Methods for solving inverse problems in mathematical physics prilepko, orlovskiy

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METHODS FOR SOLVING

Library of Congress Cataloging-in-Publication

Prilepko, A I (Aleksei Ivanovich)

Methods for solving inverse problems in mathematical physics / Aleksey I.Prilepko, Dmitry G Orlovsky, Igor A Vasin

p crn (Monographs and textbooks in pure and applied mathematics;222)

Includes bibliographical references and index

ISBN 0-8247-1987-5 (all paper)

1 Inverse problems (Differential equations)~Numerical solutions

Mathematical physics I Orlovsky, Dmitry G II Vasin, Igor A III

Title IV Series

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The theory of inverse problems for differential equations is being tensively developed within the framework of mathematical physics In thestudy of the so-called direct problems the solution of a given differentialequation or system of equations is realised by means of supplementary con-ditions, while in inverse problems the equation itself is also unknown Thedetermination of both the governing equation and its solution necessitatesimposing more additional conditions than in related direct problems.The sources of the theory of inverse problems may be found late in the19th century or early 20th century They include the problem of equilibriumfigures for the rotating fluid, the kinematic problems in seismology, theinverse Sturm-Liuville problem and more Newton’s problem of discoveringforces making planets move in accordance with Kepler’s laws was one of thefirst inverse problems in dynamics of mechanical systems solved in the past.Inverse problems in potential theory in which it is required to determinethe body’s position, shape and density from available values of its potentialhave a geophysical origin Inverse problems of electromagnetic explorationwere caused by the necessity to elaborate the theory and methodology ofelectromagnetic fields in investigations of the internal structure of Earth’scrust.

ex-The influence of inverse problems of recovering mathematical physicsequations, in which supplementary conditions help assign either the values

of solutions for fixed values of some or other arguments or the values of tain functionals of a solution, began to spread to more and more branches

cer-as they gradually took on an important place in applied problems arising

in "real-life" situations From a classical point of view, the problems underconsideration are, in general, ill-posed A unified treatment and advancedtheory of ill-posed and conditionally well-posed problems are connectedwith applications of various regularization methods to such problems inmathematical physics In many cases they include the subsidiary infor-mation on the structure of the governing differential equation, the type ofits coefficients and other parameters Quite often the unique solvability

of an inverse problem is ensured by the surplus information of this sort

A definite structure of the differential equation coefficients leads to an verse problem being well-posed from a common point of view This booktreats the subject of such problems containing a sufficiently complete andsystematic theory of inverse problems and reflecting a rapid growth and

in-iii

iv Prefacedevelopment over recent years It is based on the original works of theauthors and involves an experience of solving inverse problems in manybranches of mathematical physics: heat and mass transfer, elasticity the-ory, potential theory, nuclear physics, hydrodynamics, etc Despite a greatgenerality of the presented research, it is of a constructive nature and givesthe reader an understanding of relevant special cases as well as providingone with insight into what is going on in general.

In mastering the challenges involved, the monograph incorporates thewell-known classical results for direct problems of mathematical physicsand the theory of differential equations in Banach spaces serving as a basisfor advanced classical theory of well-posed solvability of inverse problemsfor the equations concerned It is worth noting here that plenty of inverseproblems are intimately connected or equivalent to nonlocal direct problemsfor differential equations of some combined type, the new problems arising

in momentum theory and the theory of approximation, the new types of

¯ linear and nonlinear integral and integro-differential equations of the firstand second kinds In such cases the well-posed solvability of inverse prob-lem entails the new theorems on unique solvability for nonclassical directproblems we have mentioned above Also, the inverse problems under con-sideration can be treated as problems from the theory of control of systemswith distributed or lumped parameters

It may happen that the well-developed methods for solving inverseproblems permit, one to establish, under certain constraints on the inputdata, the property of having fixed sign for source functions, coefficients andsolutions themselves If so, the inverse problems from control theory are

in principal difference with classical problems of this theory These specialinverse problems from control theory could be more appropriately referred

to as problems of the "forecast-monitoring" type The property of havingfixed sign for a solution of "forecast-monitoring" problems will be of crucialimportance in applications to practical problems of heat and mass transfer,the theory of stochastic diffusion equations, mathematical economics, var-ious problems of ecology, automata control and computerized tomography

