Ill-Posed and Inverse Problems: Dedicated to Academician Mikhail Mikhailovich Lavrentiev on the Occasion of his 70th Birthday [Reprint 2018 ed.] 9783110942019, 9783110460254

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Ill-Posed and Inverse Problems

ACADEMICIAN M I K H A I L MIKHAILOVICH LAVRENTIEV

ILL-POSED AND INVERSE PROBLEMS DEDICATED TO ACADEMICIAN MIKHAIL MIKHAILOVICH LAVRENTIEV ON THE OCCASION OF HIS 70 TH BIRTHDAY

Editors: V.G. Romanov, S.I. Kabanikhin Yu.,E.Anikonov and A.L. Bukhgeim

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Preface July 21, 2002 is a significant date for the outstanding mathematician, Lenin and State prizes winner, Academician Mikhail Mikhailovich Lavrentiev. This day he will reach his seventieth anniversary. Μ. M. Lavrentiev started scientific researches when he was a student of the Mechanics-Mathematics Faculty of Moscow State University. His first articles, giving estimates of accuracy of the solution to linear algebraic systems with a large number of unknowns, date back to 1953-1954. Having graduated from the University in 1953, Lavrentiev continued his scientific work as a postgraduate student under the supervision of Academician S. L. Sobolev. His results concerning the inverse problem of potential theory and the Cauchy problem for elliptic equations were published in 1955-1957 and formed the basis of his candidate dissertation. Since 1957 Lavrentiev has lived in the so-called Academgorodok, a special district in Novosibirsk which was intended to become a new Scientific Center in Siberia. Up till now Academgorodok has remained the heart of Siberian Branch of the Russian Academy of Sciences. On arriving in Novosibirsk, Lavrentiev got a position of senior scientific researcher at the just formed Institute of Mathematics and before long he became the chief of a laboratory at the institute. Here he obtained results which later would be regarded as classical and fundamental in the theory of ill-posed problems. Based on these results, Lavrentiev wrote his doctoral dissertation and defended it in 1961. At the same time he took part in the scientific works initiated by the national defence program and won Lenin prize for this activity. In 1963 Lavrentiev organized Department of mathematical problems of geophysics in a new institute — the Computing Center of Siberian Branch of the Academy of Sciences. At the same time he began to develop a new scientific direction — the theory and applications of inverse problems for differential equations, i.e., the problems of determining the coefficients of differential and partial differential equations from additional information concerning the direct (forward) problem solution. One of examples of the inverse problem is the problem of investigation of Earth's structure using geophysical measurements on its surface. Fruitful collaboration with colleagues from the institute of Geology and Geophysics made it possible to discuss and formulate a wide class of inverse problems of geophysics.

vi

Μ. Μ.

Lavrentiev

In 1968 Lavrentiev was elected as a corresponding member of the Academy of Sciences, and in 1981 as an academician. In 1987 Μ. M. Lavrentiev and his colleagues V. G. Romanov, Yu. E. Anikonov, V. R. Kireitov and S. P. Shishatsky were awarded the state prize for the results in the theory of inverse and ill-posed problems. For his activity as a scientist and professor Μ. M. Lavrentiev was awarded several other state prizes as well. Since 1986 Lavrentiev has been the director of the Sobolev Institute of Mathematics of Siberian Branch of the Russian Academy of Sciences. The last 15 years were extremely difficult for the institute because of perestroika and financial problems. Due to his organization experience and activity the institute holds high position in the science and is successfully developing. All leading scientific schools keep working. Many new doctoral and candidate dissertations have been defended during the last 15 years. Μ. M. Lavrentiev is the author of many fundamental scientific results in many directions of mathematics and its applications, namely, in differential equations, inverse and ill-posed problems, tomography, numerical and applied mathematics. A. N. Tikhonov, V. K. Ivanov and Μ. M. Lavrentiev created a new scientific direction of modern mathematics — the theory of ill-posed problems of mathematical physics and analysis, which is of considerable theoretical and practical value. By ill-posed problems are meant the problems which do not satisfy one of the classical well-posedness conditions such as uniqueness, existence and stability of the solution. Ill-posed problems are usually unstable with respect to small variations of data. A wellknown example is the Cauchy problem for Laplace equation. It turns out that some a priori information about the ill-posed problem solution makes the solution stable and allows one to construct approximations with an arbitrary accuracy. This fact was remarked by Academician A. N. Tikhonov in 1943 for an inverse problem of potential. Thus ill-posed problems become theoretically and practically valuable provided some a priori information concerning their solutions is given. Taking into account this fact Μ. M. Lavrentiev introduced the concept of a conditionally well-posed problem (Tikhonov well-posedness). The solution of conditionally well posed problems (or a well-posed problem in the sense of Tikhonov) is stable with respect to those variations of data which keep the solution in a priori given

Μ. Μ.

Lavrentiev

vii

set (the set of well-posedness). Thus the first question arisen in considering ill-posed problems is the uniqueness of their solution. The uniqueness of the solution garantees stability by the well-known theorem which claims that on every compact set an operator inverse to a continuous operator is continuous as well. One of the most important aspects in the theory of ill-posed problems is the construction of stable approximations in the case of noised data. Μ. M. Lavrentiev investigated the Cauchy problem for the Laplace equation and proposed a method of founding the approximate solutions. He constructed an auxiliary family of equations by adding to the original Laplace equation a differential operator of a higher order with a small parameter. The Cauchy problems for the auxiliary family of equations are well-posed and solutions tend to the exact solution of the original problem provided the small parameter is specially connected with the level of the noise and both of them approach to zero. This method was developed by Lattes and J.-L. Lions and is known as the method of quasi-reversibility. M.M. Lavrentiev proposed and justified a new effective method of solving linear and nonlinear operator equations of the first kind. In this method the information about the modulus of continuity of the inverse operator is used to construct a sequence of approximate solutions. These results were included in the wellknown book "Some Ill-Posed Problems of Mathematical Physics" published in 1962. The proposed method is widely used and is called the method of Lavrentiev regularization. The problem of continuation of an analytical function from the set of its uniqueness (for example, from a subdomain, or from some curve, or from a discrete infinite set of points which has a limit point inside the domain of analyticity) was considered by Lavrentiev in series of papers. He estimated the stability and constructed numerical algorithms for its solution. Those results were used in solving problems of continuation of solutions of differential equations which are very important for natural sciences, particularly in geophysics. Lavrentiev's results in the theory of inverse problems for differential equations and in tomography are well-known all over the world. Μ. M. Lavrentiev and his colleagues found a close connection between inverse

viii

Μ. Μ.

