,T;Xs), then such α > 0 exists that for each s' < s, equation (4.1.5) has in the interval [0,T'] С [Ο,Τ], Τ' < a(s - s') solution и e Li''"'(О, Τ'· Χ,,). Denote by J{a, /3, С) the class of operators satisfying conditions of Definition 4.1.2, substituting scale Lp{0,T-,Xs) by scale (^([Ο,Τ];^^). Assume, for brevity, ||W||T,S = ||U||C([O,T];X,)· T h e o r e m 4.1.3. Let V G J{a,a,C). 1) the solution to equation s > 0;

Then

(4.1.5) is unique in ball ||u — UO||T,í
0 exists that for each s' < s, equation (4.1.5) has in the interval [Ο,Τ'] С [Ο,Τ], Τ' < a{s - s') solution и G C{[0,T']-,Xs), where ||u - woUt',«' < r.

4.2.

DEFINITIONS A N D AUXILIARY STATEMENTS

Let Χ be a Danach space. By Lp^a{0,T-,X), ρ G [Ι,οο], α > О we denote the Banach space of functions и{т) (more precisely, of functional classes equivalent by the Lebesgue measure dr) with the values in X such that

where J is defined by formula (4.1.3). Evidently, Lp^i is an ordinary space Lp and for the other values of a, the imbeddings hold LpDLp^aDLg,

α G (0,1),

q>p/a

Lp D Lp^a

α G (Ι,οο),

q > pa.

Lg,

Chapter 4. Nonlinear equations in Banach spaces For ρ = 00, by the definition Loo,a = Loo, where sup||n(T)|U, TG (Ο,Τ). Set

85 vrai

= {ue Lp,a I II" - Щ\\р,а < r}. Let {Xs}, s G / = [0,1] be a scale of Banach spaces. For α > 0, assume Δα = {{ί,5) | 0 < i < a ( l - s ) , 0 < s < 1}. В у Л Й ( а ) , а > 0 , ^ > 0 , 7 > 0 , ρ G [1, oo] we denote the space of functions u{t) with the values in space Xs for Í < a(l — 5) such that the norm N[u] = sup

j o . t Ä ) } < 00

t,seAa

where p(i,s) = a ( l - s ) / i - l .

(4.2.1)

Further we shall see that space is the natural Lp-analog of the Nirenberg space (see Nirenberg, 1977, p.207) with norm MM=sup(p(í,s)||n(í)||,) being used for the proof of solvability of the nonlinear abstract Cauchy problem in ^([Ο,Τ];^^). Further, for simplicity, denote 5^;7(a) = B^'^'ia), The following three lemmas describe the "correlation" of J " ' with the weight ρ and Bp'^{a). Lemma 4.2.1. Let β+ ^ + l>0,

{t, s) G Aq. Then

smt)
a and J^ÌT^P-ЦТ,

s)){t)
о

(4.2.8)

holds:

(i) ve

β>α{ρ-

l)/p, 7 > 0, ρ < 00;

(ii) V E p = l;

β > a — 1/p, J > 0, ρ < 00, a > l/p for ρ > 1 and a>

1 for

(Hi) V € Βξ^, β > 0; then function w belongs to the same space as function ν (i. е. Bp,α, Вр''^, В^ correspondingly), where iV[tí;] < mN[v], m = 0(α"). Proof. Let (i) hold. Substitute in (4.2.8) s by s{t) = θ8+{1-θ){1-ί/α), where 0 is a certain, so far arbitrary, parameter from interval (0,1) and substitute s' by s. As s < s{t), {t, s{t)) G Δα, this substitution is correct. As

Chapter 4. Nonlinear equations in Banach spaces

87

a result, estimate (4.2.8), taking into account identity s(t) — s = 0p{t, s)t/a, takes a form IkWll. < co{e/ar-t-p~-{t,s)J-{\\v{r)l^,^{t))

(4.2.9)

Co = сГ(а).

(4.2.10)

Using the Holder inequality for the integral J'°': ^{ψΦ)

< [TiH^^^iJ^V')]''"'.

Ψ,Φ>ο,i/p

and assuming in it φ{τ) = ρ^(τ,θ(ί))||υ(τ)||,(ί), obtain

+ i/p' = 1 = p-l^{T,s{t)),

we

In the latter inequality we have used the norm N[v] determination and the identity p{t,s{t)) = {l-e)p{t,s).

(4.2.11)

By Lemma 4.2.1 (see (4.2.2)) for β > a/p' and using (4.2.11) we obtain J-{p-P'^{T,s{tmt)

< Γ{ρ'β -

-

вГ'^'^Пр'Р).

As a result we have
0, β < {1 + a{p a - ß - j > 0 ,

(iii) ν e i ^ ( 0 , α; Xi), /3 < α,

l))/p,

β 0, from (4.3.1) and the definition of class Ν{α,β) (see (4.1.1)) by the induction method, for • 00. Therefore, ||ω — г^Ц,/ = О, i. е. u = v.

90

Α. L. Bughgeim.

Volterra equations and inverse

problems

Existence. Let s„, sq =

lim s^ = s' < s, s„ — s„+i = dn+i > 0 n—>oo be an infinite decreasing sequence. We shall seek a solution to equation и = V{u) by the successive approximation method = V{un) beginning with element uq- Since Un+\ — Un = V{un) — V{un-i), then, formally, by the induction method, we obtain Ikn+l - «nils' < ll'Un+l (4.3.5) hn+iII.' < IIF(î^o) - í^olU [ι +

···

•

(4.3.6)

k=l

In order that this condition be correct it is necessary that for each η > 1, elements ii„_i E For this purpose, it is sufficient to choose numbers dk so that ||F(Ko)-ì.o||.[l + 5;](di---4)-^||iV'llì < r

(4.3.7)

A;=l

where ^ d k = {s-s'). fc=l

(4.3.8)

We now determine dk by the formula from Theorem 4.1.1. Then condition (4.3.8) will hold by construction and condition (4.3.7) will become the condition of Theorem 4.1.1: У(ко) £ Ysi'^'ì ^0)γ' = r/b (the finiteness of b is implied from a> β and formula (4.3.4)). For such choice of dfc, from (4.3.5), (4.3.6) follows the convergence of in Ys' to the element и G Ys'{r,uo). We show that и solves the equation и — V{u). If s' > 0, then for 0 < s" < s' sequence V{un) converges to V{u) in space Yj/, since, by inequality (4.1.1) ll^(«) -

< (s' - ^ - ^ Ί Ι ^ ν Ί Ι Ι Ι ^ -

Therefore, going in the equality Un+i = Viun) to the limit, we obtain that и solves equation и = V{u). If s' = 0, then due to the continuous dependence of number b = b{s, s') on s', there exists a sufficiently small 5i G (0, s) such that from condition V{uo) G Ys{r',uo), r' — r/6(s,0) follows V{uo) G У5(г',ио) for r' = r/6(s,si). Therefore, from the previous argument, the existence of solution и G У^Дг, kq) С Yo(r,uo) follows. The theorem is proved. •

Chapter 4. Nonlinear equations in Banach spaces

91

Proof of Theorem 4.1.2. The uniqueness of solution (4.1.5) follows from Theorem 4.1.1, for the value = c^/^T, for sufBciently small T, may be taken less then sa/e, s' = 0. Due to the fact that F is a Volterra operator, from the local (in t) uniqueness follows the uniqueness in the whole. The proof of solution existence we begin with two remarks. a) Statement 2) of Theorem 4.1.2 is sufficient to prove in such a formulation: if v{uo) e i/g"°(0,T;Xi), then such α > 0 exists that equation (4.1.5) has in the interval [Ο,ί] solution и e Τ; Χ,), t < а{1 - s). The general case is reduced to this case by changing scale parameter s. b) In order to find a solution и Ε Lp^°{0,t-,Xs) to equation (4.1.5), it is sufficient to construct in the interval [0, i] a solution υ E Lp '"(0, ¿¡Xç,) to the equation V = W{v) (4.3.9) where W{v) = V{v + F(uo)) — У{щ) and r' is chosen from the condition r' + ||F(uo) - uo\\lp(o,T;Xi) < Then, assuming и = ν + F(uo), we obtain и = У{и) for и e Lp''''°(o,t-,Xs) in the interval [0,t]. To prove the existence of the solution to (4.3.9), one must distinguish the cases (i) q>p/a,

ap>l,a£

(0,1);

(ii) q — ρ < oo, ap > 1 or a = ρ = I; (iii)

q = p = oo.

Consider in detail the case (ii). Solution to equation (4.3.9) we shall seek by the successive approximation method using for the proof of convergence constructed in Section 4.2 spaces βρ,α(α) with corresponding parameters a, β, 7, a. For this purposes, take the sequence of positive number α^, lim afe = α > О, flfe+i =afe[l + (fc + l)-2]-i,

/, = 0 , 1 , . . .

where the initial number oq < Τ will be chosen later, and consider the sequence of spaces Bp'"'{ak) with norms iVfe[u], Ph{t,s) = α^(1 — s)/t Evidently, for w G Bp''^{ak), relation < A^fe[ii;] will hold and since for t < afe_,_i(l - s), function Pfe(i, s) > {k + l)~^, then p/a and the fact that for sufficiently small Τ the norm of corresponding imbedding operators < 1. Theorem 4.1.2 is proved. • Theorem 4.1.3 is proved analogously to Theorems 4.1.2 for p = oo. The scheme of application of Theorems 4.1.1-4.1.3 to the inverse problem investigation is the following. The initial problem is transformed into the integro-differential Volterra equation of the form и = Vu and the corresponding scale of Banach spaces is chosen. In this scale operator V belongs to N{a, β), J {a, β, Lp) or J {a, ß,C), β < a. If we are interested in uniqueness and stability of initial problem, then we may use the linear alternative of these theorems (see Chapter 3) together with the Holmgren method (uniqueness and stability of initial problem are deduced from existence of solution to the conjugated problem). In applications to the inverse problems we seek the solution to equation и = Vu, as a rule, not in the whole ball with the center in щ , but rather, satisfying certain additional conditions и G M. In such a case, if UQ G M, M being invariant relative to V, and M being closed in the corresponding topology, then solution to the equation и = Vu also belongs to M. Below we shall give an exact formulation of this statement. Prom the proof of this statement, under conditions of Theorem 4.1.1, follows the corollary. Corollary 4.3.1. Let the conditions of Theorem 4.3.1 hold and M satisfy the following properties: (i) Щ G M ,

V{MnYs{r,uo))

(ii) if Vji E Mr\Ys'{r,uo) then V e M.

CM

Уз >0,

s e r ,

and Vn converges in У^/ to element

Then solution to equation и = Vu also belongs to M.

ν

G Уу(г, U Q ) ,

94

Α. L. Bughgeim. Volterra equations and inverse problems

Example 4.3.1. To find for χ E [Q,d\, d < oo the continuous function k{x) from the conditions Ut = utt + k{x)u, u{0,t)=f{t),

x>0,

t e [0, Τ]

u,(0,í)=0,

u{x,Τ) = h{x) > 0 ,

íe[0,T]

a; e [О, d].

(4.3.15) (4.3.16) (4.3.17)

Here и ζ C j t (Q) П Loo(Q), Q = (Ο,οο) χ [О, Γ], C^j is the class of twice continuously differentiable by χ and one time by t functions. Evidently, for existence of solution to the stated inverse problem k{x) (more precisely, the pair k{x), u{x,t)) the following conditions are necessary. (i) Function f{t) is analytic in ί G (О, Τ] and, in particular, such εο > 0 exists that νε(Ο,εο),

s e [0,1]

||Л|| = sup |/(ί)| < oo Ω,

Ω, = {ί | ε ( 2 - 5 ) < R e í < Τ + ε(1 + δ), |Im| < ε ( 1 + 5 ) } (ϋ) heCH[0,d\),h{0)^fiT),h'{0)

(4.3.18)

= 0.

We must show that (i), (ii) are, in some sense, sufficient. Theorem 4.3.1. Let (i), (ii) hold and d < oo. Then, for each r > 0, there exists positive r' depending only on r, d, ε, mmh{x), max|/i"(j;)| (x e [0, d]) such that for ||/ - h{0)\\i < r' there exists a unique function к 6 С([0,с?]) and twice continuously differentiable by χ function и, ||u(a;, ·)||ο < r (uniformly in X E [0, d\), that for (ΐ, χ) G [ε. Τ] χ [0,d\ equations (4.3.15)(4.3.17) hold. Proof. In (4.3.15), set t = Τ and, using condition (4.3.17), express к in terms oí u: к = {щ{х,Т) - h"{x))/h{x). Substitute this expression in (4.3.15) and integrating two times with respect to x, taking into account (4.3.16) and changing the notations a; -f^ i, Г Ή· d, we obtain the equation u{x,t) = j

{t~T)^Ux{T,x)-u{T,x)[uj:{T,d) -Η"{τ)]/Η{τ)]άτ

+ /{χ).

(4.3.19)

Chapter 4. Nonlinear equations in Banach spaces

95

The right side of equation (4.3.19) defines operator V of the class J{2,1, C) in scale У^ = C([0,T];Xs), where the norm in Xg is defined by (4.3.18). Set M = {u e C([0,T];Xo) I u{t,d) = h{t), t € [0,T]}. Then Theorem 4.3.1 immediately follows from Corollary 4.3.1. • R e m a r k 4.3.1. Analogous result arise if we substitute equation (4.3.15) by щ = F{uxx,ux,k,x,t), where F is analytic in all the arguments besides X and i^úii-^jt ^ 0· ^^ ^^^^ connection, if = 0, then, instead of scale Xg, we may use the scale of functions belonging to the Gevrey class of order 2. Operator V, arising in this case, belongs to J{2,2, C) and proof of existence we may use Theorem 4.1.3. Theorem 4.3.1 is proved by Bukhgeim (see Lavrent'ev et α/., 1980, p. 11-15). Uniqueness of solution to the similar problem was independently proved by Klibanov (1979). The other inverse problem statements for parabolic equations are in Lattés and Lions (1967), Lavrent'ev et al., (1969, 1980), Beznoshchenko and Prilepko (1977) and in bibliography cited there. The method for Theorem 4.3.1 does not depend on the type of equation. This method is naturally generalized on the multidimensional case. We only have to demand the analyticity of solution and of the desired coefficients in some of the variables. R e m a r k 4.3.2. Analog of Theorems 4.1.2, 4.1.3 is valid when substituting J°' by operator • • being the wave operator. It seems to be that if, in Theorem 4.1.1, we additionally suppose that N is completely continuous (and, so, Volterra), then the existence theorem holds also for a = β.

4.4.

INVERSE KINEMATIC P R O B L E M OF SEISMOLOGY

The inverse kinematic problem of seismology in its two-dimensional statement is reduced, as is well-known, to solution of the integral equation Φ{υ) = j L{v{x, y),y') dx = τ(χο, a^i),

xo,

e [-и, ν].

(4.4.1)

Here τ is a given function (the time of disturbance run from point (XQJO) into (xi,0)), Xo < Xi), L{v,t) = (1 -b υ is a given velocity, υ > 0, Vy > 0. Integrating in (4.4.1) implies going along the oriented curve 7(xoi ^i) '• У = yix·, a^Oi 3;i) with the beginning in the point {xq, 0) and the end in the point (ΐι,Ο). This curve solves Euler equation y" = F{v,vx,vy,y'), F = v{y'vx — Vy)L'^{v,y'). The theory of ordinary differential equations and formula (4.4.1) yield that if υ G A; > 1, υ > О, > О in a

96

Α. L. Bughgeim. Volterra equations and inverse problems

certain neighborhood of zero, then there exists such и > 0 that in domain == i-'^,'^) X (-г^, i^) the function τ = Ф(г;) e has the following properties: τίχο,χι) + T(a;i,a;o) = 0 Txq{XO,XO) < 0,

ΤχοχοχΛ^ο,Χο) < 0.