In many cases the well-posed solvability of inverse problems is establishedwith the aid of the contraction mapping principle, the Birkhoff-Tarskyprinciple, the NewtonvKantorovich method and other effective operatormethods, making it possible to solve both linear and nonlinear problemsfollowing constructive iterative procedures

The monograph covers the basic types of equations: elliptic, parabolicand hyperbolic Special emphasis is given to the Navier-Stokes equations aswell as to the well-known kinetic equations: Bolzman equation and neutrontransport equation

Being concerned with equations of parabolic type, one of the

wispread inverse problems for such equations amounts to the problem of termining an unknown function connected structurally with coefficients ofthe governing equation The traditional way of covering this is to absorbsome additional information on the behavior of a solution at a fixed pointu(x0, t) = ~(t) In this regard, a reasonable interpretation of problems the overdetermination at a fixed point is approved The main idea behindthis approach is connected with the control over physical processes for aproper choice of parameters, making it possible to provide at this point arequired temperature regime On the other hand, the integral overdeter-mination

de-fu(x,t) w(x) = ~(t ),

where w and ~ are the known functions and u is a solution of a given abolic equation, may also be of help in achieving the final aim and comesfirst in the body of the book We have established the new results onuniqueness and solvability The overwhelming majority of the Russian andforeign researchers dealt with such problems merely for linear and semi-linear equations In this book the solvability of the preceding problem isrevealed for a more general class of quasilinear equations The approximatemethods for constructing solutions of inverse problems find a wide range

par-of applications and are galmng increasing popularity

One more important inverse problem for parabolic equations is theproblem with the final overdetermination in which the subsidiary informa-tion is the value of a solution at a fixed moment of time: u(x, T) = ~(x).

Recent years have seen the publication of many works devoted to thiscanonical problem Plenty of interesting and profound results from theexplicit formulae for solutions in the simplest cases to various sufficientconditions of the unique solvability have been derived for this inverse prob-lem and gradually enriched the theory parallel with these achievements

We offer and develop a new approach in this area based on properties ofFredholm’s solvability of inverse problems, whose use permits us to estab-lish the well-known conditions for unique solvability as well

It is worth noting here that for the first time problems with the tegral overdetermination for both parabolic and hyperbolic equations havebeen completely posed and analysed within the Russian scientific schoolheaded by Prof Aleksey Prilepko from the Moscow State University Laterthe relevant problems were extensively investigated by other researchers in-cluding foreign ones Additional information in such problems is provided

in-in the in-integral form and admits a physical in-interpretation as a result of suring a physical parameter by a perfect sensor The essense of the matter

mea-is that any sensor, due to its finite size, always performs some averaging of

a measured parameter over the domain of action

Similar problems for equations of hyperbolic type emerged in theoryand practice They include symmetric hyperbolic systems of the first order,the wave equation with variable coefficients and ~he system of equations

in elasticity theory Some conditions for the existence and uniqueness of asolution of problems with the overdetermination at a fixed point and theintegral overdetermination have been established

Let us stress that under the conditions imposed above, problems withthe final overdetermination are of rather complicated forms than those inthe parabolic case Simple examples help motivate in the general case theabsence of even Fredholm’s solvability of inverse problems of hyperbolictype Nevertheless, the authors have proved Fredholm’s solvability andestablished various sufficient conditions for the existence and uniqueness of

a solution for a sufficiently broad class of equations

Among inverse problems for elliptic equations we are much interested

in inverse problems of potential theory relating to the shape and density

of an attracting body either from available values of the body’s external orinternal potentials or from available values of certain functionals of thesepotentials In this direction we have proved the theorems on global unique-ness and stability for solutions of the aforementioned problems Moreover,inverse problems of the simple layer potential and the total potential which

do arise in geophysics, cardiology and other areas are discussed Inverseproblems for the Helmholz equation in acoustics and dispersion theory arecompletely posed and investigated For more general elliptic equations,problems of finding their sources and coefficients are analysed in the situa-tion when, in addition, some or other accompanying functionals of solutionsare specified as compared with related direct problems