Lavrentiev

problems for differential equations and problems of integral geometry. One of the examples is the well-known Radon problem consisting in finding the function from its integrals over all straight lines. It turned out that curves and surfaces of integration are connected with coefficients of the differential equations and may have very complicate structure. Theory and practical applications of integral geometry were developed by Μ. M. Lavrentiev and many new results for the inverse problems for differential equations were obtained. Lavrentiev generalized the main properties of such problems and formulated a problem of investigation of Volterra operator equations. Μ. M. Lavrentiev was the first who formulated the mathematical problems of photometry. With the group of colleagues he developed this direction which has a very important applications in the interpretations of aerospace pictures, specifically in determining the relief of Earth's surface and characteristics of its optical brightness. Many theoretical results of Μ. M. Lavrentiev were applied to the problems of geophysics, mechanics, biology, ecology, medicine and other sciences. One of the main properties of his scientific approach is the deep understanding of the essence of the practical problem and his aspiration for complete theoretical investigations and practical applications. Μ. M. Lavrentiev was a superviser of more than 100 candidate (Ph. D.) and dozens of doctoral dissertations. His former students are working in many scientific and educational centers of Russia and other countries. His scientific school is very prominent worldwide. Only during the last few years under his leadership several big international conferences were organized in Sobolev Institute of Mathematics, such as International Conference of Tomography (1993), International Conference on Inverse and Ill-Posed Problems (1998), 4 Congresses on Industrial and Applied Mathematics. The teaching and organizing activity of Μ. M. Lavrentiev is very high. Prom the moment of foundation of Novosibirsk State University he is working at the Mathematical Department. He has read a lot of main and special courses for graduate and postgarduate students. He is a Chief of the chair of the theory of functions, the member of several scientific councils of the University, the Chief of a special scientific counsil on the defense of doctoral dissertations. Seven years Μ. M. Lavrentiev was the Dean of the Mathemat-

Μ. Μ. Lavrentiev

ix

ical Department. Μ. Μ. Lavrentiev is the Editor-in-Chief of the Siberian Mathematical Journal, the Siberian Journal of Industrial Mathematics and the international Journal of Inverse and Ill-Posed Problems. He was the Editor of very many Proceedings, books and textbooks. Many years he was the member of Supreme Council of the Siberian Branch of Russian Academy of Sciences and the deputy Academician-Secretary of the Mathematical Branch of the Russian Academy of Sciences. Μ. M. Lavrentiev is very active during the last years as well. Recently he has published the book "Theory of operators and inverse problems" (together with L. Y. Saveliev) , the book "Numerical modeling in tomography and ill-posed problems" (together with S.M. Zerkal and O.E. Trofimov), the textbook of mathematical analysis for the students of Novosibirsk State University in 4 parts (together with S.I. Kabanikhin and A.N. Nazarov), and many scientific papers in different journals. He has delivered lectures in many prominent Universities and scientific centers all over the world. Μ. M. Lavrentiev is very tactful and benevolent with all his colleagues and students. He has a very seldom ability to be sincerely glad to listen and discuss the new scientific results and ideas of his students and colleagues and he always tries to support all new achievements. Scientific generosity, friendliness, ability and desire to help bring him respect and gratitude of his colleagues and students. Μ. M. Lavrentiev meets his 70-th anniversary with creative power. All his collegues, students and friends wish him a good health, joy, happiness, new successes in his all-round activity. Romanov V. G., Editor-in-Chief Kabanikhin S. I., Managing Editor Anikonov Yu. E. Bukhgeim A. L.

CONTENTS Representations of functions of many complex variables and inverse problems for kinetic equations Yu. E. Anikonov

1

Uniqueness in determining piecewise analytic coefficients in hyperbolic equations Yu. E. Anikonov, J. Cheng and M. Yamamoto

13

Direct and inverse problems for evolution integro-differential equations of the first-order in time J. S. Azamatov and A. Lorenzi

25

How to see waves under the Earth surface (the Β C-method for geophysicists) Μ. I. Belishev

67

Global theorem of uniqueness of solution to inverse coefficient problem for a quasilinear hyperbolic equation A.M. Denisov

85

Identification of parameters in polymer crystallization, semiconductor models and elasticity via iterative regularization methods H. W. Engl

99

The tomato salad problem in spherical stereology R. Gorenßo

127

Two methods in inverse problem and extraction formulae M. Ikehata

145

Identification of the unknown potential in the nonstationary Schrödinger equation A. D. Iskenderov

179

Iterative methods of solving inverse problems for hyperbolic equations S. I. Kabanikhin

201

Carleman estimates and inverse problems: Uniqueness and convexification of multiextremal objective functions Μ. V. Klibanov

219

xii

Contents

Convergence analysis of a Landweber—Kaczmarz method for solving nonlinear ill-posed problems R. Kowar and O. Scherzer

253

A sampling method for an inverse boundary value problem for harmonic vector fields R. Kress

271

Approaching a partial differential equation of mixed elliptic-hyperbolic type R. Magnanini and G. Talenti

291

Complex geometrical optics solutions and pseudoanalytic matrices G. Nakamura and G. Uhlmann

305

Numerical solution of inverse evolution problems via the nonlinear Levitan equation F. Natterer

339

An inverse problem for a parabolic equation with final overdetermination A. I Prilepko and D. S. Tkachenko

345

Uniqueness theorems for an inverse problem related to local heterogeneities and data on a piece of a plane V". G. Romanov

383

On ill-posed problems and Professor Lavrentiev P. C. Sabatier

399

Regularization and iterative approximation for linear ill-posed problems in the space of functions of bounded variation V. V. Vasin

403

A posteriori error estimation for ill-posed problems on some sourcewise represented or compact sets A. G. Yagola and V. N. Titarenko

425

Multidimensional inverse problems for hyperbolic equations with point sources V. G. Yakhno

443

Ill-Posed and Inverse Problems, pp. 1-12 S.I. Kabanikhin and V. G. Romanov (Eds) © VSP 2002

Representations of functions of many complex variables and inverse problems for kinetic equations Yu. E. ANIKONOV* Abstract — In this paper, formulas for solutions of multidimensional inverse problems for kinetic equations are obtained with the use of classical representations of functions of many complex variables.

1.

STATEMENT OF THE PROBLEM. PRELIMINARY RESULTS

In this paper we establish the relation between multidimensional inverse problems for kinetic equations and representations of analytic functions of many complex variables, in particular, Bochner's, Leray's, and Calderons' representations [1, 4]. We note that the use of the above-mentioned representations of functions of many complex variables in the theory of inverse problems for kinetic equations is based on the formulas for solutions and coefficients of kinetic equations obtained by the author [2, 3]. We consider the following multidimensional nonlinear inverse problem for the kinetic equation: find complex-valued functions w(x,p, t) and X(x, t), dX/dpj = 0, j = 1 , 2 , . . . ,n, in the domains χ 6 D C W, ρ G Dx C R n , 'Sobolev Institute of Mathematics, Siberian Branch of the Russian Acad. Sei., Acad. Koptyug prosp. 4, Novosibirsk 630090, Russia. E-mail: [email protected] The work was supported by the RFBR (grant No. 02-01-00255) and by the Siberian Branch of the Russian Acad. Sei. (integration grant No. 43, 2000).