(4.4.2)

It turns out that for the functions of C"^ (C^ is the set of real analytic functions) the inverse statement holds. Theorem 4.4.1. Let τ Ε C^iü^) satisfy conditions (4.4.2). Then there exists a unique function u, determined in a certain boundary of zero, such that υ eC^^, V > 0, Vy > 0, Φ(ν) = т. Proof. Let us begin with the formulation of the algorithm for determining the function υ by r. This algorithm is given by the following sequence of transformations τ ^ wq — {f,g) wj+i — V{wj) + wq, Wj — {uj,Vj), V = limuj. Here f = τ о ψχ.^ where χχ = v?i(a;o)ío) solves equation Τ{χΰ,ψι{χο,ίο)) = ίο) T{xo,xi) being the tangent of the inclination of the curve ^{xq^xi) in the point χ = жо, i· е. T{xq,xi) = tan(arcsinm(xo,^i)), m{xQ,xi) = [1 - { { φ τ s i g n { x i - xq), g{xo) = { - д т / д х о ) ' ^ for χ = xq. Operator V acts on function vector w = {u,v), и — u{x,y,t), V = v{x,y) by the rule Fi(u;) - Jd{ua^ + uiF{v,

t) + 2L{v, t))

V2{w) = Jp{w). Here J is the operator of integrating by у from zero up to y, du = {u{x, y, t)u{x,y,0))/t, p{w) = {2 + Ux{x,y,0)v{x,y))/ut{x,y,0). To establish the algorithm and the theorem, we must introduce some notations. Set К = {x,t e С \ \x\ < a(l -b s), |i| < a(l + s)}, α > 0, s € [0,1], С being the complex plane. Let Xg be a Banach space of bounded analytic functions / with the norm ||/||s = sup |/(a;,i)|, {x,t) G К. For и, υ e Xg we have < < < s, £> being the operator of differentiation by χ or i; \\du\\s < For the function vector w = {u,v), n, υ 6 Xg, the norm ЦгуЦ^ = ||u||s-|-||v||j,. From conditions (4.4.2) and the determination of / and g, it follows that for a corresponding choice of a, the values |5(x)|, |/t(a;,0)| G [m, mi], 0 < m < mi, χ E С, |а;| < 2а. Let M(s, b) = {w = (и, υ) lu, υ E C([0, b]· Xg), - / | | , < i/, - g\\g < u}. For w E M{s,u,b) the values |ut(a;,у,0)|, 0)| E [m — ι/,τπι + и] for

Chapter 4. Nonlinear equations in Banach spaces

97

y G [О, b], |з;| < 2α and, consequently, these functions are separated from zero if г/ < m. For fixed и < m, the map V is determined in the set M{s,u,b), s e [0,1] and maps it into C([0,ò];Xj/), s' < s, where for each wi, W2 E M (s, ν, b) -

< Φ - s'r'JiWm

-

(4.4.3)

and с = c{a,l·'). Choosing b from the condition У(гг;о) G M{l,u^b), wq — {f,g), using (4.4.3) and Theorem 4.1.3 we obtain that there exists such α > 0 that equation w = V{w) + wq has a unique solution w G M(s, v, 6'), У < b, b' < a ( l — s) and w = \imwj, Wj+i = V{wj) + wq- The structure of V shows that this solution will be analytic in y. To finish the proof of the theorem, it is sufficient to show that equation Φ (υ) = г is equivalent to system w = V{w) + гоо = {f,9), f = τ о ψι^ g = —{дт/дхо)~^ for χ = XQ. This equation immediately follows from the equivalence of problem Φ (υ) — τ and the following Cauchy problem tu. +

+ F{v,

t)ut + 2L{v, Í) = 0

(4.4.4)

îi(x,0,i) = / ( χ , ί ) ,

/ = го(^1.

(4.4.5)

To prove this, choose a number г/ > 0 so that through each two points (a;o,0), (a; 1,0), жо, xi G [-ΐ',ΐ'] a unique curve 7(xo,iCi) passes and all these curves lie in the domain bounded by 7(—г/, i/) and interval [—ν,ν] of the axis у = 0. Denote by D the set of points {x,y,t) G such that there exists a curve j{xo,xi): Y — Y^^ — Ρ{ν,υξ,νγ,Υ') with Cauchy data Κ|ξ=ι = y, 1^Ίξ=ι = t and the ends xj{x,y,t) G (-г/, i^), J = 0,1, Xq < ^ < which lie in the line у = 0. Set Do = DÇ\{y = 0}, Φ : Ω^ -)• Ωι,, Φ(3;ο,α;ι) = (χι,3;ο)· Due to the supposed regularity of the ray field {7(2:0,a;i)}, equation T{xo,xi) = to is uniquely solvable relative to xi and for each (χο,ίο) £ Do' xi = φι{χο,ίο). Set / = τ о φι, ψ^ — Γ о Φ о yji, ψ = {ψι-,ψι)·, ( Τ{χο,Χ2) being the tangent of the inclination of the curve η{χο·,χι) in the point χ = xo). Diffeoniorphism φ : Do Do has the property φ о ψ = E, E being the identity map and, therefore, we may introduce the function / (given in Do) relative to diffeomorphism φ • f± = {f ± f οφ)/2. Due to the first condition (4.4.2), function / = τοφι will be cold relative to φ, i. е. /+ ξ 0. Now, let ν solve equation (4.4.1). Set u = ui + uo, where Ui{x,y,t)

=

+ (y^f

Л d^·

(4.4.6)

98

Α. L. Bughgeim. Volterra equations and inverse problems

Differentiating function и along curve j{xo{x, y, t)), {xi{x, y, t)) (which is a characteristic of (4.4.4)) we obtain (4.4.4). Condition (4.4.5) immediately follows from (4.4.6). Conversely, ifu{x,y,t) solves Cauchy problem (4.4.4), (4.4.5), then along the characteristic 7(rco, 0:1(2:0,0, i)) : у — y{x,xo,to) we obtain du{x,y,y')/dx = -2?;-Vl + (y')^ and, therefore, r-xi(xo,0,f)

Г J xo

v/1

+ {y'Yv-^ dx

= / _ ( x o , ίο) = / = r о

ψι

since /-I- = 0. Assuming 0; therefore, from (5.1.6), (5.1.7) we obtain />/3-1/4! >5

Γ Jo

{ψ" - ο\ψ'\)\\υ\\^ dt-

о

.

(5.1.8)

Returning in (5.1.8) to к = e^'^'v and using the condition φ" - €\ψ'\ > О we obtain estimate (5.1.4). Now let estimate (5.1.3) hold. By the identity 2 R e {v'Av)

= dt R e ( Α υ , ν) + R e ( ( A - Α*)ν,

υ) - R e {υ,

Α'υ)

we have h =-Re{Av,v)^-Re

0

[ Jo

{{A-Α*)ν,ν')

dt + Re

f

Jo

{v,A'v)dt.

(5.1.9)

Α. L. Bughgeim.

102

Volterra equations and inverse

problems

Evaluate the two lattet integrals, where by с we shall denote two, generally speaking, different constants depending on с from estimates (5.1.1)-(5.1.3). We have {V,A!V) \ < ε||υ||· ΙΙΛυΙΙ < с||г)||(||Лг; + βφ'ν - 8ψ'ν\\ + ||υ||) < φ | | ( | | ( Α + ν ) ^ | | + (1 + δ|^Ί)||4Ι)

Here, for estimate с||г»|| ||(Л +

we have used the inequality

slbll ||M + 5|| < -\\vf + α ΐ ΐ μ + 8φ')υ\^ a with a sufficiently small α > 0. Analogously, we have (μ

- A*)v,v')\ < elicli \\v' - ìBv + iBv\\ < C||î;|| (Цг;' - ÌBv\\ + ||Λυ||) < c(i + .

)\\vf + ^Ιΐμ

+

+ \\\ν' -

iBvf.

Adding the obtained estimates and recalling the definition of integrals II, I2 from (5.1.9) we obtain /5 >

-

^ с j ^ l + 3\φ'\)\\ν\\'dt.

(5.1. 10)

Finally, as |2Re(iSî;,

= |({(А* - А)В -{В< cllHI ΙΙΛ^ΙΙ /1 + /2 + -^з + - l-^el- Hence, combining the estimates (5.1.6), (5.1.7), (5.1.9)-(5.1.11) and taking into account ψ" -ο\ψ'\ - c/s > 1/ci for sufficiently large ci, sq, s > so, we have ¡•τ ||i;f Cl Jo

d

Returning to the function и = e (5.1.12) we obtain (5.1.4).

i

-

+

J· о

(5.1.12)

and neglecting those /1, /2 > 0 from

Chapter 5. Abstract integro-differential equations

103

Now we can prove the estimate (5.1.5). Since by (5.1.3) we have

then

holds. On the other hand, (5.1.12) yields h 0,

s^^TAl^s < holds. Ιίφ' 0 exists that for all s > 0, и Ε СЦ[0,Т]·, Η) η С{[0,т]·, Н^) the estimate + holds. If we assume that φ' ccoias)-^^

110

Α. L. Bughgeim. Volterra equations and inverse problems

therefore, from the two previous estimates we obtain

О, ί e [0,Т]. Then, for each и G D(P), the estimate

+

4P4T,S

holds true. Here ν = (dt - у/Щ^ - i \ / - ^ r ) t ¿ , s > sq, sq being sufRciently large. Proof. Proof may be achieved by the apphcation of Lemma 5.1.1 to the operators Pi and P2 from (5.4.3), (5.4.4) and taking into account estimates (5.4.5), (5.4.6). Since the operators y/±A± are self-conjugated, commutate and do not depend on i, then, in the condition φ"+α \ψ'\ > О of Lemma 5.1.1, we may assume α = 0. • Theorem 5.4.1 and Lemma 5.1.2 analogously to Theorem 5.2.1, yield Corollary 5.4.1. Under the Theorem 5.4.1 conditions, the inequality \\Pu\\0

is solved uniquely. Example 5.4.1. Consider a nonhyperbolic Cauchy problem utt=0u

+ B{x,t)ut + N{x,t)u,

i>0

u(x,0) - ΐ χ ί ( χ , θ ) = 0 . Here • η = - A^/u, x' = ( x i , . . . ,Χη-ι), ^x' is the Laplace operator by the variables x'; B{x,t), N{x,t) are bounded functions. Corollary 5.4.1 implies that u = 0.

5.5.

OPERATOR VOLTERRA EQUATIONS W I T H C O M M U T I N G KERNELS

In this section we shall prove the uniqueness theorem for solving the operator Volterra equation of the first kind {Au){t)=

í\(t,T)u{T)dT

Jo

=

f{t),

te[0,T].

(5.5.1)

Chapter 5. Abstract integro-differentíal equations

119

Here A(t, τ) is a family of normal commuting operators in a Hilbert space Я , u(t) and f(t) are functions of the scalar argument t with values in H. The applied approach is based on the full spectral theorem of Neumann and the general theory of decomposition by eigenfunctions. Our theorem, in particular, yields some of the statements of Romanov (1972) concerning the uniqueness of solving integral geometry problems. At the same time, operator equations of the type (5.5.1) are of specific interest. In contrast to Chapter 3, the method used here may be applied for investigating uniqueness problems for integral equations (with operator kernels) not necessarily of Volterra type. We must begin with some definitions and auxiliary facts. Let Ω be an open domain in R" and A{t), ί e Ω be a family of normal commuting operators acting in a separable Hilbert space Я. By the full spectral theorem of Neumann (see, for example, Moren, 1965), there exists a unitary map F of H into the direct integral of Hilbert spaces H: F:H^H

= J Ν{Χ)άμ{λ)

(5.5.2)

Fu = û = (й(Л))д£д {и, υ)^ =

ί (ΰ(λ), ί

d/i(A)

(5.5.3)

JA

such that {FA{t, X)u) (λ) = a{t, λ)ΰ(Α),

VA G Λ

where α(ί,τ) is a complex-valued (//-measurable by A) numerical function, Λ is a locally compact space with a positive measure μ, H{X) is a family of Hilbert spaces. Let X and Φ be Hilbert spaces imbedded in Η and dense in it. We shall say that A(i) e

Ζ Φ ) ,

α = (αι,...,α„),

ί = ( ΐ ι , . . . ,ί„)

i{A{t) has strongly continuous from X into Φ derivatives D^A{t) G L{X, Φ) V/3 < α (i.e. ßj < aj, j = M ) . Here D = {Di,.. Dj - d/dtj- aj, ßj being integers. Further, let C°'{Q,) be a space of complex-valued functions having continuous derivatives of order β, "iβ < a. Set Λι = {A G Λ I й(А) = 0 Vu G X}.

120

Α. L. Bughgeim. Volterra equations and inverse problems Lemma 5.5.1. μ(Λι) = 0. Proof. Let μ(Λι) > 0. Choose an element ν = 'l,

and take ν —

so, that

λ6Λι λ€Λ\Λι

Then, from (5.5.3) and the determination of Λι follows {u,v)fj = 0 \\ν\\Η =

УиеХ μ{Α,)>0

which contradicts the equality X — H. The lemma is proved.

•

Lemma 5.5.2. If A{t) 6 X Φ) and the imbedding operator I : Φ ^ Η is a Hilbert-Shmidt operator, then a{t,X) e С"'(Ω.) for almost all λ by the measure μ (i.e. for all λ G A\Ao, where μ(Αο) = 0, independent oft). Proof. Define a map F{X) by the formula F{X)

ψ{λ) = {Ρψ){λ) G Η{λ)

where F is from (5.5.2). For each λ € Л\Л2, //(Лг) = О, F{A) is a HilbertShmidt map (since / : Φ Я is a Hilbert - Shmidt map, see Moren (1965)) and, in particular, is continuous, i.e. F{X) Ε Ь{Ф,Й{Х)) for λ G Л \ ЛгConsider operator Α{ί,τ) = F{X)A{t). As A{t) G ^ " ( Ω , Χ -)· Φ), F{X) G Ь{Ф,Н{Х)) for λ G Λ \ Λ2 and Λ2 not depending on t, then A{t,T) G Χ -> Я(Л)), λ G Л \ Лз. But A{t, Х)и = α(ί, λ),

ώ(λ), иеХ

(5.5.4)

hence, by Lemma 5.5.1, α(ί, λ) G C°^{ü) for λ G Л\Ло, Ло = Л1ПЛ2. Indeed, prove, for example, the continuity of a{t,X). Prom (5.5.4) and Α(ΐ, λ) G С" (Ω, Χ Η (Χ)), λ G Λ \ Λο it follows that II {Ait + At, Χ) - Ait, Л))и||

= \ait + At, Χ) - ait, λ)| ||ΰ(λ)||

О

for Δί —> О Vu G Χ. Therefore, if we choose и such that ΰ(λ) (by Lemma 5.5.1 such an element exists if λ ^ Λι), then |α(ί + Δί,λ) — α(ί,λ)| —> Ο for Δί —>• Ο and the continuity of the corresponding derivatives of a{t, λ) is proved. Analogously, the existence and continuity of the corresponding derivatives of a is proved. The lemma is proved. •

Chapter 5. Abstract integro-diíFerential equations

121

Remark 5.5.1. As in the fundamental theorem on decomposition by the eigenfunctions (see Moren, 1965), we may assume that Φ = lim indЯ„ (the inductive limit), where are linear subspaces of H allotted by such pre-Hilbert structure, that the imbedding In • Hn ^ H is a, Hilbert - Shmidt operator Φ — H. Remark 5.5.2. The Hilbert structure of X is non-essential. Return to equation (5.5.1). Let Ω = {(ί,τ) G | О < τ < ί < Τ}, Ω be the closure of Ω. If Α{ί,τ) G C¡r{Ü, H H) and A{t,t) has an inverse bounded operator. Vi G [О,Τ], then, applying to (5.5.1) operator R — t) d/ di, we obtain the equation u{t, T) + Γ B{t, τ)η(τ) dr = f i t ) Jo B{t,T)

^

(5.5.5)

A-Ht,t)A[{t,T)

and, since \\B{t, т)\\н is uniformly (by t and r ) bounded, then (5.5.5) may be solved similar to a scalar second-order Volterra equation and the corresponding theorems of existence and uniqueness are valid (see Section 2.4). However, for some problems (for example, for the integral geometry problems, considered in Romanov (1972)), either Α[{ί,τ) ^ L{H,H), or A-^{t,t) ^ L{H,H) and the above considerations are not valid. Partially, the answer on the question of the solution uniqueness for equation (5.5.1), in this situation, gives the following: Theorem 5.5.1. Let Α{ί,τ) satisfying conditions (1). A{t,T)

G

Χ

^

be a set of normal commuting

operators,

Φ), Χ С H, Χ = H and Φ satisfies the

conditions of Lemma 5.5.2 or Remark 5.5.1 (usually X С Ф); (2). A(t,t) is a Hermite operator and A(t,t) > С > 0 Vi G [О, Τ], where с G L{H, H) is a Hermite operator not depending on t and commuting with the family Α{ί,τ) (for example, С = εΕ, ε > О, E being a unit operator; this denotes that the operators A(t, t) have a bounded inverse operator). Then equation (5.5.1) has no more than one solution u(t) G Z(2([0, Г]; Я ) .