In spite of the fact that the time-dependent system of the Stokes equations of the dynamics of viscous fluid falls within the category

Navier-of equations similar to parabolic ones, separate investigations are caused

by some specificity of its character The well-founded choice of the inverseproblem statement owes a debt to the surplus information about a solu-tion as supplementary to the initial and boundary conditions Additionalinformation of this sort is capable ofdescribing, as a rule, the indirectmanifestation of the liquid motion characteristics in question and admitsplenty of representations The first careful analysis of an inverse prob-lem for the Navier-Stokes equations was carried out by the authors andprovides proper guidelines for deeper study of inverse problems with theoverdetermination at a fixed point and the same of the final observationconditions This book covers fully the problem with a perfect sensor in-volved, in which the subsidiary information is prescribed in the integralform Common settings of inverse problems for the Navier-Stokes systemare similar to parabolic and hyperbolic equations we have considered so

far and may also be treated as control problems relating to viscous liquidmotion.

The linearized Bolzman equation and neutron transport equation areviewed in the book as particular cases of kinetic equations The linearizedBolzman equation describes the evolution of a deviation of the distributionfunction of a one-particle-rarefied gas from an equilibrium The statements

of inverse problems remain unchanged including the Cauchy problem andthe boundary value problem in a bounded domain The solution existenceand solvability are proved The constraints imposed at the very beginningare satisfied for solid sphere models and power potentials of the particleinteraction with angular cut off

For a boundary value problem the conditions for the boundary datareflect the following situations: the first is connected with the boundaryabsorption, the second with the thermodynamic equilibrium of the bound-ary with dissipative particles dispersion on the border It is worth notingthat the characteristics of the boundary being an equilibria in thermody-namics lead to supplementary problems for investigating inverse problemswith the final overdetermination, since in this case the linearized collisionoperator has a nontrivial kernel Because of this, we restrict ourselves tothe stiff interactions only

Observe that in studying inverse problems for the Bolzman tion we employ the method of differential equations in a Banach space.The same method is adopted for similar problems relating to the neutrontransport Inverse problems for the transport equation are described byinverse problems for a first order abstract differential equation in a Ba-nach space For this equation the theorems on existence and uniqueness

equa-of the inverse problem solution are proved Conditions for applications

of these theorems are easily formulated in terms of the input data of theinitial transport equation The book provides a common setting of in-verse problems which will be effectively used in the nuclear reactor the-ory

Differential equations in a Banach space with unbounded operatorcoefficients are given as one possible way of treating partial differentialequations Inverse problems for equations in a Banach space correspond toabstract forms of inverse problems for partial differential equations Themethod of differential equations in a Banach space for investigating variousinverse problems is quite applicable Abstract inverse problems are consid-ered for equations of first and second orders, capable of describing inverseproblems for partial differential equations

It should be noted that we restrict ourselves here to abstract inverseproblems of two classes: inverse problems in which, in order to solve thedifferential equation for u(t), it is necessary to know the value of some

viii Preface

operator or functional B u(t) = ~o(t) as a function of the argument t, and

problems with pointwise overdetermination: u(T) = r.

For the inverse problems from the first class (problems with evolutionoverdetermination) we raise the questions of existence and uniqueness of solution and receive definite answers Special attention is being paid to theproblems in which the operator B possesses some smoothness properties

In context of partial differential equations, abstract inverse problems aresuitable to problems with the integral overdetermination, that is, for theproblems in which the physical value measurement is carried out by a per-fect sensor of finite size For these problems the questions of existence anduniqueness of strong and weak solutions are examined, and the conditions

of differentiability of solutions are established Under such an approach theemerging equations with constant and variable coefficients are studied

It is worth emphasizing here that the type of equation plays a keyrole in the case of equations with variable coefficients and, therefore, itsdescription is carried out separately for parabolic and hyperbolic cases.Linear and semilinear equations arise in the hyperbolic case, while parabolicequations include quasilinear ones as well Semigroup theory is the basictool adopted in this book for the first order equations Since the secondorder equations may be reduced to the first order equations, we need therelevant elements of the theory of cosine functions