2

Yu. Ε. Anikonov

a < t < αϊ, if dw st"+

η

^ J=1

dw

t«|x=o = wo(t,p),

Λ =

.

.

'

.

α < < < αχ,

.

(1.1) (1.2)

ρ Ε Di.

Here Z? and Z?i are domains of the Euclidean space Κ™, η > 1; f(q,p) ψ 0, q G C 1 , ρ G D\; and wo(t,p), a < t < α\, ρ € D\, are given complex-valued functions. As in the previous author's papers, trying to approach the nonlinearities of the collisional and other terms of kinetic equations arising in applications, we do not exclude the dependence of the sought function X(x, t) on the solution w(x,p,t) in the form X(x,t) = J(w,x,t), where J is a functional independent of ρ, for example, w(x,p, t) dp Jl and \(x, t) is some function, or in the form of a more general functional

In particular, with account of this remark, the right-hand side of (1.1) may have the following form acceptable for applications (see [5]): \(x,t) ip(w,p) = \(x,t)(^

ιυ(χ,ρ,ί)άρ^

[u> — e

p2 / 2 ]

Let Μ be a A;-dimensional manifold in the complex space C"; β (y) = (βι,... ,ßn), y G M, be a continuous vector-function; and ui(y) be a differential form of fc-th degree continuously differentiable on M. Let A(q, y) and Β(ξ,η), q € C 1 , y 6 Μ, ξ Ε Μ", η Ε I T , be continuously differentiable complex-valued functions and let a differentiable function Φ(q,p), q Ε C 1 , ρ Ε R n , be the inverse function to the primitive

where ρ E D\ C M" plays the role of a parameter. Formal results of general character concerning representation of the solutions w(x, p, t) and λ (a;, t) are formulated in the following two theorems.

Representations

of functions of many complex variables

Theorem 1.1. The functions w(x,p,t) mulas

X(x,t)=

[ JΜ

3

and X(x,t) defined by the for-

A'((ß(y),y)t-(x,y),y)uj(y),

where λι _ 9A(q, y) satisfy the equation dw

dw j=1

..

[ JΜ

.

and X(x,t) defined by the for-

A'(t-(x,ß(y)),y)u(y),

where

satisfy the

.

J

Theorem 1.2. The functions w(x,p,t) mulas

X(x,t)=

.

dAiq^l equation dw

dw j=1

Λ

.

.

.

.

J

Theorems 1.1 and 1.2 are proved by direct substitution of w(x,p, t) and X(x,t) into the equation dw

dw j=1

J

w

/

χ

4

Yu. Ε. Anikonov

In virtue of Theorems 1.1 and 1.2, the solution w(x,p,t) for given .-*•")·

Theorem 3.1. Assume that for all t: \t\ < α the data wq{t,p) = f(t,p) of the inverse problem can be analytically continued in ρ into the polydisk \zj\ < Rj, j = 1,..., n. Then the functions w(x,p, t) and X(x, t) defined by

8

Yu. Ε.

the

Anikonov

formulas

w(x,p,

t) v

= Φ(

I Μ...

••·

I _ D .

I..

n

I — Ο

' dyi

7=1 Vj

I

Pi

dy n

yi

+

Vn

J·

(3-2)

= Jf

-

| ä i ι . .j ./ iι | = d

f

J

ι.,

\yι n\-Rn d

ot\

j^

3—

yj

J yi

J-

yn

(3-3)

τι ι Ix-i j=1

J

where Σ ^...AnW*?1 fcl ). . . ykn

g(t,z)=

is uniquely

determined

···*£"

from the data of the inverse

Zkl • • • 4n

W0 ( t , z ) = f ( t , z ) = Σ ^...k^t) k 1 y.^ykfl by the

formula

satisfy

the kinetic

problem

equation dw

du; j=l

and the data /(f,z)|z=p =

. .

.

.

.

^

wo(t,p).

P r o o f . B y T h e o r e m 1.2, t h e f u n c t i o n s w(x,p,t) formulas (3.2) a n d (3.3) satisfy t h e e q u a t i o n dw

dw j=l

J

w

/

a n d X(x,t)

\

defined by

Representations

of functions

of many complex

variables

We have So(i>p)=

f

...

|yi |=Äi

_g{t,y) _ j M L · dyi

[

|yn|=Än

d y,

1

Vn Vi y i ι - j=ι ς — By assumption of the theorem, the function wq {t,p) can be continued analytically into the polydisk

L·

L·

\zj\

0.

(2.1)

Yu. Ε. Anikonov, J. Cheng and M.

16

Since p,q G V, we can choose 0 = ho < h\ 0

(2.4)

w(x, y, 0) = wt(x, y, 0) = 0, wy(x,0,t)

xeR2,y>0

(2.5)

2

xeE ,i>0

(2.6)

jxj < po, 0 < t < p0.

(2.7)

= 0,

and w(x,0,t)=0,

We take even extensions in y and t of w, ρ and / , Λ. Using the same notations, we can see by (2.5) and (2.6) that w is of class Cl in R 3 χ and of class C 2 inM 3 χ (Ο,οο) and Μ3 χ (-οο,Ο), p,f Ε C(R3), λ € C(R 3 x Mt). Here and henceforth we set = {t \ t G R} for specifying a domain of t. Moreover we see that wtt(x, y, t) =Aw(x, y, t) + p{x, y)w(x, y, t) + f{x, y)X(x, y, t), χ G R 2 , y G R, t G R

(2.8)

w(x, y, 0) = wt(x, y, 0) = 0, wy(x, 0, t) — 0,

χ G R2, y G R

(2.9)

xGR2,iGR

(2.10)

\x\ < po, \t\ < p0.

(2.11)

and w(x,0,t)

= 0,

17

Uniqueness for piecewise analytic coefficients

2.2.

Second step

Here we show be continL e m m a . Let h > 0, xo G IK2, ρ = p(x,y) and F = F(x,y,t) 3 3 uous and bounded in R and R χ [Ο,οο) respectively. Let υ = v(x,y,t) G C 2 (M 3 χ (Ο,οο)) Π C ^ R 3 Χ [Ο,οο)) satisfy vtt(x, V, t) = Av(x, y, t) + p(x, y)v(x, y, t) + F{x, y, t), χ Ε Μ2, y Ε R, t > 0

(2.12)

and v(x,y,0)

= vt(x,y,Q)

= 0,

xGR2,yGR

(2.13)

We further assume that p(x, y) > 0, χ G R 2 , y G Μ and F(x,y,t)>0 F(x,y,t)>

if y/\x - x0\2 + {y - hf + t < p2, y < h, 0

t>0

if y/\x - x0\2 + {y - h)2 + t < p2, y > h, t > 0.

Then there exists μ = μ(ρ) > 0 such that if 0 < p2 < μ, then the following inequalities hold: v(x,y,t)>

0

if y/\x - x0\2 + {y - h)2 + t < p2> y > h, t > 0.