122

Α. L. Bughgeim. Volterra equations and inverse problems

Proof. Applying to (5.5.1) the map F , we obtain the family of first-order Volterra equations with the scalar kernel rt a(t, T, λ)ΰ(τ, λ) dr = f (t, λ). (5.5.6) Jo/ 0

f

Due to condition (1) and Lemma 5.5.2, α(ί, r, λ) G C¿ f ( f í ) for almost all λ by the measure μ. As С > 0, then C{X) > 0 almost everywhere by the meaasure μ, and if we prove that α(ί,ί,λ) < с(Л),

te [О,Τ]

(5.5.7)

for almost all λ by measure μ, then the formulated theorem of uniqueness follows from the corresponding theorem of solution uniqueness for the first-order Volterra equation with the scalar kernel. To prove (5.5.7), it is sufiicient to verify that the sets Λ_(ί) = { λ 6 Λ | α ( ί , ί , λ ) - ε ( λ ) О,

TER^,

ЛеЛ\Л_

Л - independent of t. Then, due to the continuity of a{t,t,X) in t, this inequality is valid Vi G [0,T], λ e Л \ Л_. But Λ_(ί) = U~=i ^n(í), Λ„(ί) = {λ e Λ I a{t, t, λ) - с(Л) < - 1 / η } . Therefore, it is sufiicient to show that μ(Λ„(ί)). Conversely, let μ(Λη(ί)). Choose й(й(Л))д^д, so that |й(А)|1я(А)

1,

λ 6 Λ„(ί)

О,

λ€Λ\Λ„(ΐ)

Then, и = F ^û e H and ||и||я = μ(Λ„(ί)). On the other hand, from (5.5.3) and the definition of Λ„(ί) we obtain {iA{t,t)-c)u,u)

= [ {a{t,t,X)-c{X)·) JAnit) < -n-V(A„(i)) < 0

||ΰ(λ)||2^

d/i(A)

which contradicts the condition (2) of our theorem. So, μ(Λ„(ΐ)) = 0. Hence, the theorem is proved. •

Chapter 5. Abstract integro-differential

equations

123

Remark 5.5.3. Uniqueness and stability problems for solving equation (5.5.1) may easily be investigated by means of the representation of the inverse operator for (5.5.6). Remark 5.5.4. Theorem 5.5.1 is valid also for the equation with a weak singularity of the form [ ^ ( í , r ) u ( r ) ( í - r ) - " d r = /(i), Jo

0 О, we may consider that Л = (0, oo). Applying F to the equation Au = / we obtain the family of the scalar Volterra equations i\t Jo

- τ)λ)ΰ{\, τ) dr = fix, t),

λ > 0.

(5.5.10)

Α. L. Bughgeim.

124

Volterra

equations

and inverse

problems

Using the Laplace transformation roo

{Lu)ip)=

e-P^u{t)dt

Jo

equation (5.5.10) we solve explicity. Indeed, if A;(i) =

J^(iÀ), then

where c^ == + 3/2), Reiv > - 1 , ρ > 0. Equation (5.5.10) is an equation of the convolution type k * û = f , и = η / 2 — 1, where functions k, й are continued by zero for negative t. As L { k * u ) = ( L k ) • ( L u ) , then

{LÛ){X,p) = c - ' p - H p ' +

(5

5Д1)

Let η be odd. Then m = (2г/ + 3)/2 is an integer. By the identity

and the fact that LJ^u = p'^Lu, LD"/ = for /('=)(0) = 0, к = 0, n - 1 , and formula (5.5.11) after applying to it the operator we obtain +

(5.5.12)

Now let η be even. Using the formulae

+ д2){2.+3)/2 ^ ^„+1(1 ^ д2р-2)п/2+1(р2 ^ д2)-1/2 analogously to the previous working we obtain from (5.5.11)

Ht, λ) =

+ X'jY'^'

Jo{X{t

- τ ) ) Ό ^ ' Ϊ { τ , λ) dr. (5.5.13)

Formulae (5.5.12), (5.5.13) and the fact

Jo{X{t

- r ) ) = A-V2(i

- r)-i/V(A(i - r))

where φ is a bounded function, φ{0) φ О yield the theorem statement for Α ι { ί , τ ) = 0. In the general case, converting by means of formulae (5.5.12), (5.5.13) the principal part of A corresponding to the first term in (5.5.8), by

Chapter 5. Abstract integro-differential equations

125

estimate (5.5.9), we come to the operator Volterra equation of the second order и + Bu = fi with the operator kernel B{t, τ), satisfying the estimate

Prom the results of Chapter 2, it follows that {E + is bounded in Lp([0,r]; H^). Therefore, Theorem 5.5.2 is vahd in the general case. • Remark 5.5.5. It is easy to show that the following multidimensional integral equation of the first order [ 4

θ { Ί - τ - \ Χ - ξ \ ) o ( x , ξ, t, T) Η{ξ, τ) άξ d r = t{x, t)

Jo JR" may be reduced to the equation considered in Theorem 5.5.2. Here χ,ξ e Ш^, 9{t) is the Heaviside function. As an example of an integral equation with an operator kernel of the non-Volterra type, consider the following integral geometry problem. Example 5.5.1. To find a function и{р,т), ρ > 0, τ e [О,Τ] by its integrals along the curves T{r,t) with the weight αο{ί,ίφ{τ)): /

Jr{r,t)

αο{ί,ίφ{τ))η{ρ,τ)άτ

=

f{r,t),

r>0,

¿е[0,Го

Here r ( r , i ) = {p^t \ ρ = r a i ( í , τ 6 [О,Τ]}, function αϊ > ci > О, the number ci is independent of i, r, and the system of functions {ψ^{τ)} is complete in ¿2(0,Τ). Consider this problem in the operator form Au=

f

Jo

A{t, T) U{T) dr = f{t)

t E [0, Τ]

A{t,T)v{r)=ao{tMr))v{rai{tMr)))

(5.5.14) (5.5.15)

υ £ Η, where Я is a Hilbert space with the norm roo

1Н1я=/

Jo

Иг)|^г-Чг.

We seek the solution to (5.5.14) in ¿2([0,Т];Я). It is easy to see that A{t, T) form a normal commuting family of bounded operators in H. The

126

Α. L. Bughgeim.

Volterra equations and inverse problems

map F : Η H, Η = L2{-oo,oo), case, is the Mellin transformation {Fv){X)=v{X)

=

diagonalizing operators Α{ί,τ),

in our

1

^ \/2π Jo

1 F-^v = v{r) = - =

r°°

v{\y^dr

V 2π J-oo

(A(t^v)(A)=a(t,T,X)v(X) a{t,T,X) =

αο{ί,ίψ{τ))α[^{ί,ίφ{τ)).

Theorem 5.5,3. Let aj{t,u) G in a neighborhood of the point (0,0), aj(0,0) = 1, ; = 0,1, m = Βηαι{0,0) φ 0. Then, solution to equation (5.5.14) is unique in 1,2([0,Г];Я). Proof. Let Au = 0. Then rT

[

ao{t,

ίφ{τ))ΰ{Χ, τ) d r = 0.

(5.5.16)

Jo

Applying to (5.5.15) the operator D^ and setting í = 0, we obtain, by the induction method Ρη(λ) /

ΰ{Χ,τ)φ'4τ)άτ

= 0,

η = 0,1,2...

(5.5.17)

Jo

where Pn{X) — (imÄ) + . . . is the n-th degree polynomial evaluated explicitly. As the set of all real zeros of is no more than countable, then, for almost all λ G R ^ from (5.5.16) we have ΰ(λ,τ) ξ 0 in ¿2[0,Г]. Consequently, u Ξ 0 and the theorem is proved. • In the particular case φ{τ) — c o s t , τ G [О,π], when functions aj are analytic in a certain neighborhood of zero, the solution u{p, τ) is continuous and satisfies in this neighborhood the Holder condition. Theorem 5.5.3 was proved by another method by Romanov (1972, p. 14).

Chapter 6. Multidimensional Inverse Problems In this chapter we shall establish uniqueness theorems to determine coefficients or a right side of a differential equation by given traces of the solution. The method of proof is based on the receiving of a priori estimates with the weight taking the maximal value on those manifolds, where the traces of solution are known. Thus, the uniqueness of inverse problems is proved following one and the same scheme (according to Carleman) as the Cauchy problem uniqueness. Our presentation is based on the work of Bukhgeim (1981b). Similar results were obtained independently and simultaneously by Klibanov (1981). The results of the above works were published in Bukhgeim and Klibanov (1981).

6.1.

INVERSE PROBLEMS, COMMUTATORS AND A PRIORI

ESTIMATES

The first (standard) step, when investigating the uniqueness and stability of inverse problems, is to reduce the initial, generally nonlinear problem to a hnear problem. In the general case, this as achieved by Theorem 1.6.1. As a result we obtain a linear problem for determining a pair of functions u, f from the conditions Pu^f,

Qf^g

(6.1.1)

Here Ρ is an operator of the 'direct' problem, Q is an 'informative' operator describing the law of the right side variation; ^ is a given element, u, f are described elements of the corresponding functional spaces. Applying the

128

Α. L. Bughgeim.

Volterra equations and inverse

problems

operator Q to the first equation (6.1.1) we obtain QPu — g, or Pqu = [P,Q]u + g

(6.1.2)

where [P, Q] — PQ — QP is the commutator of Ρ and Q. The sense of commuting is in the following. As a rule, apart from the fact that Qf = g we know nothing about / ; therefore, it is easier to investigate Q on the direct problem solutions u, which satisfy the corresponding boundary conditions. Besides, in the typical applications, the operator Q does not spoil the part of boundary conditions which describe the domain of operator Ρ definition. As a result we obtain original factorization of the inverse problem (6.1.1) into the product of two 'direct' problems generated by Ρ and Q. In particular, in the trivial case [P, Q] = 0 and the initial problem falls into two simpler problems Pv = g and Qu = υ. Equation (6.1.2) shows that the investigation of uniqueness and stability of the inverse problem is reduced to the investigation of the properties of the operators P, Q, [P, Q]. Usually these properties are described by means of certain a priori estimates. We shall illustrate the above concept by the following examples. E x a m p l e 6.1.1. In the domain Ω = {x, ί | a; G (0, T), |i - T| < Γ - ж} we need to determine the smooth functions u{x, t), a{x) from the conditions utt - Uxx - a{x)u = 0, u{0,t) = h{t), u{x,x)

u^i0,t) = l/2,

{x,t)eO.

= 0,

ie[0,2T]

xe[0,T]

(6.1.3) (6.1.4) (6.1.5)

Problem (6.1.3)-(6.1.5) generates the nonlinear operator equation A{u,a,h) = 0 relative to an unknown pair u, a (the function h is given). Assuming that we have two solutions ( u i , a i ) , («2,02) which corresponds to the function h, acting by the Theorem 1.6.1 scheme we obtain the linear problem for the functions и = щ — U2, a = ai — a2: Pu = utt - Uxx - a2U = uia = f u(0,t) = 0,

Ux{0,t) = 0,

и(з;,а;)=0,

t e [0,2Г]

χ e [0,Т].

(6.1.6) (6.1.7) (6.1.8)

The right side / = ui{x,t)a{x) satisfies the equation uif¡ - u[^f = 0. As ui{x,x) = 1/2, then, for Τ sufficiently small, by the continuity of ui 0 in Ω and, consequently, Qf = {Dt-i

Inni),)/ =

Α ( « Γ V ) = 0.

(6.1.9)

129

Chapter 6. Multidimensional inverse problems Let 11^t=

Σ

ΙΙ^^'^ΙΙμω).

D =

{Dt,D,).

\a\ 0 and the solution u{x, y, t) of the equation auu — Uxx — AyU = 0, assuming that и is even by x,

132

Α. L.

Bughgeim.

Volterra

equations

and

inverse

problems

u{x, y, 0) = 0, ut(x, y, 0) = Je (a;, y) {δ^ being the delta-like function with support in the ball of radius ε). As additional information we have the function u(0,0, t ) = h { t ) . In this c a s ^ = {Qo, Q i , . . . Q„), Qof = P A Í P " V ) = 0, Q j f = p D y ^ { p ~ ^ f ) = 0, j = l , n , ρ = D^uiPassing as in Example 6.1.1 to the linear problem we obtain PQjU — [ P , Q j ] u , j — 0, n. For j = I , η the principal part of the equation QjU — P ~ ^ [ P , Q j ] u = 0 is D y . u . By means of this fact, the information on the solution ιι(Ο,Ο,ΐ) = 0 is extended on the plane χ = 0, and in the equation PQqu = [P,Qq\u the derivative by y are eliminated. As a result we obtain the oneOdimensional problem with the parameter y. Repeating, for the most part, the considerations of Example 6.1.1, we obtain the uniqueness theorem in the whole. The application of such approach to the initial nonlinear problem results in the local existence theorem. This problem was also investigated by Yakhno (1980) using other methods. Remark 6.1.2. The results of Example 6.1.3 were derived by Mukhometov (1977). The problem of recording the function f{x) in the domain Ω by an arbitrary integral along the vector field и trajectories were reduced to the problem in this example. We shall now formulate the results of Hörmander (1965) on the weight a priori estimates for the general differential operators. We shall begin with the definitions. Let Ω be a bounded opened set in K^, ψ he Ά real function of the class

\a\ 0 exist that for all и G C o ^ ( f i ) , r > To the estimate + ar-'\\u\\l^

< c\\Pu\\l,

(6.1.19)

holds. Theorem 6.1.2. Let the principal part of the operator Ρ have real coefficients and Vip{x) Φ 0 for χ e Ci. Assume, that Φ(χ,ζ,τ) > О, where ζ = ξ+ίτ Vip(x), о ^ τ e χ e й, ξ e W and ζ satisfies the characteristic equation Pm(x,() - 0. Let, besides, Ф(а;,^,0) > 0 if χ e Ü and 0 ^ ξ e M" satisfy the equations Ρτη(χ,ξ) = 0, (νςΡηι(χ,ξ),νφ} = 0. Then such constants c, To > 0 exist that for all и G τ > tq the estimate t I I u I I ^ , < c\\Pu\\l,

(6.1.20)

holds.