A systematic study of these problems is a new original trend initiatedand well-developed by the authors

The inverse problems from the second class, from the point of sible applications, lead to problems with the final overdetermination Sofar they have been studied mainly for the simplest cases The authors be-gan their research in a young and growing field and continue with theirpupils and colleagues The equations of first and second orders will be ofgreat interest, but we restrict ourselves here to the linear case only Forsecond order equations the elliptic and hyperbolic cases are extensively in-vestigated Among the results obtained we point out sufficient conditions

pos-of existence and uniqueness pos-of a solution, necessary and sufficient tions for the existence of a solution and its uniqueness for equations with aself-adjoint main part and Fredholm’s-type solvability conditions For dif-ferential equations in a Hilbert structure inverse problems are studied andconditions of their solvability are established All the results apply equallywell to inverse problems for mathematical physics equations, in particu-lar, for parabolic equations, second order elliptic and hyperbolic equations,the systems of Navier-Stokes and Maxwell equations, symmetric hyper-bolic systems, the system of equations from elasticity theory, the Bolzmanequation and the neutron transport equation

condi-The overview of the results obtained and their relative comparison

are given in concluding remarks The book reviews the latest discoveries

of the new theory and opens the way to the wealth of applications that it

is likely to embrace

In order to make the book accessible not only to specialists, but also

to students and engineers, we give a complete account of definitions andnotions and present a number of relevant topics from other branches ofmathematics

It is to be hoped that the publication of this monograph will stimulatefurther research in many countries as we face the challenge of the nextdecade

Aleksey I PrilepkoDmitry G OrlovskyIgor A Vasin

Preface ooonl

Inverse Problems for Equations of Parabolic Type

1,1 Preliminaries

1,2 The linear inverse problem: recovering a source term

1,3 The linear inverse problem: the Fredholm solvability

1.4 The nonlinear coefficient inverse problem:

recovering a coefficient depending on x

1.5 The linear inverse problem: recovering the evolution of

a source term

12541

5460

2 Inverse Problems for Equations of Hyperbolic Type

2.1 Inverse problems for x-hyperbolic systems

2.2 Inverse problems for t-hyperbolic systems

2.3 Inverse problems for hyperbolic equations

of the second order

717188

106

3 Inverse Problems for Equations of the Elliptic Type 1233.1 Introduction to inverse problems in potential theory 1233.2 Necessary and sufficient conditions for the equality of

3.3 The exterior inverse problem for the volume potential withvariable density for bodies with a "star-shaped" intersection 1393.4 Integral stability estimates for the inverse problem

of the exterior potential with constant density 1523.5 Uniqueness theorems for the harmonic potential

of "non-star-shaped" bodies with variable density 1663.6 The exterior contact inverse problem for the magnetic

potential with variable density of constant sign 1713.7 Integral equation for finding the density of a given body

3.8 Uniqueness of the inverse problem solution for

3,9 Stability in inverse problems for the potential of

a simple layer in the space R~, n >_ 3 197

xi

xli Con ~en ~s

4 Inverse Problems in Dynamics of Viscous

4.2 Nonstationary linearized system of Navier-Stokes

4.3 Nonstationary linearized system of Navier-Stokes

4.4 Nonstationary nonlinear system of Navier-Stokes

4.5 Nonstationary nonlinear system of Navier-Stokes

4.6 Nonstationary nonlinear system of Navier-Stokes

4.7 Nonstationary linearized system of Navier-Stokes

equations: adopting a linearization via

4.8 Nonstationary linearized system of Navier-Stokes

equations: the combined recovery of two coefficients 281

5 Some Topics from Functional Analysis

5.1 The basic notions of functional analysis

5.2 Linear differential equations

5.3 Linear differential equations

5.4 Differential equations with

5.5 Boundary value problems for elliptic differential

6 Abstract Inverse Problems for First Order Equations

and Their Applications in Mathematical Physics 3756.1 Equations of mathematical physics and abstract problems 3756.2 The linear inverse problem with smoothing

overdetermination: the basic elements of the theory 3806.3 Nonlinear inverse problems ,with smoothing