(2.15)

In Appendix, we will give the proof. Taking the even extensions in y of u = u(q), ρ and G, we obtain u G C 2 ( R 3 χ (Ο,οο)) Π C 1 ( R 3 χ [Ο,οο)) and ' utt{x,y,t)

= Au(x,y,t)

+ p(x,y)u{x,y,t)


0

ku(z,y,0)

= ut{x,y,0)

= 0,

(1.1')

xGK ,yGR. 2

Setting xo = 0 and h = 0, by (1.3) we apply Lemma and obtain \{x,y,t)

= u{q){x,y,t)>

0,

V|x|2 + y2 + t < p3, y > 0, t > 0

with a sufficiently small p3 > 0. We can further assume that pz 0 and C > 0 such that \f(x,y)\ > Cya if \F \x — XQ\2 + y2 < p± and y > 0 (e.g. Lemma 1.2 (p.4) in Anikonov [1]). Without loss of generality, we may assume that p4 < Consequently, since / is even in y, we obtain f(x,y)

> 0

if \/\x — XQ\2 + y2 < pA and y Φ 0

(2.18)

or if V\x-x0\2

f(x, y)< 0

+ y2 < Pi and y φ 0.

(2.19)

Without loss of generality, we can assume (2.18). Otherwise, we set \{x,y,t)

=u(p)(x,y,t),

w(x,y,t)

= u(q)(x,y,t) - u{p){x,y,t),

(2.3')

and it is sufficient to apply the present argument to u>tt(x, y, t) =Aw(x, y, t) + q{x, y)w(x, y, t) + (-f{x,

y))\{x, y, t),

2

x£R ,y>0,t>0.

(2.8')

We return to the case (2.18). We can take small p^ > 0 such that y/\x - x0\2 + y2 < p4 implies \/\x\2 + y2 < P3 by |a;o| < ΡΆ· Therefore, in terms of (2.16) and (2.18), we have f{x,y)X(x,y,t)

>0,

y/\x - x0\2 + y2 + t < Pi, y φ

0,

t>

0.

(2.20)

By (2.20), setting h = 0, we apply Lemma to (2.8) and (2.9), so that there exists a sufficiently small ps > 0 such that W(x, y,t) > 0 if — XQ\2 + y2 + t < P5> y > 0 and t > 0. Therefore w(x,0, t) > 0 if \x — xo| + 1 < ps and t > 0. This contradicts (2.11). Thus (2.17) holds. Since / is analytic in {(x,y) I χ € M2, 0 < y < hi], we see that f(x,y)=

0,

i£R

2

,

0 0. 2.5.

P r o o f of (2.23)

Applying (2.21) in (2.8), we obtain wtt{x,y,t)

= Aw(x,y,t)

and

+p(x,y)w(x,y,t),

w(x, 0, t) = Wy(x, 0, t) = 0,

χ G I 2 , 0 < y < hi, t G Μ |s| < Po, |i| < Po.

Then noting that ρ is analytic in {(x, y) \ χ G M2, 0 < y < hi}, we can apply the global Holmgren theorem (e.g. Theorem 1 (p.42) and Corollary 8 (p.49) in Rauch [13]), so that (2.23) is seen. 2.6.

Fifth step

We will consider the problem in R 2 x {y > hi}: wtt(x, y, t) =Aw(x, y, t) + p(x, y)w(x, y, t) + f(x, y)\{x, y, t), χ G E 2 , y > hi, t G M, u;(a;,y,0)=«; i (a;,i/,0) = 0,

(2.24) xeR2,y>hu

(2.25)

and w(x, hi,t)

= wy(x, hi,t)

= 0,

\x\ < po - hi, \t\ < po - hi.

(2.26)

We choose pe > 0 so small that {(x,y,t)

I y/\x\2 + (y-hi)2

+

C {(a:,y,t) \

t0} + y 2 + t < p0, y > 0, t > 0}.

(2.27)

By (1.3) we see G(x,y,t)>0,

Λ/\Χ\2

+ (y-hi)2

+ t < p6, t > 0.

(2.28)

20

Yu. Ε. Anikonov, J. Cheng and M. Yamamoto

Hence we can apply Lemma to λ = u(q) in (1.1'), so that there exists a sufficiently small p-j > 0 such that ρτ < po — hi and X(x,y,t)>

0,

\/M 2 + ( y - / i i ) 2 + i 0.

y>hu

(2.29)

Next we will prove f(x, y) = 0

if |a;| < P7 and hi < y < h2.

(2.30)

Contrarily assume that / does not vanish identically in {(x,y)

| |a:|
0

f{x, y) < 0

if y/\x — £o|2 + (y — hi)2 < p8 and y > hx

if

- x0\2 + [y - hi)2 < p8 and y > hv

(2.31)

(2.31')

By the same argument in taking the case (2.18), we can assume that (2.31) holds. On the other hand, we can assume that {(x,y,t) \ 2 2 2 2 y/\x — £o| + (y — hi) + t < PS, t > 0} c {(x,y,t) | y/\x\ + {y-hi) +t < P7, t > 0}, because |xo| and p& can be arbitrarily small. Therefore by (2.21), (2.29) and (2.31), we obtain = 0 f(x,y)X{x,y,t)


0

if and

^\x-x0\2 y < hi,

+ (y-hi)2

+ t

0

y/\x — xo\2 + (y — h\)2 + t < ps, y > hi,

(2.32)

t > 0.

Applying Lemma to (2.24) and (2.25), and noting (2.32), we obtain w(x,hi,t) > 0 if \x — zol a n d t > 0 are small. This contradicts (2.26). Thus the proof of (2.30) is complete. Continuing this argument until hi > po with some I > N, we complete the proof of our main result.

APPENDIX. PROOF OF LEMMA Without loss of generality, we may assume that h = 0 and xo = 0. We set Z = (x, y) G R 3 with X e R 2 and y € M. By the Kirchhoff formula (e.g.

Uniqueness for piecewise analytic coefficients Vladimirov [16]), υ — v(z,t)

is given by

ρ({Μί,ί-|*-ί|) ι

r

21

F&t

άξ

\z

(l)

zeR\t>o.

Let us set K(p2) =

{{z,t)\\z\+t0}.

Then we can directly verify that (ξ,ί-\ζ-ξ\)€Κ{ρ2)

it (z,t)e

Κ fa) and \ζ-ξ\0 ~ 4π4π1\ζ-ξ\β V ~

by (8). Therefore (7) implies the conclusion (2.15) of the lemma. Acknowledgement This paper has been written during the stay of the first named author in Japan and the stay has been supported partly by Sanwa Systems Development Co., Ltd (Tokyo, Japan). The second named author is partly supported by NSF of China (No. 19971016) and the research fund from Fudan University in Shanghai. The third named author is partly supported by Sanwa Systems Development Co., Ltd (Tokyo, Japan). The authors thank Professor Alexandre L. Bukhgeim for his valuable comments.