6.2.

T H E O R E M S OF D E C O M P O S I T I O N

Let us return to the problem (6.1.2) in the general statement. Let Ω be an open bounded set such that Ω = Ω П {(^ > 0}, where í/j G C°°(]R"); P , Q are

134

Α. L. Bughgeim. Volterra equations and inverse problems

linear diíTerential operators of the order m and I correspondingly with coefficients from (7°°{Ω). Assume that Γ = 0Ω \ {(/? = 0} and s С Ω are hypersurfaces of the class C°° and a solution и to the equation PQu = [P, Q]u -b g belongs to and satisfies the following boundary conditions: =

k = 0,m-l·

k = Ö J ^

(6.2.1)

where d¡j is the operator of differentiating along the unit normal to Г or S correspondingly. We shall say that Ρ and Q commute in the main if the order of [P,Q] is not greater than m + I - 2. Set C^^ = Π {u | d^lgu = 0, k = Одй}. Theorem 6.2.1. Let such numbers c, TQ > 0 exist that for each τ > TQ the estimates Vu e < c\\Pu\\l, (6.2.2) Vn e

\\u\\lm+l-2
ìp > 0, г/) ξ 1 for φ > s, φ = О ίοι s' > φ > О, where О < s' < s. After the closure procedure for the function фи, we may apply estimate (6.2.3). Analogously, due to the fact that Г is characteristic relative to Q, we may apply estimate (6.2.2) to Q^u. As PQu = [P, Q]u, then фи satisfies the inhomogeneous equation PQφu ^ [Р,Я]фи + f , where / = [φ, [P,Q] - PQ]u. Applying estimate (6.2.3) to the function фи and (6.2.2) to the function Qφu we have < с\-'\\РЯфи\\1, ^ c^r-'\\[P, Q]φu + fWlo TQ less than BYomhere (6.2.4) As φ = 1 ΐοτ φ > s, then f = 0 if φ > s and, consequently maximum of the weight ψ in the norm ||/||^o i® equal to s, i.e. ',0 < e^"||/|lL(n).

Chapter 6. Multidimensional inverse problems

135

Notice that the norm \\фи\\1 in Ω is greater than this norm in Qs = Ω η {(^ > s}, minimum of the weight is equal to s. So, we obtain е'^'ИЦш

< \\Ф 0 exist, that for τ > TQ the estimates Vu e < c\\Pu\\l, Vn G

m Q M l , < c\\Qu\\l^_,

(6.2.5)

hold. If Г is characteristic relative to Q and ker Q — {0} on the functions и e satisfying conditions (6.2.1), then the problem (6.1.2), (6.2.1) is uniquely solved. In the general case, [P, Q] is in operator of the order I + m — 1 and, therefore, the norm ||[P,Q]|| = sup ||[P, (5]u||t,o is finite for = 1, r >0. Theorem 6.2.3. Let such constant c, tq, α > 0 exist that for all τ > TQ, the estimates Vu e Co~(Ω)

r\\u\\lm-l

+ « ^ " l ^ l l ' m < 4Pu\\lo

(6-2.6)

< cWQuWl^

(6.2.7)

Vu e C^i^ra-1 hold. If Γ is characteristic relative to Q and 2c^[P,Q]f

1.

estimate

(6.2.9) holds,

where

с

is

independent

o f u,

τ,

Ü,

Τ

=

s u p |x„|

f o r ж G

Ω.

Proof. Let Qu = v. Then u{x) = e-'-i^) r^ rXn b { x ) =

Jo

t) dt,

Jo

χ GΩ

a { x ' , t ) d t

and, therefore, \ D ' ' u f < (

V

Y ]

β О is an integer. Further, let (V(a, V 0 exist, that for all τ > TQ, и e Cq^^ the estimate r\\u\\l^ < cWQuWl^.

(6.2.11)

Proof. By the standard methods (see Hörmander, 1965, Chapter 8) Theorem 6.2.4 is reduced to the particular case m = 0, Q = (a, Vu), the domain Ω being small. Because of the invariance of the theorem conditions relative to a change of variables, we may assume that α = ( 1 , 0 , . . . , 0). As a result, Theorem 6.2.3 is implied from the following lemma. • Lemma 6.2.2. Let φ G С'^{[-Т,Т]), where ψ"{ί) < О for φ'{ί) = 0. Then Vv G С^(-Т,Т) such that v{t) = О for φ'{ί) = О and for sufficiently large To the estimate
TO

holds. Proof. The localization and the change of variables reduces the lemma to the case φ(ί) = —í^, i.e. actually to the double application of the first Theorem 2.3.3 estimate (for t G [0,Τ] and t G [ - T , 0]). • Remark 6.2.3. The condition (V(a, Vt/?), a) < 0 is opposite to the condition which is necessary for estimate (6.2.11) validity Vu G ^^¿"(Ω) (see Hörmander, 1965, Chapter 8).

6.3.

E X A M P L E S OF I N V E R S E P R O B L E M S THAT A D M I T DECOMPOSITION

Let us begin with an example of the general form. Example 6.3.1. Consider two boundary-value problems PjUj^

aa{x)D''uj+aÌ^D''U2)~^Pi, we have Ζ)„Ρκ = 0. So, in this case, under the additional condition = 0, к = 0,m — 1, the solution to

140

Α. L. Bughgeim. Volterra equations and inverse problems

the inverse problem is unique without the assumption of the independence of Pm on Xji- However, for m > 1 this statement is over deter mined. Now consider the problem of determining all or some of the coefficients of the second-order classical operators. Example 6.3.2. Let Ω be a bounded domain in R" with a boundary Γ of the class C°° situated locally at one side of Γ. Consider the two boundaryvalue problem 4 + Lju^ = 0, иЦх,0)=д{х),

Here Lj =

X en, и{{х,0)=0,

ΐ 6 [0,Τ] χ

(6.3.5) en

is a uniformly elliptic operator

>

О О, ξ e üa e С{й), where 4 = 4 α G Ai{a \ |a| < 2} \ A2, A2 Ç {a \ |q;| < 2} is a certain nonempty multi-index subset, u^ e χ [о, τ]), g e C^(fì) are function vectors of the dimension IA2I, 1^2! being tne number of the elements of A2; ν is the unit normal vector to Γ. Denote by B{x) a matrix of the order \A2\ x IA2I, columns of which are the function vectors a e A2· Theorem 6.3.2. Let detB{x) Γ X [0, Т]. Then fli - al for Va.

0, χ e Ù and u^ ^ u^ for {x,t) 6

Theorem 6.3.2 may be proved analogously to Theorem 6.3.2. For this purpose we have to continue the function u{x, t) evently into domain ί < 0. The corresponding estimates for Ρ = Dj + Li are proved in Theorem 6.1.2. Remark 6.3.1. If in (6.3.5) we substitute ul^ by —ul^ (the hyperbolic case), then for this problem the theorem analogous to Theorem 6.3.2 holds under the following conditions. Number Τ = Τ(Ω) is to be sufficiently large, the sucface dO. x [0, T] and similar to it imbedded into Ω χ [О, Τ] cylinders are strongly pseudoconvex in the sense of Hörmander (1965) relative to the operator Ρ = D^ — Li. In other words, for these domains and the operator Ρ the estimate (6.2.2) holds. If in (6.3.5) we substitute u^j by (the parabolic case) and condition (6.3.6) by ω^(χ,Τ') = g{x), 0 0 defines the smooth strongly-convex surface. The inverse problems of determining coefficients of hyperbolic equations, invariant relative to the group of shifts or rotations and equation Pu — f connected with them were investigated by Romanov (1972). Without assumptions on the invariance, the solution uniqueness for the equation Pu = f in the class of piecewise-analytic functions was proved by Anikonov (1978). In this chapter we obtain the necessary and sufíicient conditions, when Pu — f has a finite-order regularizator. We also give the classification of the equations by their stability, for the uniqueness theorems are proved.

144

Α. L. Bughgeim.

Volterra

equations

and

inverse

problems

The basic results for this chapter are in Bukhgeim (1975 a,b,c; 1978 a) and in Lavrent'ev and Bukhgeim (1973). 7.1.

STATEMENT OF THE PROBLEM

First, let us recall the definition of the Dirac function, concentrated in the smooth surface in M". Let a function vector p(x) = ( p i { x ) , . . . ,pk{x)) belong to C°°(K") and vectors Dpj, j = l,k, к < nbe linearly independent for ρ = 0. Then the equation ρ = 0 determines a nondegenerate surface of the dimension n —/г. Set \Dp\ = {det{Dpi, Dpj))^/"^, where (·, • ) is the scalar product in E". From the geometrical point of view, is the parallelepiped volume stretched on the vectors Dpj and, therefore, is not zero for ρ = 0. The generalized Dirac function δ{ρ) ξ δ{ρι)δ{ρ2)... ¿(pfc) ^.nd the Heaviside function θ(ρ) = θ{ρι)... 9{pk) are determined by the formulae =

(ГЛ.1)

(7.1.2)

ί θ { ρ ) η ά χ = ί udx. J Jp>0

Here и e άσ is a EucUdean element of the surface ρ = 0 area. The notation ρ > 0 denotes that Pj{x) > 0, j = l,k. The derivatives with respect to ρ of θ{ρ) and δ{ρ) are determined so that the ordinary formulae of differentiation hold true. In particular, if I = (Zi,... ,lk) is a, multi-index and δ^^Hp) =

...

=

Ό^δΙρ)

holds, then, the formulae DJδ(p{x))-^ΣS^^^{p)DJpι

(7.1.3)

l¡|=i

must hold. Here

Dj

= d/dxj^

pi

=

{l,p)

= Pk{x)

for Í = (0,..., 1 , 0 , . . . , 0). к

Multiplying both parts of the assumed equality (7.1.3) by Djp mi ( m i , . . . , mjt), |m| = 1 and summing by j from 1 to η we obtain {Dpm,Döip))

-

· {Dpm,Dpi). \1Ы

—

(7.1.4)

Chapter 7. Multidimensional integro-differential equations

145

As the vectors Dpj, |/| = 1 are linearly independent for ρ = 0, then their Gram determinant det(Z)pm, Dpi) is not zero for ρ = 0 and system (7.1.4) is uniquely solvable relative to which are to be determined: def

Integrating this formula, we obtain for every I ς |α|=μ|

(7.1.5)

where the coefficients CQI are uniquely determined by means of ρ and its derivatives of the order The derivatives of the Heaviside function with respect to ρ are determined analogously. In this connection we obtain that for / = ( 1 , 1 , . . . , 1) the equality = δ{ρ) holds. These definitions also yield where α is a scalar function from C°°(M"), α > 0. For integrals containing the generalized functions δ{ρ), θ{ρ) and its derivatives the following formulae hold: integrating by parts, changing variables, differentiating by the parameter under the integral sign, the change of the integration order. In this case we need only the linear independence of function gradients entering in Θ, the sufficient smoothness and compactness of supports of corresponding functions (or the rapid decreasing in infinity which provide the convergence of the corresponding ordinary integrals). By definitions (7.1.1), (7.1.2), (7.1.5) the proof of these statements is reduced to the corresponding theorems for the integrals from smooth functions. Furhter we shall use these formulae unconditionally. Remark 7.1.1. For more detailed investigaiton of the properties of the functions δ{ρ) and θ{ρ) see Gel'fand and Shilov (1958a). Now, we shall pass to the problem statement. Let Ω and G be bounded opened sets in K" and correspondingly, щ > η, ρ, а^, к = —1,тЪе real functions from C°°{G x Ω), where Dyp φ 0 for p{x,y) = 0, {x,y) Ε G χ Ü. For и 6 assume m

(Pn)(:r)= J ]

,

/afc(x,y)¿W(p(x,y))K(2/)dy,

xeG

(7.1.6)

146

Λ. L. Bughgeim.

where

Volterra equations and inverse

problems

Ξ θ{ρ). Formulae (7.1.1), (7.1.2), (7.1.5) yield (7.1.7)

for each integer s > 0, m > —1, s + m > О, where ||· ||s is a norm in C^{G) or in correspondingly. (For и G (7^(Ω) we assume that \\u\\s = sup|D'^n(x)|, |a| < s, χ e Ω.) The prime object of this chapter is finding the necessary and sufficient conditions, where Pu = f has a regularizator R of the finite order (i.e. R G for the corresponding s and I), and also, when a priori estimate of the norm \/u e Mt\\u\\^ < cWPuW^^

(7.1.8)

holds. Here s, si, t are fixed nonnegative integers, ί > s, α G (0,1], Mt = {и G Со°°(0) I ||u||t < m}. Now we consider some examples of equations Pu = f . E x a m p l e 7.1.1. Let m — — 1, ni = n. Then the equation Pu = / is the multidimensional analog of the first-order Volterra equation. E x a m p l e 7.1.2. Let m — 0, ni = n, A_i Ξ 0, OQ = \Dyp\. Then, by (7.1.1), (7.1.6) Pu =

ηάσ = f{x) Jp=0

and we obtain the problem of recovering the function и by its integrals from the family of hypersurfaces Г(х) : p{x,y) = 0, χ e G. If we consider ρ as a function vector of dimension k, then for m = 0, ni = n, a_i = 0 we obtain the problem of determining a function и by its integrals with the weight oo by the hypersurfaces Г(х) of the dimension k. For 0 < A; < η such problem is called the integral geometry problem (the terminology of Gel'fand (I960)). This problem has been investigated by many authors (see John, 1958; Gel'fand, 1960; Lavrent'ev and Romanov, 1966; Lavrent'ev et al, 1969, 1980; Romanov, 1971, 1972, 1973; Bukhgeim, 1972b, 1973, 1974, 1975 a,b,c, 1978a; Garipov and Kardakov, 1974; Anikonov, 1976, 1978; Mukhometov, 1977, 1978; Uspenskii, 1977; Bernshtein and Gerver, 1978; Mukhometov and Romanov, 1978; Beil'kin, 1979; Gel'fand et al, 1980; and the bibliography cited there). E x a m p l e 7.1.3. Let v{x,x^,t)

be a solution to the Cauchy problem

DcV ~vtt - Av - c{x)v = g{x,x'^,t)u{x),

i>0

(7.1.9)

Chapter 7. Multidimensional integro-differential equations v(x,0)=0,

vt(x,0)=0.

147 (7.1.10)

Here Δ is the Laplace operator by x, c{x), g{x, x^,t) are given functions. To determine a function u{x) with a support in the ball |χ| < r by the solution V trace in the cylinder |x| = r, t E [Ο,Τ] for x^ = x: =

te [Ο,Τ].

(7.1.11)

The fundamental solution VQ for the operator Dc, i.e. the problem • vq = ¿(x — a;'')á(í), = ^ solution has a form υο =

ΖΤΓ

-

- x Y ) + a{x, x\t)e{t^

- |x -

(7.1.12)

where a{x, x^,t) is expressed by means of c{x). If 5 is a smooth function not depending on then, inversing by means of (7.1.12) the operator Dc and taking the trace of / we obtain that the inverse properties (7.1.9)-(7.1.11) is equivalent to Pu = f with m = —1, p{t,x,y) = t — \x — y\, t E [Ο,Τ], |a;| = r and the weight functions ao, a_i expressing in terms of g and a. If g{x,x'^,t) has a form analogous to (7.1.12), for example,

then, after simple transformations, we obtain the equation 10{t'-4\x-y\')u{y)dy

+ J a{x, y,t-\x-

7.2.

y\)e{t^ -4\x-

yf)u{y) dy = f{x, t).