6.4 Inverse problems with.smoothing overdetermination:

6.5 Inverse problems with singular overdetermination:

semilinear equations with constant operation

6.6 Inverse problems with smoothing overdetermination:

6.7 Inverse problems with singular overdetermination:

6.8 Inverse problems with smoothing overdetermination:

6.9 Inverse problems with singular overdetermination:

6.10 Inverse problems with smoothing overdetermination:

semilinear hyperbolic equations and operators

7 Two-Polnt Inverse Problems for First Order Equations 489

7.2 Inverse problems with self-adjoint operator

7.3 Two-point inverse problems in Banach lattices 514

8 Inverse Problems for Equations of Second Order 5238.1 Cauchy problem for semilinear hyperbolic equations 5238.2 Two-point inverse problems for equations

8.3 Two-point inverse problems for equations

9 Applications of the Theory of Abstract Inverse Problems

9.3 The system of equations from elasticity theory 591

METHODS FOR SOLVING

MATHEMATICAL PHYSICS

A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS

Rutgers University University of Delaware New Brunswick, New Jersey Newark, Delaware

EDITORIAL BOARD

M S Baouendi University of California,

San Diego Jane Cronin Rutgers University

Anil Nerode Cornell University Donald Passman University of Wisconsin, Madison

Jack K Hale Georgia Institute of Technology

Fred S Roberts Rutgers University

S Kobayashi University of California,

Berkeley

David L Russell Virginia Polytechnic Institute and State University Marvin Marcus

University of California,

Santa Barbara

W S Massey Yale University

Walter Schempp Universitiit Siegen Mark Teply University of Wisconsin, Milwaukee

MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS

1 K Yano, Integral Formulas in Riemannian Geometry (1970)

2 S Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings (1970)

3 V S Vladimirov, Equations of Mathematical Physics (A Jeffrey, ed.; A Littlewood, trans.) (1970)

4 B N Pshenichnyi, Necessary Conditions for an Extremum (L Neustadt, translation ed.; K Makowski, trans.) (1971)

5 L Nadcietal., Functional Analysis and Valuation Theory (1971)

6 S.S Passman, Infinite Group Rings (1971)

7 L Domhoff, Group Representation Theory Part A: Ordinary Representation Theory Part B: Modular Representation Theory (1971, 1972)

8 W Boothby and G L Weiss, eds., Symmetric Spaces (1972)

9 Y Matsushima, Differentiable Manifolds (E T Kobayashi, trans.) (1972)

10 L E Ward, Jr., Topology (1972)

11 A Babakhanian, Cohomological Methods in Group Theory (1972)

12 R Gilmer, Multiplicative Ideal Theory (1972)

13 J Yeh, Stochastic Processes and the Wiener Integral (1973)

14 J Barros-Neto, Introduction to the Theory of Distributions (1973)

15 R Larsen, Functional Analysis (1973)

16 K Yano and S Ishihara, Tangent and Cotangent Bundles (1973)

17 C Procesi, Rings w~th Polynomial Identities (1973)

18 R Hermann, Geometry, Physics, and Systems (1973)

19 N.R Wallach, Harmonic Analysis on Homogeneous Spaces (1973)

20 J Dieudonn~, Introduction to the Theory of Formal Groups (1973)

21 / Vaisman, Cohomology and Differential Forms (1973)

22 B.-Y Chert, Geometry of Submanifolds (1973)

23 M Marcus, Finite Dimensional Multilinear Algebra (in two parts) (1973, 1975)

24 R Larsen, Banach Algebras (1973)

25 R O Kujala and A I_ Vitter, eds., Value Distribution Theory: Part A; Part B: Deficit and Bezout Estimates by Wilhelm Stoll (1973)