Uniqueness for piecewise analytic coefficients

23

REFERENCES 1. Yu. E. Anikonov, Multidimensional Inverse and Ill-posed Problems for Differential Equations. VSP, Utrecht, 1995. 2. Ju. M. Berezanskii, The uniqueness theorem in the inverse problem of spectral analysis for the Schrödinger equation. AMS Transl. Ser. 2 (1964) 35, 167-235. 3. A. L. Bukhgeim and Μ. V. Klibanov, Global uniqueness of a class of multidimensional inverse problems. Soviet Math. Dokl. (1981) 24, 244-247. 4. V.l. Dmitriev, Inverse problems in electrodynamical prospecting. In: Ill-posed Problems in the Natural Sciences. A. N. Tikhonov and Α. V. Goncharsky (Eds). Mir Publishers, Moscow, 1987, 77-101. 5. V. B. Glasko, Inverse Problems of Mathematical Institute of Physics, New York, 1988.

Physics. American

6. O.Yu. Imanuvilov and M. Yamamoto, Global uniqueness and stability in determining coefficients of wave equations. Commun. in Partial Differential Equations (2001) 26, 1409-1425. 7. O.Yu. Imanuvilov and M. Yamamoto, Global Lipschitz stability in an inverse hyperbolic problem by interior observations. Inverse Problems (2001) 17, 717-728. 8. V. Isakov, Uniqueness of the continuation across a time-like hyperplane and related inverse problems for hyperbolic equations. Commun. in Partial Differential Equations (1989) 14, 465-478. 9. V. Isakov, Inverse Problems Springer-Verlag, Berlin, 1998.

for

Partial

Differential

Equations.

10. Μ. V. Klibanov, Inverse problems and Carleman estimates. Problems (1992) 8, 575-596. 11. Μ. M. Lavrentiev, Some Improperly Posed Problems of Physics. Springer-Verlag, Berlin, 1967.

Inverse

Mathematical

12. Μ. M. Lavrent'ev, V. G. Romanov, and S. P. Shishatskii, Ill-posed Problems of Mathematical Physics and Analysis. American Mathematical Society, Providence, Rhode Island, 1986.

24

Yu. Ε. Anikonov,

J. Cheng and M.

Yamamoto

13. J. Rauch, Partial Differential Equations. Springer-Verlag, Berlin, 1991. 14. V. G. Romanov, Integral Geometry and Inverse Problems for bolic Equations. Springer-Verlag, Berlin, 1974. 15. V. G. Romanov, Inverse Problems of Mathematical ence Press, Utrecht, 1987. 16. V. S. Vladimirov, Equations Moscow, 1984.

of Mathematical

Hyper-

Physics. VNU Sci-

Physics. Mir Publishers,

17. M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems. J. Math. Pures Appl. (1999) 78, 65-98.

Ill-Posed and Inverse Problems, pp. 25-65 S. I. K a b a n i k h i n a n d V. G. Romanov (Eds) © V S P 2002

Direct and inverse problems for evolution integro-differential equations of the first-order in time J. S. AZAMATOV* a n d A. LORENZI* Abstract — In this paper we study first-order operator integro-differential equations of the form (*) u'(t) — Aiu(t) — f* h(t - s)A 2 u(s) ds = f(t) , in the bounded time interval [0, T], under the assumption that operator A2 dominates A\. We prove existence and uniqueness results for the solutions of both direct and inverse problems related to (*). The inverse problem consists of recovering, in addition to it, also the kernel h appearing in the integral term. The results so found are applied to integro-differential partial equations governing thermal materials with memory related to spatial cylindrical domains.

1.

INTRODUCTION

T h e first aim of this p a p e r consists of dealing w i t h the following (direct) firstorder integro-differential abstract Cauchy problem in a real Hilbert space Η endowed w i t h the scalar product (·, ·) and n o r m || · ||: t € [0,T], u(0) = u0, where Τ > 0, u0 € Η a n d functions / : [0, T] ->• H, h : [0, T]

(1.1) (1.2)

R are given.

* Novosibirsk t Milan. The author is a member of GNAMPA of the Italian Istituto Nazionale di Alta Matematica (INdAM). Partially supported by the Italian Ministero dell'Istruzione, dell'Universitä e della Ricerca.

J. S. Azamatov and A. Lorenzi

26

As far as operators Aj : T>(Aj) C Η — Η , j = 1,2, are concerned, we will assume, for the time being, that they are linear and closed. We recall that direct problems of the form (1.1), (1.2) were intensively studied and solved in the eighties in the framework of Banach spaces by many authors (cf. e.g. [2], [3], [4], [5], [6], [7] and the references thereupon) under the basic assumptions that h is not differentiable and either T>(A\) C V{A2) or Αχ = Ο. On the contrary, the aim of this paper consists of investigating the simpler case of differentiable kernels h, but under the more difficult assumption V(A2)CV(A1),

(1.3)

involving the domains of A\ and A2. In other words, condition (1.3) states that the differential operator Dt — A\ is not the dominant part in the integro-differential equation (1.1). On the contrary, the dominant part is the integro-differential operator Dt — h* A2u, where * stands for convolution, i.e. h* f(t) = f j h(t — s)f(s) ds. To reduce problem (1.1) to a Cauchy problem with a dominant differential part we are forced to differentiate both sides in (1.1) to obtain the following second-order Cauchy problem, where ho = h(0): u"{t) - AIU'(t)

- h0A2u{t)

- [ h'(t Jo

u(0) = uo,

S)A2U(S)

ds = f'(t),

t G [0, T], (1.4)

«'(0) = Aiuo + /(0) := «χ.

(1.5)

Along with the direct Cauchy problem (1.1), (1.2) we will be dealing also with the identification problem u'(t) - AlU{t) - [ h(t — Jo

S)A2U(S)

φ[«(ί)] = ί/(ί),

ds = /(£),

t

E

[0,T],

(1.6)

«(0) = u0,

(1.7)

te[0,T],

(1.8)

consisting of determining the pair (u,h), when we are given the additional information (1.8), g : [0,T] — Ε and Φ being, respectively, a given function and a linear continuous functional defined on the whole of H. We note that, when V{A2) D V{A{), problem (1.6)-(1.8) was studied and solved in the last fifteen years by many authors both in the framework of Banach spaces (cf., e.g., [9], [15], [17], [16]) and in the framework of Hilbert spaces (cf., e.g., [11], [12], [13], [20]). In the latter case a deeper investigation

Evolution

integro-differential

27

equations

was carried out in the case A\ — A2 = A for a large class of self-adjoint operators A and linear functionals Φ. Our aim is to investigate the identification problem (1.6)—(1.8) under the less usual condition (1.3) in the framework of Hilbert spaces. Of course, problem (1.6)—(1.8) will be reduced to the following second-order identification problem, where ho — h(0): u"(t)

- AlU'(t)

-

h0A2u(t)

- f h'(t-s)A2u(s)ds Jo

= f'{t),

u{0) = u0,

u'(0) = A1u0

Φ[Μ(ί)] = »( —κ min (mi, ho) for some κ € [0,1).