NECESSARY CONDITIONS

Set S = define a set

= {1/ e M" I \u\ ^1},

u+{y) = {i^eS\i^

S+ = Sn{u

= Dyp{x,y)/\Dyp{x,y)\,

> 0}, η > 2. For y G Ω

χ Ε G,

ρ = 0}.

The geometric sense of is the following. Fix a point уо E Ω. and consider all the surfaces Г(а;) = {у G Ω | p{x,y) = 0} which contain уоThen the unit normal to Γ(χ) in уо (directed into ρ > 0) crosses out on the sphere S the set и+(уо). The closed set ш+(уо) may be considered as the multidimensional analog of the 'Volterra direction'.

148

Α. L. Bughgeim. Volterra equations and inverse problems

Definition 7.2.1. We shall write < S+ if in the sphere S = there exists a unit sphere of the dimension η - 2 such that ω+ η φ 0. Otherwise, we shall write > Remark 7.2.1. If ω-^. is connected, then < S-f. if and only if such rotation V of the sphere S exists, that С In particular, f o r n = 2 this condition is equivalent to μ(ω+) < π, where μ is the Lebesque measure in S^. For η = 1 it will be 5 = { - 1,1} and, by definition, we assume that < s+ = 0. Remark 7.2.2. The relation ω+ < of variables in the operator P.

is invariant relative to the change

Theorem 7.2.1. In order that Pu = f , where Ρ is defined by (7.1.6), has a regularizator of a fìnite order, it is necessary that > 5+ Vy e Ω. The following theorem shows that if k e r P = {0}, but in at most one point уо = Ω the set w+(yo) < then for in the well-posedness set Mt of the form 11 •i^llt ^ there is no exponential estimate of stability. Theorem 7.2.2. In order that (7.1.8) hold it is necessary that S+ Vu e Ω.

>

Remark 7.2.3. Theorems 7.2.1 and 7.2.2 are proved in Bukhgeim (1975c). They are valid also in the case when the coefficients Ofc(x, y) of Ρ are matrix and u is a vector. The examples of Ρ of the form (7.1.6), where k e r P = {0} but is no exponential estimate, are in Bukhgeim (1972b), Romanov (1972), and Garipov (1974). Howevere, in these works, the absence of the exponential estimate was not connected with the condition < 5+, which, on the one hand, shows its importance but, on the other hand, it easily verified. To prove Theorem 7.2.1 and 7.2.2 we need Lemma 7.2.1. Let ш+(уо) < y'^ G Ω. Then there exists a neighborhood Qy^ С Ω of the point y'^ such that Vy 6 Ω^^ the set < In other words, the relation ш+(у} < S^T^ is continuous in y. Proof. As w_|_(y°) < then there exists a unique sphere С such that Π w+(y'') = 0. Without loss of generality we may

149

Chapter 7. Multidimensional integro-differential equations assume that S " ' ^ = П {vi ^ 0}. Then the condition u;+(yo) < equivalent to the condition i),op(a:,yO)|>a>0,

Vx 6 Г ( у ° )

is (7.2.1)

where a is independent of ж. We must show that inequality (7.2.1) (possibly with the smaller constant a) holds for all у close to уо and χ G Γ*(у). Assume the contrary. Then there exists such sequence that = 0 (i.e. e r*(y'=)), y'^ ^ yO and 0,

Dy^.Pi^^y')

for

к ^ oo.

As x^ belongs to the compact set G, we may assume that x^ also converges to a certain point e G. Tending к to infinity, we obtain p{x'^,y^) = 0, D.,op(x,y) = 0, which contradicts (7.2.1). The lemma is proved. • Proof of Theorems 7.2.1 and 7.2.2.. Let (7.1.8) hold and such y° e Ω exist that w+(y°) < S+. Then, by Lemma 7.2.1, there exists a neighborhood Ωο of уо such that u+{y) < S+, Vy e Ωο. (7.2.2) Using the estimate, construct the function и for which (7.1.8) has failed. The function и we shall seek in the form λ > 1, fc > О

u{y) =

where φ e (^¿^'(Ωο), ι/* > О in Ωο, = 1 in a certain neighborhood Ω^ С Ωο of the point y^. For к = t and for the corresponding constant c, the value ||u||i < m and, besides, \\ul > (7.2.3) Calculate now ЦРиЦ^^. By condition (7.2.2) and, if necessary rotating the coordinate system we obtain for

\Dy,p{x,y)\>a>0

р{х,у)=0,

у € Ωο

(7.2.4)

In the formula which defines P, make the change of variables zi — p{x,y), = У2, · • Zji — Уп- For the chosen function и with the sufficient small support we have m ^ (Pn)(x)= Σ / a^(x,y(z))â('^(z:)cAIII.

„

X exp[iÀy„(z)]i/>(y(z)) ¡Dy^p(x,y(z))r^

dz.

150

Α. L. Bughgeim. Volterra equations and inverse problems

Integrating several times by parts and using (7.2.4) we obtain \\Pul^ < cnX-"

(7.2.5)

for each integer iV > 0. Substituting estimates (7.2.3), (7.2.5) into (7.1.8) and tending λ ^ oo for AT > ί - s we obtain a contradiction. Hence Theorem 7.2.2 is proved. Now, for (7.2.2), let the equation Pu = f have a regularizator of a finite order, i.e. RPu = u + Nu, ||iiPu||o < c\\Pu\\ for a certain s > 0. As N is completely continuous, then for a sufficiently small domain Ωο С Ω, the norm of the contraction of this operator into C(iîo) may be done less then 1/2; therefore Vu e 6'ο*'(Ωο) we obtain estimation ||u||o < 2c\\Pu\\s which contradicts (7.2.2). So Theorem 7.2.1 is proved. • Remark 7.2.4. If in (7.1.6), ρ is a function vector of the dimension I < n, then for the estimate (7.1.8) it is necessary that for Vy G Ω, the contraction of Dyp : E" ^ М', generated by the Jacobi matrix Dyp{x,y), into each ¿-dimension subspace has in a certain point of the surface p{x,y) the rank less than I. Example 7.2.1. (Application of Theorem 7.2.2 to the problem of the wave field continuation from the time-like manifold). Consider the following Cauchy problem: • w{x, i) = 0,

χ G R,

t e (О, Τ)

(7.2.6)

where G = Γ χ (Ο, Τ), Γ being a piece of a hypersurface in R" of the class C°°) / ь /2 be given. Set г(;(з;,0) = uo{x), wt(x,0) = ui(x). Then, by the Kirchhoff formula (n > 3, odd), we have

+ i y ¿(("-3)/2)(|:,_y|2_í2)^i(y)dy

(7.2.8)

and, therefore, the Cauchy problem (7.2.6), (7.2.7) may be considered as the problem of determining the initial data u = (uq, ui) from the equation Pn = f ,

/ = (/o,/l)

Chapter

7. Multidimensional

PQU

=

W\Q

integro-differential dw Pi = ^^

= /о,

equations

151

G

where P, by the formula (7.2.8), has the form (7.1.6), wherep = \x - y|^— (χ,ΐ) € G = Г X (О, Г ) . The set w+(y), in this case, is simply the 'survey sector' of the point у from the surface Г. Thus, Theorem 7.2.2 follows. T h e o r e m 7.2.3. In order that the problem of the wave fìeld continuation into a domain Ω' С Ω from a time-like manifold Γ has an exponential stability estimate, it is necessary that У у G Ω' the set со+{у) > 5-|-, where ы+(у) is а 'survey sector' of у from Г. When the surface Г has a convex directrix, then ω^ (у) > S for the points sufficiently close to Г. As was shown by Shishatskii (1973), in this domain Ω' we have an exponential estimate; therefore, the conditions of Theorem 7.2.2 can not be weakened. 7.3.

S U F F I C I E N T CONDITIONS

In this section we shall establish that under the condition u;+(y) ξ Vy 6 Ω and the corresponding conditions on the functions ρ and the equation Pu — f has a finite order regularizator, where for analytic /, p, Ofc the solution и is also analytic. Using this fact we prove the uniqueness theorems. The constructing of the regularizator is based on reducing the problem to the following first-order integral equation: {Ки)[х)

= / Jü

¡ — \x - y\

= /(ж)

a; e Ω = { x e K" I |ж| < с}.

(7.3.1)

Here Ä" is a smooth function, α > 0, η > 3, α ^ n(mod 2) for a > n. Set Ra = ΔΛ-^'/^Δ'

(7.3.2)

where Δ is the Laplace operator in R " , J

\x-y\ уГ

for /3 > 0 and Λ0 = E. i Numbers I and β are uniquely determined from the equation a = 2(Z + l ) - A β e [0,2) (7.3.3)

152

Α. L. Bughgeim. Volterra equations and inverse problems

I being a nonnegative integer. Let Ωί = {ж G K" I |a;| < C + Í}, Qt = {x,r,e\r>0, ee

Ωο = Ω x,x +

reeüt}·

Theorem 7.3.1. (i) Let a function К belong to for a certain t > 0, K(x,0,e) 0, Va; e Ω, u independent of e. Then for some constant Co and R = we have RKu = it + ЛГк, Vu G where Ν is a completely continuous operator in (7(Ω_/ι), h = 0 for β = 0, h>0 for β>0. (ii) If we additionally assume that К e C^iQt), Ku e ^^(Ω) and иеС{й), thenueC^iÙ). This theorem was proved in Kahane (1965) for the case when K{x, r, e) = Ko(x,x + re). In the general case the statement (ii) is proved i Bukhgeim (1975b). The proof is based on reducing the equation K u = / to the secondorder equation и + Nu = R f . Continuing this equation into the complex neighborhood Ω of the domain Ω we may show that the contraction iVj of N into Ωε = ΩΠ{ — y| < e, ж, у G С"} has in the analytic in Ω^· and bounded in its closure space of function a norm less than unity, The, the analyticity of и follows from the representation и = {E + Ns)~^[Rf + (iVg - N)u] and the analyticity of the function in the square brackets. The details of the proof are in Kahane (1965). Now, write the equation Pu = / , so that the condition explicitly enters in the definition of P: m

ξ

.

{Pu){,y,x)^

ak{u,x,y)ô^'Hpi,y,x,y))uiy)dy = f{i^,x)

хеп,

Dy^O Dyp\^=. =

(7.3.4)

for p = 0 (7.3.5)

Conditions (7.3.5) denote that the hyperspace T{u,x) : p{u,x,y) = 0 contains the point χ and has in this point the normal u. Further we shall asuume that n>3, 0 0 set m » {Peu){u,x)

smoothness

Ух

independent

continuous

that

(7.3.2)

the 0,

then

RPu = u + Nu η

n, ^

^ =

0. In the first term we may change the order of integration, since the vectors и = DL,{\v\ — and Di,p are linearly independent for - y\ > ε/2, ρ = 0, |i/| = 1 and, besides, Dyp 0, if ρ = 0. As a result, for ε —)• 0 we obtain equation (7.3.1), where a — η — m — 1, m K { x , r , e ) =

Υ^ k=-l

r ^ - ' K k { x , r , e )

(7.3.10)

154

Α. L. Bughgeim. Volterra equations and inverse problems Kkix,r,e)

^p{u,x,x + re)\ ak{u,x,x + re) du r am{i^,x,x) /

= I r = \y-x\,

e - ^ . (7.3.11) r When deducing these formulae, we have used the homogeneity of the functions ¿C^): = We now verify the Theorem 7.3.1 condition (i). As α = η - m - 1, then, by (7.3.3) 2/ + 2 = n - m - 2 , /3 = 0 for the odd η and 21 + 2 — η — m, β — 1 ioi the even n. Since then p{i',x,x + re)r~^ — / Jo

{Drp){v,x,tr,e)dt

and from the smoothness of the functions p, ak, it follows that К G C^'"*"^. Formulae (7.3.5), (7.3.10), (7.3.11) yield K{x,0,e)=

f

δ^"^H{e,u))dωn-ι

where dun-i is the Euclidean element of the sphere S^'^ area. As the measure dw„_i is invariant relative to the rotations, then K[x^ 0, e) is independent of e. Setting e = (1,0,..., 0), we obtain K{x,Q,e) = Г J-l

ί dun-2 Js^-2 Í - 0

(ω„_2 being the area of since m is even, 0 < m < n — 2. Reference to the Theorem 7.3.2(i) completes the proof. • By the formula (7.3.7) and the fact for a small ball Ω_/ι = {ж | с — /ι > |a;| } the norm of the operator N contraction into (7(Ω_/ι) may be made less than unity. Corollary 7.3.1. Let и G and с - h> ОЪе sufficently small. Then, under the Theorem 7.3.2 conditions, the solution to Pu = f is unique and the estimate ||u|| < ci\\APu\\2i+2 < сгЦРиЦгг+г holds, where 2/ -Ь 2 = η — τη for even η and 2Í -Ь 2 = η — m — 1 for odd η.

Chapter 7. Multidimensional integro-differential equations

155

Theorem 7.3.3. Let the Theorem 7.3.2 conditions hold and, besides, the functions ρ and Ok, к — l,m be analytic in Qt, i > 0. Then Pu = f is uniquely solved in We are interested in Lemma 7.3.1. Let Ω be an arbitrary opened set in K", UQ : M" ^ M"® be a sequence of function vectors (a = ( a i , . . . , a„) is a multi-index) of the form η

Ua+ej = AjUa +

Bj^DkUa,

j =

(7.3.12)

k=l

where ej = (0,0,..., 1 , 0 , . . . , 0),

=

and matrices Aj, Bj^,

3

are analytic in Ω. Then, for each compact set К С 0,, there exists a constant с such that for Va, < c(c|a|)'"', χ e K. (7.3.13) Proof. Proof of the lemma follows from the formula (7.3.12) and the fact that V € C^m

^ \D''v[x)\
) = {P,yU,v). Then = 0 for и E ωο. On the other hand, w = {u, P^v) or, in detail, w{v)= j u{y)!^ ¿

J

ak{iy,x,y)ô'^''\p{u,x,y))v{x)e{c-\x\)dx^dy.

By the theorem conditions the integral in braces and the function w are analytic by u. Consequently, w = 0. Due to the density of polynomials ν in ¿2(Ω)„it will be P^u = 0 for all и G i.e. Pu = 0. Prom here follows that u Ξ 0, which was to proved. • Remark 7.3.1. Theorems 7.3.2 and 7.3.4 in the local statement are proved in Lavrent'ev and Bukhgeim (1973). Theorem 7.3.3 is proved in Bukhgeim (1975b). All the theorems of this section are vahd also for η = 2. For the corresponding conditions of decreasing on infinity these theorems are propagated for the case of the unbounded domain Ω. The case of the odd m, 0 < m < n — 2is reduced to the even m by differentiating.

Chapter 8. Inverse Problems of Wave Propagation and Scattering In this chapter we consider the uniqueness and stability of solutions to the inverse problems for the Helmholtz equation, and for the equations connected with it (the wave equation and the equations system of elasticity theory). The problems are considered both in exact and in approximate formulations.

8.1.