26 K.B Stolarsky, Algebraic Numbers and Diophantine Approximation (1974)

27 A Ft Magid, The Separable Galois Theory of Commutative Rings (1974)

28 B.R McDonald, Finite Rings with Identity (1974)

29 J Satake, Linear Algebra (S Koh et al., trans.) (1975)

30 J S Golan, Localization of Noncommutative Rings (1975)

31 G Klambauer, Mathematical Analysis (1975)

32 M K Agoston, Algebraic Topology (1976)

33 K.R Goodearl, Ring Theory (1976)

34 L.E Mansfield, Linear Algebra with Geometric Applications (1976)

35 N.J Pullman, Matrix Theory and Its Applications (1976)

36 B.R McDonald, Geometric Algebra Over Local Rings (1976)

37 C W Groetsch, Generalized Inverses of Linear Operators (1977)

38 J E Kuczkowski and J L Gersting, Abstract Algebra (1977)

39 C O Chdstenson and W L Voxman, Aspects of Topology (1977)

40 M Nagata, Field Theory (1977)

41 R.L Long, Algebraic Number Theory (1977)

42 W.F Pfeffer, Integrals and Measures (1977)

43 R.L Wheeden andA Zygmund, Measure and Integral (1977)

44 J.H Curtiss, Introduction to Functions of a Complex Variable (1978)

45 K Hrbacek and T Jech, Introduction to Set Theory (1978)

46 W.S Massey, Homology and Cohomology Theory (1978)

47 M Marcus, Introduction to Modem Algebra (1978)

48 E.C Young, Vector and Tensor Analysis (1976)

49 S.B Nadler, Jr., Hyperspaces of Sets (1978)

50 S.K Segal, Topics in Group Kings (1978)

51 A Co M van Rooij, Non-Archimedean Functional Analysis (1978)

52 L Corwin and R Szczarba, Calculus in Vector Spaces (1979)

53 C Sadosky, Interpolation of Operators and Singular Integrals (1979)

54 J Cronin, Differential Equations (1980)

55 C W Groetsch, Elements of Applicable Functional Analysis (1980)

57 H.I Freedan, Deterministic Mathematical Models in Population Ecology (1980)

58 S.B Chae, Lebesgue Integration (1980)

59 C.S Rees et aL, Theory and Applications of Fouder Analysis (1981)

60 L Nachbin, Introduction to Functional Analysis (R M Aron, trans.) (1981)

61 G Orzech and M Orzech, Plane Algebraic Curves (1981)

62 R Johnsonbaugh and W E Pfaffenberger, Foundations of Mathematical Analysis (1981)

63 W.L Voxman and R H Goetschel, Advanced Calculus (1981)

64 L J Comvin and R H Szczarba, Multivariable Calculus (1982)

.65 V L Istr~tescu, Introduction to Linear Operator Theory (1981)