(2.3)

For this purpose we need to introduce the triplet (Η, W\, W2) of real Hilbert spaces satisfying the following properties: HI Wj «-» Η Μ· W* is a Hilbert triplet, j = 1,2, where W* denotes the dual space to Wj; H2 W2 W\ with ||υ||ιν, > ||v|| for any υ € W\ and any ν Ε

> IMIwi for

HS Wj ( j = 1,2) are separable Hilbert spaces which are dense in H. As far as operators A\ and A2 are concerned, we assume the following: H4 Aj e C(Wj-,W*)

( j = 1,2) and

— (AjV,w)j

= a,j(v,w),

Vv,w G Wj, j = 1,2,

where (·, ·)0 denotes the pairing between W* and Wj, j = 1,2, and a j '· Wj x Wj M, j = 1,2, are continuous positive definite Hermitian forms satisfying Mu,u;)|

< Mj\\v\\Wj\\w\\wj, Mv,w € Wj,

mjWvWwj
j} e j which is orthonormal and complete in Η, as well as orthogonal and complete in W\ and W2 when the latter are endowed with the scalar products αϊ and a2> respectively. Moreover, {vj}ej consists of (common) eigenvectors of A\ and A2: —AmVj = ßmjVj

for any j e J, m = 1, 2.

Remark 2.1. The Hilbert spaces W2+j

= {uE

Wj : AjU

G H},

j = 1,2,

(2.4)

endowed with the respective graph-norms, satisfy the continuous embeddings W3 -»· Wi,

Of course, Aj G C(W2+j]H),

WA M- W2.

j = 1,2.

Moreover, only when dealing with our identification problem, we shall assume H6 WA M· W3 M· W2 • Wi.

Remark 2.2. According to property Η5 operators A\ and A2 commute. As far as the data are concerned, we list the following assumptions related to the Sobolev spaces Ws'p((0,T); Ζ), Ζ being a Hilbert space. Such assumptions are not independent, since they will be chosen according to the problem to be dealt with: H7 h{0) > 0, h'(0) < 0;

H8 I G W 1 , 1 ((0,T); i f ) , ζ G W ^ f t O . T ) ; Wi), u0 G WA, ui,z(0) G Wy, H9 f G W2'2((0,T)-H),

h G W 2 ' 2 ((0,T);R), U0 G W4, m G

H10 Φ G H*· Hll

i) ho > 0, hx < 0, ii) / G Wa'2((0,T)-W4), g G W 5 ' 2 ((0,T); E), iii) UQ,U\,VQ,V\, A\u0, A2U0, A2VQ G W4, iv) U2,AIUI,M[Mu\ + /(0)],

J. S. Azamatov and A. Lorenzi

30

A2V1 Ε W3, where ho is defined by (1.15) and ui = Amo + f(0), u2 = Am

+ h0A2u0 + /'(0),

hi = {Φμ 2 η ο ]} _ 1 {5 { 3 ) (0) - Φ[ΛΙ«2 + hoA2Ul + /"(0)]}, v0 = Ai[Aim

+ /(0)] + h0A2u0 + /'(0),

«ι = Ai{Ai[Aiui + /(0)] + h0A2uQ + /'(Ο)} + hoA2[AlUo

+ /(0)] + hiA2uo + /"(0).

In this section we are going to prove the following existence, uniqueness and continuous dependence theorem for the solution to our direct differential problem. Theorem 2.1. Under assumption H1-H5, H9 and (2.3) problem (2.1), (2.2) admits a unique solution u € W 2 '°°((0,T);i7) η Wl'°°({0,Ty,W2) Π W 2 ' 2 ((0,T); W\) satisfying the following estimate: IMIvy2.°°((0,t);J/)

+

IMIv^KO.t);^) + ΙΙ"ΙΙν^!.«>((0,ΐ);ΐν2)

< CI[II^O||2m,4 + Il«i||2w, + \\m\\ + ||*(0)|&, + IIH&i. The positive constant C\ depends on κ, mi,m2,

E ( 0 , T ) . (2.5) M2, ho, fco, only.

Remark 2.3. From assumptions H2, H4 and Theorem 2.1 we immediately deduce that AlU' 6 W ^ f t O . T ) ; W f ) and A2u G W 1 ' o o ( ( 0 , r ) ; W?) M· W 1 , o o ((0,T); W*). Before proving Theorem 2.1 we recall the following definition according to [14]. Definition 2.1. A weak solution to problem (2.1), (2.3) is any function u belonging to W 1 , o o ((0,T); W\) x Z°°((0,T); W2) and satisfying u(0) = u0 and the following equation for all V G Φ, where (·, ·) denotes the scalar product in Η and the Banach space Φ is defined by

Evolution integro-differential equations

31

Φ = { ψ € ^ 1 ( ( 0 , Τ ) ; ^ 1 ) η Ι 1 ( ( 0 , Τ ) ; ^ 2 ) : ψ(Τ) = 0}: Jο [(1(ή,ψ(ί))+αι{ζ(ή,-ψ{ή)]άί.

+ Λ 0 αι(«(ί), V»(*))] } dt - tiiV(O) = J

(2.6)

Proof. In order to solve problem (2.1), (2.2) we will use the FaedoGalerkin method with the special orthogonal basis { d j } j e j introduced in H5. Let us consider the Cauchy problem for the following ordinary secondorder differential system: m « m j W + Σ ««,»(*) [°i («»,«,·) + Möi.üj·)] i=1 m + Y^amti(t)[hoa2(vi,Vj) + k0ai(Oi,Oj)] i=l j = l,...,m,

= (l(t),Vj) + ai(z{t),Oj), a m ,i(0) = (uQ,Vi) := ßu

a'mi(0) = (ui,Vj) := ji,

i = l,...,m.

(2.7) (2.8)

According to the definitions of ßi, 7j and assumptions Hll we get 771 m ^ ^ ß i ü i — t u o in W4, —> ui in W3 as m —>• +00. i=l i=l Introduce now the function sequence m m (t) = Σ α "Μ i=1

u

defined by ·

(2·9)

Multiplying both sides in (2.7) by a'mj(t) and summing with respect to j over {1,... , m}, we obtain

+ (1/2) A [h0a2(um{t),um{t))

+fc 0 Ol(«m(i).«m( (1 - «)τηι||υ||^ + («mi + fc0)|M|2 > (1 - ^mM^,

Wv G Wu (2.11)

h0a2(v,v)

+ koai(v,v)

> h0( 1 - κ)α2(ν,ν)

+ (κΗ0 + Α; 0 )αι(υ,υ)

|22 > (1 - K)h0m2\\v\\ I wW2 2>,

Vu e W2.