THE INVERSE PROBLEM OF WAVE PROPAGATION IN LAYERED MEDIA

Some problems of the wave propagation in layered media may be reduced to the following boundary-value problem (see, for example, Brekhovskikh,

1973): +

= f{x,z),

χ = (a;i,a;2) e K^,

/o[n] = u - h a l i t i lh[u]=0,

z = h

= 0

ze[0,/i]

(8.1.1)

(8-1-2) (8.1.3)

where α is a certain complex number; lh[u] is, generally speaking, a nonlocal boundary operator uniquely determined by the complex reflection coefficient V{a). Reflection is performed from the half-space ζ > h {a being the angle of the flat wave incidence), c{z) is the speed of the perturbation propagation in

158

Α. L. Bughgeim. Volterra equations and inverse problems

the layer ζ G [О, h]. Often the function c{z) and the physical characteristics of the half-space ζ > h, which determine the reflection coefficient, are unknown, and therefore before we can solve the direct problem (8.1.1)-(8.1.3), we must solve the inverse problem: to ñnd the functions c{z) and F(a) (or one of them if the other is known). The data are some experimentally measured functionals of the function u{x, z). In this section we give one approach for solving this inverse problem. First we determine the function c(z) on the basis of kinematic information on the arrival and scattering times of wave in the general directions. This problem, naturally, may be called the inverse kinematic problem of scattering. In contrast to the kinematic problem of seismology, the suggested model is applicable only in those media, where we may generate the narrow localized 'cord' solutions u(x,z). (In the terms of equation (8.1.1)) this statement corresponds to the case ω ^ oo, and the special right side f{x, z)). The compensation for deficiency is the opportunity of unique determination of the wave guide (the opportunity of experimental confirmation of the data we have used follows, for example, from Kollistratova, 1959). First we prove the theorem of uniqueness for the inverse problem solution in the exact statement, and then we obtain explicit formulae of its solution for the linearized case. Determining, thus, the function c{z) by the function =

(8.1.4)

(zi being a fixed number from the interval (0, h), r = / = δ{χ)δ{ζ — Zi), Zi G (О, h), δ is the Dirac function), then by conditions (8.1.1), (8.1.2), (8.1.4) we re-establish the boundary operator Z/i[n] (i.e. the reflection coefficient F(a) as the function of the angle α e [0, π/2]). The solution to this problem is expressed in terms of the Bessel transformation g{\) from the function 9{r). Mathematically the inverse kinematic problem of scattering is reduced to the following nonlinear integral equation of the first kind: Í Уг(х) c(z)

=

xe[a,b].

(8.1.5)

Here r(a;) = Γι (ж) U Г2(а;) is a curve consisted of two intervals intersecting geodesies r j ( x ) of the Riemann metric di of the form

Chapter 8. Inverse problems

of wave propagation

and scattering

159

of the plane (r, z) outgoing correspondingly from the points (0,0) and (l(x),0) at the angles αι(χ) and a2(x)· The acute angle θ(χ) between ri(a;) and Γ2(2;) in its points of intersection has the sense of a scattering angle. The question arises, under which conditions on the functions αϊ (ж), a2(x), l{x) we may uniquely determine the speed c{z) by the times t{x), X e [α, 6] as is shown in the following theorem (Theorem 8.1.1) the sufficient conditions for unique determination of c{z) are the following a) Conditions on l{x), aj{x)

{j = 1,2), χ G [α, 6]: l{x),aj{x)

eC^[a,b]

0 О

(8.1.8)

α'ι(χ) + α'2(χ) + /'(χ) > 0

(8.1.9)

b) Conditions on the class of speeds c{z): cGqO,oo) 0 < φ ) 0

(8.1.11)

there exist positive numbers ho, H, ho < Η such that the function h{x) constructed by the functions c{z), aj{x), l{x) as follows 2

rh{x) rn

(n2(z) - cos2

dz = l{x)

(8.1.12)

Vcosaj(x) / n{z) = c(0)/c(z)

where

(8.1.13)

satisfy the conditions h{a) do — max(ai,al).

> 0 and thus,

the inverse

Φ

161

(8.1.16) ^ is

determined

class

Proof of Theorem 8.1,1. Use the Snellis law and the fact that c{z) is known in the interval [0,/lo] 2 Rewrite equation (8.1.12) in the form rh{x)

/

Jh{a)

2

V n ^ ( г ) ( n 2 ( г ) - c o s ^ α J ( x ) ) - ^ / ^ d z = /l(x)

(8.1.17)

^

rh(x) 2 rn,(X) / У^cosαj(г;)(n^(г) Jh{a) ^

— cos^ 0!j(a;))~

'

dz —

(8.1.18)

/2(2;)

where /1, /2 are the known functions rn^aj rKa)

/i(x)

= c{0)t{x)

¡2{χ)

= l{x)

-

/

rn.(a) -

/ Jo

2' yn\z){n^{z)

- cos^

dz

COS aj{x){n'^{z)

- cos'^

dz

' Y

(8.1.19)

(8.1.20)

functions n{z), h{x), a; > α to be determined. Introducing the change of variable ζ = h{s), dz — h'{s)ds (due to Lemma 8.1.1 this change is correct because h'{s) > 0) and differentiating equations (8.1.17) and (8.1.18) by x, we obtain the system of Volterra equations of the second kind W{x)+

[ Ja

F{x,s,W{s))ds

= g{x).

(8.1.21)

Here Wix)

=

Vix)J

9{x)

=

2 V{x)

= J2cosaj{x)h'{x){n^{h{x))

- cos"^

(8.1.22)

162

Α. L. Bughgeim. Volterra equations and inverse problems

and F is a smooth matrix 2 χ 2 of the variables x, s, W. From Lemma 8.1.2 it follows that the functions and h'{x) are uniquely expressed in terms of functions U{x) and V{x)\ therefore, the theorem follows from the uniqueness theorem for system (8.1.21). • Remark 8.1.1. The theorem of uniqueness remains valid for /IQ = 0. In this case, instead of co(z) (see (8.1.11)) it is sufficient to take the number c(0). Remark 8.1.2. Condition (8.1.7) on the functions aj{x) may be substituted by the following: 0 < ai(a;) < π/2,

0 < α2(χ) < π/2.

The problem of numerical solution of equation (8.1.5) in the case 02(2;) = π/2 was considered in Bukhgeim and Zenkova (1977). Here we give a simpler and more stable algorithm based on the linearization (8.1.5). Namely, for n{z) such that |1 — n{z)\ 1 the approximate solution c{z) = c{0)/ñ{z) is given by the formula ñ{z)

=c{0)/c{z) = 1 + (c(0)/0 cos2a[(sina(l + sina)~^)^(i(a)

-to{a)) (8.1.23)

for a2{x) = π/2, α = αϊ, and ñ{z)=ciO)/c{z) = 1 + (c(0)/2/) cos^ α[(sma){t{a) - to{a))]'^

(8.1.24)

for ai{x) Ξ «2 = a. Here, in the first case, íq = + sina)/(c(0) cos a), and in the second ÍQ = '/( h is expressed in terms of g(À) as follows

ig(À)(a;2/c2(/i) - λ2)ΐ/2 _ 1 λ=

c{0)/c{z)

•

Sina.

By the well-known expression for the resolvent of boundary-value problem (8.1.25)-(8.1.27) we may show that q{X) is determined in terms of з(А) by the formulae: ς{Χ)

= -η/Η·Α{μ)/Β{μ),

Α{μ)=

μ

-n^/h!^

• X^

(8.1.28)

¿ j,k=l

Β{μ)

=θ{χι +

-Χο)η2{χι,μ)ηι{χο,μ)+θ{χο-Χι)ηι{χι,μ)η2{χο,μ) πι{μ)ηι{χι,μ)υι{χο,μ)-ςι{μ).

Here θ is the Heaviside function, 9ι{μ)

=5(λ)

164

Α. L. Bughgeim. Volterra equations and inverse problems Xj = Zjn/h,

i = 0,1

m(/i) = -(u2(0,m) +Я4(0,μ))/(ul(0,^^)

+ Ηη[{0,μ)),

Η -

ah/π

and the functions Uj{x,ß) satisfy the equation -y"{x) - φ{χ)ν{χ) = μν{χ), φ{χ) = -π^/h^-ω^/ο^

x G [О, π]

(8.1.29)

{ζ)

with the initial data ηι{π,μ) = 1,

η[{'π,μ) = 0

(8.1.30)

η2(π,μ)=0,

4(7Γ,μ) = - 1 ·

(8.1.31)

Formula (8.1.28) holds for all μ such that Β{μ) φ О, where = -Λ^/π^μ does not belong to the spectrum of (8.1.25)-(8.1.27). The method of numerical calculation of solutions κχ (λ, г) and П2 (λ, г) of equation (8.1.29) with the boundary conditions (8.1.30), (8.1.31) is described in Bukhgeim and Zenkova (1981). Remark 8.1.3. The results of this section are taken from Bukhgeim and Zenkova (1981). Theorem 8.1.1 has an analog in the multidimensional case с = c{x,z). In contrast to the multidimensional inverse kinematic problem of seismology, considered in Chapter 4, this problem is well-posed in the local statement from C^ into for corresponding к and m. This is so, because the curves Γ(2;) from equation (8.1.5) have an angle point in the vertex. In the linearized statement we obtain an inversion formula analogous to (8.1.24).

8.2.

THE PROBLEM OF D E T E R M I N I N G THE R I G H T SIDE OF THE LAMÉ EQUATIONS

In this section we solve the explicit form of the problem of determining the right side of the equations system for elasticity theory in the half-space, given the oscillation conditions of the free boundary. This problem may be reduced to the problem of determining the initial data of the wave equation, or, which is the same, to the nonhyperbolic Cauchy problem. Begin with the notations and one auxiliary statement. Let χ — (χο,χ'), у = (yo, у') G M", a;', y' e , η > 3 be even. Set p(x) = | x f - x o

(8.2.1)

Chapter 8. Inverse problems of wave propagation and scattering (P/)(x) =

I

- y))f{y) dy,

f e

165 (8.2.2)

As in Chapter 7, represent the action of the generalized function δ''{ρ) on the test function / by the integral. Formula (8.2.2) is equivalent to the equaUty {Pf){x) = ( -

J^_Jiy)\DyP{x-yr'da.

Here da is a Euclidean element of Γ(χ) = {у G M" | p{x — у) = 0}. Proposition 8.2.1. The closure of Ρ in

is a unitary operator.

Proof. It is sufficient to establish the following equalities: r / I L · . = II/IIL,,

V/GCo~(M")

ЩР) = ¿2

(8.2.3) (8.2.4)

where R{P) is the range of values of the operator Ρ : L2 ¿2· As is a tempered distribution, then for a finite function / , by the theorem on the Fourier transformation of the convolution, we have {Fu) = n-ir^-msii-m) where

(p) /

(8.2.5)

m =f

After simple calculations we obtain = ;r("-i)/2exp{i[7r(n - l)/2 - l^f/(4^0)]}

(8.2.6)

whence ( ^ ' Ж О ! = 1/(01-

(8.2.7)

Formulae (8.2.5)-(8.2.7) and the Parseval equality for the Fourier transformation yield (8.2.3), (8.2.4). The proposition is proved. • Set Pf = / , / G L2. Corollary 8.2.1. The inversing formula f{x) =

I

ip*{x - y))f{y) dy

(8.2.8)

166

Α. L. Bughgeim. Volterra equations and inverse problems p*(x)^p(-x)

= ¡xf + xo

holds. Really, as Ρ is a unitary operator, then Ρ ^ = Ρ*. Transferring in the scalar product {Pu,f), u, f E C^, the operator Ρ with u on / it is easy to show that P*f is given by (8.2.8) ( / G C^). For an arbitrary / G ¿2, formula (8.2.8) is meant in the sense of the limit in L2 for ε -)· 0 of the expression P*fe, where Д G CQ", /е / in ¿2· Let a function ω{χ, t), χ = {χ = {xi,..., Xn) G К", a;„ > 0} satisfy the conditions = u{x,0)=f{x),

a;((x,0)=0

=

XGR",

for

> 0,

t>0

(8.2.9)

ω^Χ^^^

= 0. (8.2.10)

Introduce the operator Sc which with each / G C^ (K" ) assigns the function u{x',t) = ω ( χ ' , 0 , ί ) , where ω solves (8.2.9), (8.2.10), x' = {χχ,... ,Xn-i). Remark 8.2.1. As — 0) then continuing w evently by the variable Xn we obtain the Cauchy problem for even by Xn function ώ {ώ is an even continuation of ω). This remark we shall use further. For η = 1, from the d'Alambert formula, it follows that Scf{x) = /(ci). From here, for each с > 0 roo

/ \f{x)fx-Ux= Jo

roo

/ Jo

\u{t)\4-Ut.

In other words, the operator Sc, considered as an operator acting in the space L2(K+,P), p{x) — is isometric (in our case, even, unitary). The basic results of the section is the establishment of an analogous property for the operator Sc in the case of even odd dimension η > 3. Theorem 8.2.1. (i) The closure of Sc in ¿2(Κ+,ρ), = ^ñ^ variable Xn > 0 for the map Sc goes to t > 0) determines the isometric operator (which we shall denote by the same symbol). (ii) The function u(x',t) from £2(Щ,р) belongs to the range of values R(Sc) of the operator Sc : ¿2(К+,р) ι-)· ¿2(Κ+,ρ) if and only if

Chapter 8. Inverse problems of wave propagation and scattering

167

HcU — 0, where He is bounded in L2(Wl,p) and

^ I ¿{(n-l)/2) d^' _ yj2 _

_

¿y, (8.2.11)

Proof, (i) To prove (i) it is sufficient to verify the identity ,p) = \\f\\L2(Ri,p)

^f e CO^Í^f^")

(8-2.12)

(since ) is dense in L2(K+, p))· For / G a solution to (8.2.9), (8.2.10) as is known, is given by the formula (see, for example, Gel'fand and Shilov, 1958a) ω(α;,ί) =

J

whence, setting

= 0, we obtain

- yp - c ¥ ) / ( y ) dy

u{x't) = Scf = Xj Make the change of variables xq =

+ y2-cV)/(y)dy.(8.2.13) уо = y^ and set

/1(2/о,у')=УО'^'/(У',УУ')

(8.2.14)

ui{xo,x') =

(8.2.15)

Then the equation Scf = и becomes P+ fi = щ, where is the contraction of the unitary operator Ρ onto the subspace L2ÍWI) С ¿2(Μ"). Formulae (8.2.1), (8.2.2) yield that ¿2(^4.) is the invariant subspace of P; therefore, P+ : ¿2(Κ" ) Ы Щ ) is determined correctly. As ||P+/i||l2(R!|:) = ||/i||i,(Rn), then, returning from the variables Жо, yo to the initial variables we obtain (8.2.12) (ii) Prom the (i) proof it follows that a) u e R{Sc) e P(P+) (и, f G ¿2(1^4-, ρ) щ, f i E L2{Wl)), where /1, щ are determined by (8.2.14), (8.2.15). Since P+ is contraction of the unitary operator Ρ onto the subspace L2 (M" ), then

168

Λ. L. Bughgeim.

Volterra equations and inverse

b) w e R{P+) ^ {Р*щ){хо,х')

Ξ О, for xo < 0

problems = 0)·

Using a), b), formula (8.2.8) for P* {xq = —x^ for xq < 0) and returning from the function to и we obtain (ii). The theorem is proved. • Corollary 8.2,2. (The reversion formula). Let Scf = u. Then fix)

= XI

+

-c^í2)u(y^í)dy'dí. (8.2.16)

Proof. By the fact that Sc is isometric,

ξ

where 5*1^(50) :

R{Sc) ^ Z2(K+,P)· Since Scf is given by (8.2.13), it is easy to verify that the right side of (8.2.16) is exactly S*u. • Now, we shall go to the main theme of this section — solving the inverse problem for the Lameé equations. Let a function vector u{x,t) = (ui, ií2, из), ж G (generalized) Cauchy problem

be a solution of the

d^u Xn Ξ — - (λ + 2μ) grad(div u)/p +μ rot(rot и)/ρ = a{t)f{x) 4 < o = 0.