66 R.D J~rvinen, Finite and Infinite Dimensional Linear Spaces (1981)

67 J K Beem and P E Ehrlich, Global Lorentzian Geometry (1981)

68 D.L Armacost, The Structure of Locally Compact Abelian Groups (1981)

69 J W Brewer and M K Smith, eds., Emmy Noether: A Tribute (1981)

70 K.H Kim, Boolean Matrix Theory and Applications (1982)

71 T W Wieting, The Mathematical Theory of Chromatic Plane Ornaments (1982)

72 D.B.Gauld, Differential Topology (1982)

73 R.L Faber, Foundations of Euclidean and Non-Euclidean Geometry (1983)

74 M Carmeli, Statistical Theory and Random Matrices (1983)

75 J.H Carruth et aL, The Theory of Topological Semigroups (1983)

76 R.L Faber, Differential Geometry and Relativity Theory (1983)

77 S Barnett, Polynomials and Linear Control Systems (1983)

78 G Karpilovsky, Commutative Group Algebras (1983)

79 F Van Oystaeyen andA Verschoren, Relative Invariants of Rings (1983)

80 L Vaisman, A First Course in Differential Geometry (1964)

81 G W Swan, Applications of Optimal Control Theory in Biomedicine (1984)

82 T Petrie andJ D Randall, Transformation Groups on Manifolds (1984)

83 K Goebel and S Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings (1984)

64 T Albu and C N~st~sescu, Relative Finiteness in Module Theory (1984)

85 K Hrbacek and T Jech, Introduction to Set Theory: Second Edition (1984)

86 F Van Oystaeyen and A Verschoren, Relative Invariants of Rings (1984)

87 B.R McDonald, Linear Algebra Over Commutative Rings (1964)

88 M Namba, Geometry of Projective Algebraic Curves (1984)

89 G.F Webb, Theory of Nonlinear Age-Dependent Population Dynamics (1985)

90 M R Bremner et aL, Tables of Dominant Weight Multiplicities for Representations of Simple Lie Algebras (1985)

91 A E Fekete, Real Linear Algebra (1985)

92 S.B Chae, Holomorphy and Calculus in Normed Spaces (1985)

93 A J Jerd, Introduction to Integral Equations with Applications (1985)

94 G Karpilovsky, Projective Representations of Finite Groups (1985)

95 L Narici and E Beckenstein, Topological Vector Spaces (1985)

96 J Weeks, The Shape of Space (1985)

97 P.R Gribik and K O Kortanek, Extremal Methods of Operations Research (1985)

98 J.-A Chao and W A Woyczynski, eds., Probability Theory and Harmonic Analysis (1986)

99 G D Crown et aL, Abstract Algebra (1986)

100 J.H Carruth et aL, The Theory of Topological Semigroups, Volume 2 (1986)

101 R S Doran and V A Belfi, Characterizations of C*-Algebras (1986)

102 M W Jeter, Mathematical Programming (1986)

103 M Airman, A Unified Theory of Nonlinear Operator and Evolution Equations with

Applications (1986)

104 A Verschoren, Relative Invadants of Sheaves (1987)

105 R.A UsmanL Applied Linear Algebra (1987)

106 P Blass and J Lang, Zariski Surfaces and Differential Equations in Characteristic p >

0 (1987)

107 J.A Reneke etaL, Structured Hereditary Systems (1987)

108 H Busemann and B B Phadke, Spaces with Distinguished Geodesics (1987)

109 R Harte, Invertibility and Singularity for Bounded Linear Operators (1988)

110 G S Ladde et aL, Oscillation Theory of Differential Equations with Deviating ments (1987)

Argu-111 L Dudkin et aL, Iterative Aggregation Theory (1987)