(2.12)

Integrating with respect to time the first and last sides in (2.10) and taking advantage of (2.11), (2.12), we easily get the following inequalities for any t e [0,T] and any ε e (0,1]: ||t4(f)ll2 + 2(1 - K)mi

f Jo

ds + ( l - K)h0m2\\um(t)\\2W2

< 2 [ [||Z( s )||||^( S )||+M 1 |K S )|| l y 1 lK(s)|k 1 ]d S + c m Jο < 2 r||/(s)||||^(s)||ds + M l £ - 2 Jo Jo

fwzMW^ds

+ Μι ε 2 Γ ||ι4(β)||^ ds + cm, Jo

(2.13)

where (cf. Η4) cm =

||ul,m r + (h0 + \k0\)M2\\u0 ,m Ww2'

(2.14)

From (2.13) with ε = [(1 — «JmiM-f 1 ] 1 / 2 we obtain the inequality i K n W f + t l - « ) ™ ! / lKn( 5)llvvi ds + (1 - K)/i0m2||um(i)||^2 Jo 0 and let assumptions H1-H5, H9 and (2.3) be Then for any h e L2((0,

fulRlled.

(3.21) belongs to

T ) ; H)

the solution u to problem

W l i 2 ((0,r); W 3 )nL 2 ((0,T);

W4) andsatisßes

the

(3.20), following

estimate for all t € [Ο, Τ] : 2

2

P l u l l w i . 2 ( ( 0 , r ) ; H ) + \\Mu\\L2{{0tT).H)

/

2

2

ι

< Ci (J|tio|lw4 + IKl I I l 2 ( ( 0 , T ) ; H ) J · (3.23)

P r o o f . Using the representations (cf. H5)

l(t) = ^^l n (t)vj,

Φ) = j£J

j€J

we obtain the following equalities in W*: -Aiu'(t)

=

-A2u(t)

=

j€J

^ß2,jUj{t)vj, jeJ

μτη,ί > 0,

771=1,2,

j Ε J.

Consequently, problem (3.21), (3.22) is equivalent to the following sequence of scalar Cauchy problems, where j G J: Aiij^

W +

+ k0ßij)uj{t)

= hj{t) -

/iijfc'

* Uj(t),

t e [0,

Γ],

(3.24) Uj{0) =

(3.25)

uoj.

Introduce now the function sequence { v j } j £ j defined by Vj ( ί ) = -μ2βί

(t),

t G [0, T],

je

J.

(3.26)

Observe then that from (3.24)-(3.26) we deduce the following sequence of Volterra integral equations in [0,T], j € J,

W )

+

= OSZ)iκ) j€J

/ , —_ i1 ι < 2(1 - ^o'ΐΙ^ΙΙο,ι,σ,κ)ι 2'(Α 0 Ρ2«ο|| 2 + Λ0 2 ΙΙ'ιΙΙο,2 >σ> *)·

(3.31)

—

Finally, from (3.31) and inequalities (3.12) we get βχρ(-2σΤ)|Μβι2ι0ιΗ < 2(1 - Λ ο ' Π ί ο , ι , σ * ) " 2 ^ " 1 ! ! ^ ! ! 2 + Λ02||/Ι||§,2,Ο,Η)·

(3-32)

Consequently, from (3.32), definition (3.26) and Remark 2.1 we conclude that u belongs to L 2 ( ( 0 , T ) ; and A2u satisfies \\A2U\\12AH x

< 2

exp

(1 -

Κ

1

(2σΤ) ΠΙο,ι,σ,κ) ~2

fa1

U2U01|2

+

Proceeding similarly, we can show that u 6 L2((0,T);

h f

||Zi || 2

2 A H

) .

(3.33)

and A\u satisfies

Ρι«ΙΙο,2,ο,Η < 2 e x p ( 2 a T ) χ ( ΐ - / ι 0 Ί ΐ ^ Ί ΐ ο , ι > σ > κ ) " \ _ 1 ( Ι | ^ ι ^ ο | | 2 +νΊΚιΙΙο,2,ο,Η)·

(3·34)

Evolution integro-differential

equations

43

Finally, from equations (3.21) we deduce that A\u' belongs to L 2 ((0, T); H) and satisfies \\Aiu'\\0i2AH

< (ko

+

||fc'||o,iIff1R)Piu|lo,2,o,ff + h0\\A2u0\\2 + ll^ll 0,2,0 ,H· (3.35)

From (3.33)-(3.35) we conclude that u belongs to W 1 > 2 ((0,T); W3) Π L 2 ( ( 0 , T ) ; Wi) and satisfies estimate (3.23). • Combining Theorems 2.1 and 3.1 and Lemma 3.1, from definition (3.8) we easily derive the following regularity results for the solution u to problem (3.6), (3.7). Theorem 3.2. Let assumptions H1-H5 and H7 be satisfied. If the data fulßll conditions H9, then problem (1.4) —(1.6) admits a unique solution u 6 2 2 w 2 ' ° ° ( ( 0 , Τ)] Η) Π w ' ((0, T ) ; Wi) Π W r l ' o o ( ( 0 , Τ ) · w2) η ^ ( ( O , T ) ; Wz) Π

L2((0,T);W4). 4.

IDENTIFICATION PROBLEMS EQUIVALENT TO (1.9)-(1.11)

In this section first we derive a suitable formulation for our identification problem (1.9)—(1.11). For this purpose we introduce the auxiliary function v{t)=u"{t)

i,p2)) - exp (-2σί)] d^ 2 ~2σ/

ex

P( _ c r ^i)lf(ei)\&Qi f

= ^(^βχρ(-σρ)|/(ρ)μ^2.

βχρ(-σρ 2 ) |/(02)| dftj •

(4.45)

Evolution

5.

integro-differential

49

equations

P R O O F OF T H E O R E M 4.1

To solve the fixed-point system (4.34), (4.35) we introduce the following closed set X(r,a)

= [W2,oo((0,T);H) η

in W(a)

Wi)] x

2

Η^ ((0,Τ);Μ) : Χ Μ η(ο)

€[^σ,Η + μ 2 / " ( ο ) ΐ ι 2 ^ ) 1 / 2 + ^ο|χ|-Ί|Φ|ΙΙΙ/Ιΐ2 1 2,^((2Τ) 1 /2|| 9 || 1 ) 2 ι σ > κ

II

+

ν^|9(ο)|),

(5.13)

L7(q*w',q*w)\\li2,(T,R < C5\m\\L2(q*w',g*w)\\2,2,ff,H

+

\\L5{q*w',q*w)\\i,2,a,R

< ||Φ||||ς|| 0> 2, σ , Κ [^ 5 σί /2 (1 + ΤγΙ2σ-1'2 χ [τ(Μ\ΐ2,σ,Η

+ I M I W J

+ 2Ce]

+ IKWII2 + ΙΜ0)||^]1/2.