(8.2.17) (8.2.18)

Here 0!(i) is a scalar function equal to zero for ί < 0; / = (/1,/2,/з); λ, μ are the Lamé constants; ρ > О is a constant density Equation (8.2.17) describes the distribution of waves in isotropic and homogeneous elastic medium. The problem of determining the shift vector и from (8.2.17), (8.2.18) and the given right side α · / is well known. We are interested in the inverse problem. Problem 8.2.1. Given the functions w|x3=o = a(a;', t),

Wx3li3=o =

to determine f{x) from (8.2.17), (8.2.18) Problem 8.2.2. Let Lu = ait)f{x),

> 0,

Í> 0

Chapter

8. Inverse

problems

of wave propagation

u(x,0) = u t ( x , 0 ) = 0 ,

and scattering

169

а;з>0.

In the plane хз = 0, a function и satisfies the boundary condition (free boundary condition) lu = 0,

хз^О

Q

(lu)·^ = Adivu + 2μ-

дхз

It is required by the function oscillation) to re-establish f{x)

(дщ

dui\

(дщ

du2\

= a{x', t) (condition of the free boundary (α(ί) is given).

Problem 8.2.2 by the change of variables may be reduced to Problem 8.2.1. Really, let u e C ^ { x 3 > 0 , t > 0) be a solution to Problem 8.2.2. Prom the conditions lu = 0, ιιΙ^,^-ο = α we find the function «ijlj-j^o ~ Setv = u - хзЬ. Then ν e C"^ (x^ > 0, ί > 0) and Lv = af - g (хз > 0), where g = Ь[хзЬ] and = 0. Denote by ν the even continuation of ν into the subspace Хз < 0. As Vijl^^^g = 0, then ν e C^ (t > 0) and, therefore, Lv = af — g. The function g is known; consequently, setting ω = v+p, where Lp = g, ξ 0 we obtain Problem 8.2.1: Lu = af, ξ 0. Because of this remark, furhter, we shall confine ourself to the Problem 8.2.1. To formulate the algorithm for solving the Problem 8.2.1, we need some notations and lemmas. For an arbitrary function vector / G C^(M^), set î+{x) = \{î{x\x3) + î{x\-x^)).

χ'^{χχ,χ2)

Since /+(х',0) = /(x',0), (/+),з(х',0) = О and f{x)

(8.2.19)

= (/+),з + (/,3)+, then (8.2.20)

= U{x)+ Jo

Thus, by the functions /+ and (/13)^, from formula (8.2.20) we uniquely determine the function /. Define the class (7°°'" of the function vectors as follows: /6С~(Г),

lim|x| • 1/(^)1 < oo

(8.2.21)

170

Α. L. Bughgeim. Volterra equations and inverse problems V·/, Vx/€Co°°(r)

(8.2.22)

( s u p p V ' / U s u p p V X /) η { x I X3 = 0} = 0

(8.2.23)

The class of functions which satisfy the first two conditions we denote by Let J^Q be the set of matrices F{x) = {Fij{x)) {i,j = 1,2, Fij be real functions, F2j = {F2j,F^j), F^j be function vectors) such, that Fij{x',x^) = Fij{x',-X3),

FijEC^,

x'^{xi,x2)

(8.2.24)

JF^O

(8.2.25)

supp F η { x I хз = 0} = 0.

(8.2.26)

In the condition (8.2.25), J is the operator that associates with the matrix F the vector dx3,

F22 dx3, V · F2)

F2{x) = F2i{X) + Г Jo

(8.2.27)

F22 dxs.

(8.2.28)

The class of matrices which satisfy only (8.2.24), (8.2.25) denote by J^o,i· Introduce the map A, which maps / G to the matrix F = {FIJ)^ i,j = 1,2 by the formula Af = F,

Fix) = f / J • /J · ] . V ( V x / ) + (Vx/x3)+y

(8.2.29)

Here V = {д/дх1,д/дх2,д/дхз), V · / - div/, V x / = rot/, and the operator + is determined by (8.2.19)

= df/dx^,

Lemma 8.2,1. The operator A bijectively maps the set

onto FQ.

Proof. To prove С FQ is is sufficient to verify that JAf = 0 (conditions (8.2.24), (8.2.26) immediately follow from the Af definition). Using (8.2.29), (8.2.20) we obtain Г Jo

FI2 dx3 = Г Jo

(V ·

dx3 = V · / - (V • /)+.

As the right side of the equality is finite, then the left side is finite also; in particular, roo

f

Jo

Fi2dx3 = 0,

i = l.

Chapter 8. Inverse problems of wave propagation and scattering

171

The case г = 2 is considered analogously. Further, (8.2.20), (8.2.28), (8.2.29) yield F2 = V x / and, since V · (V x / ) = 0, V/ e then V · F2 = 0. So, JAf = 0 and the inclusion Ç J^o is proved. We now show that for each matrix F G To, there exists a unique function vector / 6 such that Af = F. Define / by the formula fix) - - ( 4 π ) - ^ V j

j

dy + (4π)-ι Vx

where

^ ^

dy

(8.2.30)

rX3

Fi{x) = Fii{x) +

Fi2{x)dx3,

F e To-

(8.2.31)

Jo As /0°° Fi2 da;3 = 0, г = 1,2, F e C ^ , then f ^ ' d^s also belongs to C ^ and, therefore, F¿ G C^, where, by conditions (8.2.26) supp Fj η {x I хз = 0} = 0. Formula (8.2.30) implies V - / = Fi,

VX/-F2

lim |x| · |/(a;)| = (4π)-ΐ| [ (VFi - V x F2) dy\ < 00. |x|->oo J

(8.2.32) (8.2.33)

To obtain (8.2.32) from (8.2.30) it is sfufficient to use following identities V-(V(^) = Αφ, V - ( V x V ) = 0, V x ( V x ^ ) = V(V-V)-AV'; (Δ is the Laplace operator) and the condition VF2 = 0. From (8.2.31)-(8.2.33) it follows that / € and Af = F . If Л / = 0, / € then V · / = 0, V χ / = 0; therefore, V x (V x / ) = Δ / = 0, i.e. / is a harmonic vector function in and, as |/(a;)| 0 for |a;| 00 (see (8.2.21)), then / ξ 0. The lemma is proved. • Prom the Lemma 8.2.1 proof we see that, simultaneously, we prove the following Corollary 8.2.3. The operator A bijectively maps the set Cj*^'*^ onto

L e m m a 8.2.2. c\ = {Χ + 2μ)/ρ,

The identity AL = LA holds, where L = μ!p.

©Dca,

172

Α. L. Bughgeim. Volterra equations and inverse problems

Proof. The equality ALu = LAu, и 6 follows from the well-known identities V · Lu = Dg^V · u), V x Lu = •ci(V x u) and the fact that the operator + commutâtes with Dc. • Let Lu = 0,

a;GM^

u{x,tí)=f{x),

i>0

ut(a;,0) = 0.

(8.2.34) (8.2.35)

Lemma 8.2.3. For each f G there exists a unique solution to the Cauchy problem (8.2.34)-(8.2.35) u{x,t), which belongs to for each t > 0. Proof. Prom Lemmas 8.2.1, 8.2.2 and Corollary 8.2.3 it is sufficient to show that for each matrix / G Jo there exists a unique solution U{x, t) G of the following Cauchy problem: LU = 0 U\t=o =

Ut\t=o = 0

where U G for each t. As ¿ = ®üc2 and the solution to the wave equations with the initial data from C^ is again (for a fixed t) a function of the class C^ (moreover, to the even by x^ initial data there corresponds the even by χ s solution), then, to prove the lemma it is sufficient to verify that JU{x,t) = 0. Let us prove, for example, that f ^ Undx^ = 0. Indeed, due to the evenness character of Un by хз rxì г^з / Dei Ì7i2da;3 = Пс1 / Ui2dx3 = 0 Jo Jo holds, and, since, the initial data corresponding to f^^ Uu dxs are finite {F E J^o) then the solution is finite by χ and, in particular, {JU)\ — /Q°° U12 dxs = 0. The other conditions, which guarantee the realization of JU{x,t) = 0 are verified analogously. The lemma is proved. • Introduce the map S with the domain of definition !Fo, which associates with F E To the matrix t/(.'.i)=fj·" Vvxn

J·""') V X u.. ;

(8.2.36)

Here u{x,t) is a solution to the Cauchy problem (8.2.34), (8.2.35) and / = A~^F. By Lemmas 8.2.1, 8.2.3 the operator S is defined correctly. The basis of solving Problem 8.2.1 lies in the following.

Chapter 8. Inverse problems of wave propagation and scattering

173

Problem of 8.2.1'. By the given matrix U{x',t), to find a matrix F from the equations SF = U. Let S ^ Sc, ® Sc2, H = He, θ Яс2 {Sc and He are defined by (8.2.13), (8.2.14)) be operators acting in the Hilbert space L2,p of matrices with the norm ^ i i ^ i i - Σιι^ϋΐιΙ(Μη„ρ)·

(8.2.37)

i,j=l

Identifying the even by x^ matrix F E Tq with its contraction onto the half-space xz > 0, we may assume that Jo Ç L2,p· The closure To by the norm (8.2.37) we denote by T. The Problem 8.2.Γ solves the following Theorem 8.2.2. (i) The operator S may be extended by continuity to the isometric operator S : Τ ¿2,ρ, i.e. ||SF|| = ||F||,

VF e J .

(ii) The matrix U{x',t) from L2,p belongs to the range of values of the operator SHU ^ 0 and JS*U = 0. (Hi) IfSF

= U, then F = S*U (the inversing formula).

Proof. First we show that the contractions of S and S onto To coincide, i.e. SF = SF,

Fe To-

(8.2.38)

By the definition S F = U, where U is given by (8.2.36). On the other hand, by the definition of A and + we have U Therefore, using Lemma 8.2.2, the matrix may be constructed by solving the Cauchy problem LAU = 0, = F, [AU)fl^^Q = 0. Lemma 8.2.3 and Corollary 8.2.3 yield AU € .^од and, in particular, (^?^)а;з11з=о ~ 0 (evenless). From the equahty L — ФПсг) (^^)i3li3=o ^c definition (see (8.2.9), (8.2.10)) it follows that U = (5^ Θ ^ c j F - SF. So, (8.2.38) is proved. By the formula (8.2.38) the statements (i)-(iii) of the theorem become the immediate corollaries of Theorem 8.2.1 (for η = 3) and the remark that F e Τ F e L2,P and JF = 0. (The operators of differentiation entering in J for an arbitrary F G ¿2,ρ are understood in the sense of the generalized functions). The theorem is proved. •

174

Α. L. Bughgeim. Volterra equations and inverse problems

Remark 8.2.2. Uniqueness of the Problem 8.2.1 solution occurs without any restriction on / in infinity. We shall outline a brief proof of this fact. Really, if / is a harmonic function in M", then, applying to (8.2.17) the Laplace operator (which commutâtes with L) we obtain L{Au) — LAf = 0, = 0. From here, Au = 0 and the uniqueness of the Cauchy problem for the Laplace estimate follows, since (w, are given functions. The case of an arbitrary smooth function / may be reduced to this, since the functions V • /, V X / are uniquely determined wothout any conditions in oo (the corresponding problem may be reduced to the problem • с ω = О, ~ ""xalis^o ~ Ргои! the fact that V - / = V x / = Oit follows that Af — 0, i.e. / is a harmonic vector. This result is carried over the generalized functions /. We now apply Theorem 8.2.2 to solve Problem 8.2.1 explicitly. Denote by u" the solution to the equation Lu°' = a • f , ξ 0. Recall that in Problem 8.2.1 from the pair of functions = 0 we need to determine f{x) (α(ί) is given, o; Ξ 0 for ί < 0). Introduce the Volterra operator of convolution with the kernel a: (Vff)(t) =

Γ a{t - т)д{т) dr. Jo

If u solves (8.2.24), (8.2.25), then, by the Duhamel principle {Vu){x,t) = Dtu''{x,t) holds. Therefore, knowing the vector

we may find

(for V to be reversible, it is sufficient, for example, that a{t) ^ 0, a Ε C[0, GO)). By the vector {u,ИХЗ)!^,^^^), from the equation Lu — 0 it is easy to determine and, consequently, Thus, by Theorem 8.2.2 we can define the inverting A, obtain /. Therefore, form Theorem 8.2.2 follows Corollary 8.2.4. Let α G C[0,oo), a φ 0, f Ε Then solution to Problem 8.2.1 is unique, where finding vector / involves the following:

^

fix).

Chapter 8. Inverse problems of wave propagation and scattering

175

We can show that determining the right side of the form α · / in the case when a is also unknown is nonunique. By Lemma 8.2.3, it is sufficient to construct such an example for the wave operator. Example 8.2.1. Let Duj = aj(t)fj{x), where aj =

i = 1,2,

í > 0,

UJIJ^Q

ξО

fj(x) e sup f j с {χ I |a;| < a,

|2;| > 6 ,

a > b} = G.

We must show that if the functions aj, f j are connected as follows αι(ί) = D¡a2{t),

f2{x) = Δ / ι ( χ )

(8.2.39)

then u = ui —U2 vanishes everywhere outside the set G X [Ο,Γ + α]. So, the desired example will be constructed. By the definition of и and g = aifi — «2/2 we have Ou = g, = whence, by the Kirchhoif formula u{x, t) = (4π)-ι J g{y, t - \x - y\)\x - y f ^ dy

(8.2.40)

holds. We now calculate Uoo(ai, σ) = lim|j.|^oo |x| · u(|x| · ω, |x| + σ), where ω G R^, |ω| = 1, σ > 0. Going in (8.2.40) to the limit we obtain ηοο(ω, σ) = (4π)-ι j g{y, σ + {ω, y)) dy.

(8.2.41)

Substituting (8.2.39) into (8.2.41) we have Κοο(ω,σ) = (4π)-ΐ J ϋΙα2{σ +

{ω,y))fι{y)dy

- (4π)-ι У α2(σ + {ш,у))АуМу)

dy.

Integrating by parts we verify that ηοο(ω,σ) ξ 0. From the results of Priedlander (1967), it follows that u{x,t) = 0 for |a;|>a,

t>a

+ T.

(8.2.42)

Since, on the other hand, • ω = 5 = 0 for |а;з| < ò, then, by (8.2.42) and the Holmgren theorem it is easy to obtain that u{x, t) — 0 everywhere outside G X [Ο,Τ + а]. The example is constructed.

176

Α. L. Bughgeim. Volterra equations and inverse problems

Remark 8.2.3. Results in this section are based on the paper of Bukhgeim and Kardakov (1978). Theorem 8.2.1, although simple, had a long path of development. First, the system of necessary and sufficient conditions on the function = u{x',t), when there exists a solution to the Cauchy problem for equation (8.2.9) was obtained by Hadamard (see Hadamard, 1978, p. 272-280) in 1916. His method has based on solving the equation Scf = и by the method of moments. As Hadamard had noticed (see Hadamard, 1978, p. 281) his solution was not satisfactory because of the complicated confounded conditions which could be independent, or not. The advantage of the Hadamard method is that it may be applied in the local statement, i.e. when the function и is given in the finite part of the plane = 0. It allows us to clarify the character of Cauchy problem ill-posedness in the local statement and establish, in some sense, the function и quasi-analyticity. The Hadamard theorem was again obtained in Lavrent'ev et а/. (1969) in connection with investigation of the inverse problem for the telegraph equation in the linearized statement. The equation Scf — и is practically the problem on the re-establishment of the function, knowing its means by the spheres of an arbitrary radius, centers of which belong to the hyperplane Xn — 0. Other statements of this problem were considered by Garipov (1974) and Kajstrenko (1975). Formula (8.2.16) for η — 3 was obtained in another way by Garipov (1975), and by Garipov and Kardakov (1973), there the equation Scf = и was studied in the space C^. Isometricity of and the inversing formula, which gives the solution to the equation P+/i = ui was obtained in Bukhgeim (1973). Thus in the papers of Bukhgeim (1973) and Garipov (1975) the well-posedness of this problem established. Note also, that the statement (ii) of Theorem 8.2.1 is closely connected with the well-known theorem of Dajson, Witeman, Gording (this theorem is, for example, in Vladimirov (1964, p. 379)).

Remark 8.2.4. We have considered only the case of the odd n. For even η the corresponding formulae are obtained by means of the standard Hadamard method of derivation.

8.3.

STATEMENT OF THE INVERSE SCATTERING T H E B A R R I E R PROBLEMS

ON

Let Ω be an unbounded domain in R" with a compact simply connected complement Ω' = R" \ Ω and with sufficiently smooth boundary Γ. Such a domain we shall further call the exterior domain. Consider in Ω the solution

Chapter 8. Inverse problems of wave propagation and scattering

177

to the Dirichlet problem for the Helmholtz equation: Δη + λ \ = 0,

хеП

(8.3.1)

îi|p = exp(iA(u, χ)).

(8.3.2)

For |a;| ->· oo, the function и = η{χ,ω, λ) satisfies the radiation condition и=

(7 jXI

- iAti =

^ oo.

(8.3.3)

Here a; is a unit vector in R", giving the direction of incidence on the barrier Ω' of the plane wave exp(iÀ(a;, a;)). The real number λ has the physical meaning of frequency; и is the field scattered on the barrier Ω'. For the given Ω', the problem of determining the scattering field from the conditions (8.3.1)-(8.3.3) is a well-known direct problem of scattering theory. The inverse problem is in determining the boundary Г under the condition that u, or from (8.3.1)-(8.3.3), is known in a certain set G of the points (χ,ω,λ): U\Q = ¡{χ,ω,λ), {χ,ω,\)Εθ. (8.3.4) Solving the direct problem (8.3.1)-(8.3.3) by the method of the potential theory (for definiteness, we shall consider the case η = 3) we obtain that the inverse problem (8.3.1)-(8.3.3) is equivalent to the determination the surface Γ from the system of integral equations =

{x,u,X)eG

μ{χ, ω, A) + (2π)-ι J μ{η, ω, =

\χ - уГ' χ er.

(8.3.5)

dSy (8.3.6)

Here, δξ is the diff'erentiating operator along ξ, where ξ = ξ{y) is the exterior unit normal to Γ in the point y. Often in applications, the distance between the points of observation χ — rO {r > 0, θ e and the surface Γ is so large that it may be considered as infinite. In this case, the known function is the function ν{θ,ω,λ)=

lim r->oo

(8.3.7)

instead of the function U\Q = / , where the parameters {Θ, ω, λ) pass through a certain set G С χ χ Here, = {α; G R" | |ω| = 1},

178

Α. L. Bughgeim.

Volterra equations and inverse

= K^ η {λ > 0}. Setting in equation (8.3.5), χ = we obtain -iA I

problems

and tending r to oo

dSy = ν{θ,ω,\),

{θ,ω,λ)€θ.

^ (8.3.8) Let Γω = {y G Γ I {ω,ξ{ι/)) > 0}. Using the optics terminology, we may say that Г - ^ is a part of the surface, illuminated by the incidence of a plane wavefront in the direction ω; Г^ is a part of the surface visible from the direction Θ. Intuitively, it is evident that for λ sufficiently large the approximate solution μ for the equation (8.3.6) is given by the formula

where Γ+ = П Γ^, хг+ is the characteristic function of Г+. Substituting this formula into (8.3.8), we obtain the following approximate equation: f

Jr+

=

{θ,ω,λ)Εθ.

(8.3.9)

The same equation to within a constant factor, depending only on λ, is obtained for every dimension η > 2. If in (8.3.9) we fix the direction ω of the incident wave and change the direction Θ, then, in the plane or the three-dimensional axially symmetric case, equation (8.3.9) for the strictly convex domain Ω' reduces to a nonlinear Volterra equation of the first kind. This case was explicitly considered in Bukhgeim and Konev (1979). If in (8.3.9) the function ν{—ω,ω, λ) is known, then we obtain the equation ίλ(2π)-^ Í

Jr+

(-ω,ξiy)}e^^^ 0, θ{τη) = О, m < 0. Elementary estimates show that (8.4.13)

\X\l)-X\l2)\ Co > 0.

(8.5.11)

To continue the proof we need the following abstract lemma. Lemma 8.5.1. Let Αχ e C{X,Y), X, Y be Hilbert spaces. If the operators Αχ analytically depend on λ in the disk |λ| < fc and Vu G X, и 1 ker Αο, ||Aau|| < m||Aoíi||, (8.5.12) there m |λι • λ2 · · · \N¡ < k^, О < |λι| < |À2| < · • · < \XN\ < к, then there exists constant с such that for Vu e X, и ± ker Aq N

||Аоп||

пЦЛл^пЦ.

(8.5.14)

j=i Set

{ΑχΗη,Αρηο) FniX) = ^ :7: Г P o "ni

(8-5-15)

where (·,·), || · || are the scalar product and the norm in Y correspondingly. Then Fn{X) is analytic in the disk |λ| < к, where from the CauchyBunyakovsky inequality and estimate (8.5.12), it follows that \Fn{X)\ 1. Other geometric problems, connected with the inverse scattering problem in the high-frequency statement, are investigated in Anikonov (1978). Remark 8.5.2. In the class of strictly convex surfaces, where η{ξ) > r and for the corresponding t{s) > s the estimate ||u||t < c||it||s < R holds and, by the stationary phase method we may show that, beginning with a certain λχ > 0, the function и may be uniquely determined from the equation Фдм = / (λ being fixed, A > λι). An analogous remark holds also for the inverse scattering problem in the explicit statement. In this case we have to use that, for ί —>• oo, the inverse scattering problem may be reduced to the Minkovskii problem (see Majda, 1976).

8.6.

PROBLEMS OF COEFFICIENTS D E T E R M I N A T I O N

Let u{x,x^,X)

solve the equation Au + X^a{x)u = ô{x-x^),

(8.6.1)

such, that u = UQ + V, exp(iA X Щ = -4π |a; -

)

(8.6.2)

and V satisfy the radiation condition (8.3.3) (n = 3). Assume that a{x) = l-|-ò(a;), where b is a continuous function with a compact support, contouring in a bounded domain Ω € The function щ , solving equation (8.6.1) for α = 1 is the incident wave. The function v, when b is not zero, is the scattering wave. We require, by the scattering in the inverse direction wave v{x,x,X)

^g{x,X),

хеПо,

Xe{Xo,Xi)

(8.6.3)

to re-establish the function b{x). Here, Ωο is a given domain in Ωο η Ω = 0, λο < Al. Under the above assumptions, the following theorem holds.

189

Chapter 8. Inverse problems of wave propagation and scattering

T h e o r e m 8 . 6 . 1 . The function b{x) is uniquely determined by the function g{x, λ). P r o o f . By (8.6.1) and (8.6.2), the function υ satisfies the equation (Δ + = + Prom here, inverting the operator Δ + λ^ and using the radiation conditions, we obtain

Απ J

\х-у\

(8.6.4) By the analyticity of w by λ and given g, we may find the function f{x) = - limx_>.oo λ). As υ(χ, = 0, then, taking into account (8.6.3), (8.6.4), we have \x-y\

\y)dy

= f{x),

хеПо.

The uniqueness of solution to this equations follows from the following Riesz theorem (see Riesz, 1938). T h e o r e m 8 . 6 . 2 . Let ö e L i ( R " ) , supò С Ω. If for a certain a, such that a φ 2 + 2 m , α фпЛ-Ъп, m = 0,1,2..., IAAB)ix)

= I

d

y

= О,

Χ E ПО

where Ω, Ωο are bounded nonintersecting domains in M", then ò ξ 0. P r o o f . As the function AQ6 is analytic by ж in R " \ Ω, then, due to uniqueness of the analytic continuation Aab = 0 for |a;| > r, r being sufficiently large. Calculating lim|x|_yoo {Aab){x), we have

/

b{y) dy = 0.

Since A = (α-2)(α-η) then, applying the Laplace operator A to Actb we have (Аа_2б)(а;) = 0, |x| > r. Prom here, for j = l,n, |ж| > r we obtain

= XjAa-2b - Aa-2Ìyjb) = - A a - 2 { y j b ) = 0.

190

Α. L. Bughgeim. Volterra equations and inverse problems

Dividing this equality by

and tending |a;|

oo we have

J yjb{y) dy = 0. Analogously, for each multi-index β Idy = 0 and, therefore, b = 0. Theorems 8.6.2 is proved.

•

Setting in it η = 3, α = 1 we prove Theorem 8.6.1.

•

Remark 8.6.1. Theorem 8.6.1 is the variant of the Lavrent'ev theorem (see Lavrent'ev et ai, 1980 and bibliography cited there). The proof of this theorem, based on the Riesz theorem is new. It seems to be that, usin Lemma 8.5.1, we may confine ourself by the representation of g for a finite number of frequencies λ = λι, À2,..., Лдг. In conclusion of this section we shall consider one inverse problem in the nonstationary statement and its simplest cases of degeneration. Suppose we have to find the smooth functions u{x,t), a{x), given the conditions utt - Au-

a{x)u = 0,

u{x, 0) = fix),

ut{x, 0) = 0,

u{x', 0, t) = 9{x', t),

t>0

(8.6.5)

xn>0

(8.6.6)

a;„ > 0,

u^Jx', 0, t) = 0.

(8.6.7)

Here x' = {x\,... ,x„i). Set v{x, t) = - f 7Г Уо ^ + τ

r) dr.

(8.6.8)

Then, it follows from (8.6.5)-(8.6.8), that vtt + Av + a{x)v = 0 v{x,0) = f{x), v{x',0,t)^h{x',t),

г;((гс,0)=0

(8.6.9) (8.6.10) (8.6.11)

where the function h is connected with g by the integral transformation (8.6.8). Convergence of the integral in (8.6.8), for a smooth finite function f{x) follows from the law of the exponential decrease of energy (see, for

Chapter 8. Inverse problems of wave propagation and scattering

191

example, Lax and Fillips, 1971). Let f(x) 0 for χ e Q, Q — {x 6 R" I |a;| < r}. Show that in the set Q the coefficient a(x) is determined uniquely. Continue the function ν evenly by t and consider equation (8.6.9) in the domain Ω = {{x,t) | = r^ - \xf - a^t^ > 0, Xn > 0}. For sufficiently large a, the function v{x,t) 0 in Ω, since f{x) 0 in Ω. The Carleman estimate for D^ + A follows from Theorem 6.1.1, where as the weight function ψ we may take the function ψ = exp Χφ - I, for sufficiently large λ. Thus, the uniqueness of problem (8.6.5)-(8.6.7) solution follows from Theorem 6.2.3. Now, let v{x,t) = t^vo{x,t), where vq 0 in Ω and β is a fixed non-zero real number. In this case, the problem may be reduced to the integro-differential inequality, which in the abstract form is as follows: \\Pu\\ < c(||u|| + t^-'j'^iMr-^)

+ Cii^-I J((||An|| + llu'lDr""^).

Here, Ρ Ξ dt-A{t)—iB{t), Λ are the operators from Section 5.1, the function u{t) e Ж2^([0,Т];Я1/2)п12([0,Т];Я1), u(0) = 0, с, ci are certain constants not depending on t, where ci = 0 for A ξ 0, ci > 0 for ||Λυ|| < ο(||Αυ|| + ||υ||), \/v G H. In this case, for μ < 1/2, < 0, α > 3/2, this inequality is uniquely solved. To prove this we may use Lemma 5.1.1 and the following lemmas, which are proved analogously to Theorems 2.3.2 and 2.3.3. Lemma 8.6.1. Let μ < 1/2, φ G ^^[Ο,Γ], φ'{ί) < 0. Then

Lemma 8.6.2. Under the Theorem 2.3.3 conditions for μ < О the following estimate holds:

II^IIt,. =

^ > 0.

Remark 8.6.2. The reduction of a hyperbolic equation to an elliptic one is well-known. For the inverse problems this method had been systematically used by Reznitskaya (see Lavrent'ev, et ai, 1980 and the bibliography cited there). The idea of applying the Carleman type estimates for the proof of the uniqueness of inverse problem solution baes on its reduction to the Cauchy problem for integro-differential inequalities was stated and discussed by the author together with M. V. Klibanov and S. P. Shishatskii in 1975.

192

Α. L. Bughgelm. Volterra equations and inverse problems

The first attempt at such a kind was made by Beznoshchenko and Prilepko (1977). Here, the inverse problem for the parabolic equation was reduced to the Cauchy problem for the integro-differential equation. There the case of determining the right side of the special form f{x, t) = a{x) was considered. This case of separatioon of variables does not allow us to investigate the coefficient inverse problems. The general case [P,Q] φ 0 was considered independently by Bukhgeim (1981b) and Klibanov (1981). Subsequently, this method was extended by M. V. Klibanov for the case of determining the coefficients of nonlinear equations. The author obtained the Carleman type estimates for the difference schemes, which allow us to investigate inverse problems in the difference statement.

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Leningrad. Otdel. Mat. Inst. Steklova 84, 3-6 (in Russian). English trans.: Soviet Math. 21, No. 3. A. V. Belonosova and A. S. Alekseev (1967). A statement of the inverse kinematic problem of seismology for two-dimensional continuously inhomogeneous medium. In: Some Methods and Algorithms for Interpriting Geophysical Data. Nauka, Moscow, 137-154 (in Russian). Yu. M. Berezanskii (1958). The uniqueness theorem in the inverse problem of spectral analysis for the Schrödinger equation. Trudy Moscov. Mat. Obshch 3, 3-62 (in Russian). English trans.: Amer. Math. Soc. Transí. 2 (1964) 35. I. N. Bernstein and M. L. Gerver (1978). A problem of integral geometry for a family of geodesic and an inverse kinematic problem of seismology. Dokl. Akad. Nauk SSSR 243, 302-305 (in Russian). English trans.: Dokl. Earth. Sci. Sections (1978) 243. N. Ya. Beznoshchenko and A.I. Prilepko (1977). Inverse problems for parabolic equation. In: Problems of Math. Phys. and Math. Nauka, Moscow, 51-63 (in Russian). A. S. Blagoveshchenskii (1970). An inverse problem for the wave equation with unknown source. In: Problems of Math. Phys. Vol. 4- Leningrad University Press, Leningrad,. 27-39 (in Russian). English trans.: Topics in Math. Phys. No. 4, Plenum Press, New York, 1971. L. M. Brekhovskikh (1973). Waves in Layered Media. Nauka, Moscow. M.S. Brodskii (1957). One problem of Gelfand. Uspekhi Mat. Nauk 12, No. 2, 129-132 (in Russian). M. S. Brodskii (1961). On the triangular representation of completely continuous operators with a single spectrum point. Uspekhi Mat. Nauk 16, No. 1, 135-141 (in Russian). M.S. Brodskii (1965). On one-blockness of the integration operator and on the Titchmarsh theorem. Uspekhi Mat. Nauk 20, No. 5, 189-192 (in Russian). M.S. Brodskii and G.E. Kisilevskii (1966). The criterion of one-blockness of Volterra operators with nuclear imaginary components. Izv. Akad. Nauk SSSR. Ser. Mat. 30, No. 6, 1213-1228 (in Russian).

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