(5.14)

W e have used here the elementary inequality α 1 / 2 + 6 1 / 2 < [2(a + 6)] 1 / 2 . L e t us now point out some basic properties of operators Nj,

j =

1,2,

defined in (4.40), (4.41). Prom (5.1), (5.3), (5.4), (5.7), (5.8), ( 5 . 1 2 ) - ( 5 . 1 4 ) and the inequality ( Σ ^ - χ Uj)2 < 5

< 5Μο\\22,2,σ ,Η + 5 | | ξ ο Ι Ι 1 H-Il^ycoxi^

Α

aj

+ §^

2

easily o b t a i n the estimates

w e

||Λ

2

ω ο

||

2

+

\\A3fXAfiiH

^^(ll^olli^.^+M^olli^^.v^J+ll^oWlP+ll^oCo)!!^^-^2] :=C7(r,a),

(5.15)

where C7 is a positive constant independent of (r, σ). Analogously, we get \\Ν2(ξ,η)\\ΐΜ 2, σ ,Μ + σ 5 | | Φ | | ( ( 7 1 Τ ) 1 / 2 ( 2 σ ) - 1 / 2 χ φ 2 μ 2 Η 0 | | 2 + \\A2f\\hfi ,H

+ P2/"(0)||2Wl)1/2

+ /ϊο|χΓ 1 Ι|Φ|ΙΙΙ/ΙΙο,2, σ ,#(2Γ) 1 / 2 Γ + Ι|Φ||Τ 1 / 2 || ς ο ||ι,2, σ , Κ ((1 + Tfl2C,C\l2a-ll2

+ 2 Ce)r

J. S. Azamatov and, A. Lorenzi

52

+ | | Φ | | Γ { ^ ν ν 2 [ Γ ( | κ ΐ Ι ^ , σ , „ + K l l ? , ^ ) + IK(0)ll 2 + lko(0)||2Wl]1/2 + 2C6T^(\\w'Q\\l2^H + ||Φ||^1/2[(1 + T)ll2cbc\/2a~1'2 ~C8(r,a).

+

\\wQ\\l2^Wl)1'2}

+ 2C6]r2 (5.16)

Observe now that for any two pairs (£1,771), (£2,%) G equalities hold:

0,

a G R+;

(6.7)

G ω χ R^,

(6.8)

72 > αο{χ) > 7ι > 0,

Vz G ω. (6.9)

56

J. S. Azamatov and A. Lorenzi

Then, as far as operator Β is concerned, we will assume that θω χ (0,^) or Γ2 = ω χ {0,£} we can prescribe either of the conditions Bu = u or Bu = Dvu, where ν denotes the outward unit vector to Γι UT2, i.e. Vj(x,y) = Σ^·=χ nj(x,y)aij(x), n(x,y) for the outward unit vector to Γι U Γ2. Finally, functional Φ will be defined by the equation Φ [it] = / JΩ

on Γ χ = boundary conormal standing

(p(x,y)u(x,y)b{x)dxdy,

with , V) stands for Dirichlet conditions on both Γι and ωχ {ji}, j = 0,1: Bu — u on Γχ U Γ 2 ; ii) (V, Af ) stands for Dirichlet condition on Γι and Neumann condition on ω x {ji}, j — 0,1: Bu = u on Γχ, Bu = Dvu on Γ2; Hi) (Af,T>) stands for Neumann condition on Γχ and Dirichlet condition on ω x {j£}, j = 0,1: Bu = Dvu on Γχ, Bu = u on Γ 2 ; iv)

(λί,λί) stands for Neumann conditions on both Γχ and ω χ {j£}, j = 0,1: Bu = Duu on Γχ υ Γ 2 ·

Moreover the symbol (/, 0), with I = V or I — M, stands for no condition on Γ 2 and for Dirichlet or Neumann condition on Γχ. A similar meaning has the symbol (0, K), with Κ = V or Κ = Μ. Let us now define the Hilbert spaces H, W\ and W2: L2(Q;b(x)dxdy),

H = W1(I,K)

L2((0,£);HQ(U))

={

for I = T>, Κ G

{T>,AF},

L 2 ((0,£); Ηι(ω)) for I = Αί, Κ G {V,Af}, for (I,K) = (O,V),

{H%(Cl) L2((0,iy,Hb(u,))nL2(a>-,Hl(0,i))

W2(I,K)={

2

>

(6.11)

ι

2

for

(Ι,Κ) = (Ό,ΛΓ),

1 ((0,ί)·,Η (ω))ηΣ (ω·,Η^(0,ί))

for (I,Κ) = (Af,V),

Hl(Cl)

for (Ι,Κ) = (Αί,Λί)·

(6.12)

Evolution

integro-differential

Remark 6.2. Since μ 3 < b(x) < I/ (Ω) and their norms are equivalent.

57

equations

for any χ Ε ω, Η coincides with

2

Introduce now the two bilinear forms aj, j = 1, 2, defined by ai(v,u)

= j

Γ

N

(

a

i,jDivDjU

+ a^vu^j

άχdy,

Vu,veWi{I,K), a2(v,u)

= / ( 2_] ai jDivDju JnK iJ=ι '

(6.13)

+ abDyvDyU + üqvu] d x d y ,

Vu,v EW2{I,K).

(6.14)

As a consequence, we obtain the bounds αι(υ,?;) = / ( V^

a,i jDivDjv

+ aov2 ) dx dy

> ^\\D x v\\ 2 L 2 m ) . L 2 ( u ) ) +7llMli2((0,πϋη(/ΐι,7ι)||υ||£2((0|/);Ηΐ(ω)),

VveWi{I,K),

(6.15)

and + α μ 3 | | Ζ ^ | | | 2 ( Ω ) + 7ι IMl£(n)

a2(v, ν) > μι\\DxvfL2{n)

> ππη(μι,αμ3,7ι)|Η|Ηι(Ω),

Vv EW2{I,K).

(6.16)

Let us then introduce the second-order Sobolev spaces related to Dirichlet and Neumann conditions Ηΐ{ω)

= Η2(ω) η

Η%,(0,£) = Η^(ω) H^(0,£)

Η^(ω),

H2{0,£)nH^(0,i),

= {u Ε Η2(ω) : D„u = 0 on δω} , = {u Ε H2(0,£)

Finally, define the spaces Wz{I,K) WIT

κ)=ί^((0,£);Ηΐ(ω))ϊονΙ

: Dyu{ji)

= 0,

and Wi{I,K) = ν,

j =

0,1}.

by

Κ Ε {Τ>,Λί},

J. S. Azamatov and A. Lorenzi

58

(I,K)= for ( Ι , Κ ) = £))) for (I, K ) = for ( Ι , Κ ) =

ϋ 2 ((0,/);4Μ)ηΙ 2 ( ω ; ^((0,ί)))

2

2

(u-,H]s((0,t))) W 4 ( I , K ) = LL ((0,ey,H^))nL ((0, έ)·,Η^(ω)) ΓΊ fff, 2 2 {Σ ((0,έ)·,Η^(ω))ηΐ (ω·,Η^((0,ή)) 2

((0,

for

(V,V), (V,M), ( N , V), (M,N). (6.18)

Observe now that the following equalities hold for any Ι , Κ 6 {D,Af}:

— (v,A\u)h — —J n/ ν V] Di(aijDju)dxdy+ i,j=ι ' ν r r = —

/

ni