112 T Okubo, Differential Geometry (1987)

113 D.L Stancl and M L Stancl, Real Analysis with Point-Set Topology (1987)

114 T.C Gard, Introduction to Stochastic Differential Equations (1988)

115 S.S Abhyankar, Enumerative Combinatodcs of Young Tableaux (1988)

116 H Strade and R Famsteiner, Modular Lie Algebras and Their Representations (1988)

117 J.A Huckaba, Commutative Rings with Zero Divisors (1988)

118 W.D Wallis, Combinatorial Designs (1988)

119 W Wi~staw, Topological Fields (1988)

120 G Karpilovsky, Field Theory (1988)

121 S Caenepeel and F Van Oystaeyen, Brauer Groups and the Cohomology of Graded Rings (1989)

122 W Kozlowski, Modular Function Spaces (1988)

123 E Lowen-Colebunders, Function Classes of Cauchy Continuous Maps (1989)

124 M Pavel, Fundamentals of Pattern Recognition (1989)

125 V Lakshmikantham et al., Stability Analysis of Nonlinear Systems (1989)

126 R Sivaramakrishnan, The Classical Theory of Arithmetic Functions (1989)

127 N.A Watson, Parabolic Equations on an Infinite Strip (1989)

128 K.J Hastings, Introduction to the Mathematics of Operations Research (1989)

129 B Fine, Algebraic Theory of the Bianchi Groups (1989)

130 D.N Dikranjan et aL, Topological Groups (1989)

131 J C Morgan II, Point Set Theory (1990)

132 P Biler and A Witkowski, Problems in Mathematical Analysis (1990)

133 H.J Sussmann, Nonlinear Controllability and Optimal Control (1990)

134 J.-P Florens et al., Elements of Bayesian Statistics (1990)

135 N Shell, Topological Fields and Near Valuations (1990)

136 B F Doolin and C F Martin, Introduction to Differential Geometry for Engineers (1990)

137 S.S Holland, Jr., Applied Analysis by the Hilbert Space Method (1990)

138 J Okninski, Semigroup Algebras (1990)

139 K Zhu, Operator Theory in Function Spaces (1990)

140 G.B Price, An Introduction to Multicomplex Spaces and Functions (1991)

141 R.B Darst, Introduction to Linear Programming (1991)

142 P.L Sachdev, Nonlinear Ordinary Differential Equations and Their Applications (1991)

143 T Husain, Orthogonal Schauder Bases (1991)

144 J Foran, Fundamentals of Real Analysis (1991)

145 W.C Brown, Matdces and Vector Spaces (1991)

146 M.M Rao andZ D Ren, Theory of Odicz Spaces (1991)

147 J S Golan and T Head, Modules and the Structures of Rings (1991)

148 C Small, Arithmetic of Finite Fields (1991)

149 K Yang, Complex Algebraic Geometry (1991)

150 D G Hoffman et al., Coding Theory (1991)

151 M.O Gonz~lez, Classical Complex Analysis (1992)

152 M.O Gonz~lez, Complex Analysis (1992)

153 L W Baggett, Functional Analysis (1992)

154 M Sniedovich, Dynamic Programming (1992)

155 R P Agarwal, Difference Equations and Inequalities (1992)

156 C Brezinski, Biorthogonality and Its Applications to Numerical Analysis (1992)

157 C Swartz, An Introduction to Functional Analysis (1992)

158 S.B Nadler, Jr., Continuum Theory (1992)

159 M.A AI-Gwaiz, Theory of Distributions (1992)

160 E Perry, Geometry: Axiomatic Developments with Problem Solving (1992)

161 E Castillo and M R Ruiz-Cobo, Functional Equations and Modelling in Science and Engineering (1992)

162 A J Jerd, Integral and Discrete Transforms with Applications and Error Analysis (1992)

163 A CharlieretaL, Tensors and the Clifford Algebra (1992)

164 P Bilerand T Nadzieja, Problems and Examples in Differential Equations (1992)

165 E Hansen, Global Optimization Using Interval Analysis (1992)

166 S Guerre-Delabd~re, Classical Sequences in Banach Spaces (1992)

167 Y, C Wong, Introductory Theory of Topological Vector Spaces (1992)

168 S H Kulkami and B V Limaye, Real Function Algebras (1992)

169 W C Brown, Matrices Over Commutative Rings (1993)

170 J Loustau andM Dillon, Linear Geometry with Computer Graphics (1993)

171 W V Petryshyn, Approximation-Solvability of Nonlinear Functional and Differential Equations (1993)

174 M Pavel, Fundamentals of Pattem Recognition: Second Edition (1993)

175 S A Albevedo et al., Noncommutative Distributions (1993)

176 W Fulks, Complex Variables (1993)

177 M M Rao, Conditional Measures and Applications (1993)

178 A Janicki and A Weron, Simulation and Chaotic Behavior of c~-Stable Stochastic Processes (1994)

179 P Neittaanm~ki and D 77ba, Optimal Control of Nonlinear Parabolic Systems (1994)

180 J Cronin, Differential Equations: Introduction and Qualitative Theory, Second Edition (1994)

181 S Heikkil~ and V Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations (1994)

182 X Mao, Exponential Stability of Stochastic Differential Equations (1994)

183 B.S Thomson, Symmetric Properties of Real Functions (1994)

184 J E Rubio, Optimization and Nonstandard Analysis (1994)

185 J.L Bueso et al., Compatibility, Stability, and Sheaves (1995)

186 A N Michel and K Wang, Qualitative Theory of Dynamical Systems (1995)

187 M.R Damel, Theory of Lattice-Ordered Groups (1995)

188 Z Naniewicz and P D Panagiotopoulos, Mathematical Theory of Hemivadational Inequalities and Applications (1995)

189 L.J Corwin and R H Szczarba, Calculus in Vector Spaces: Second Edition (1995)

190 L.H Erbe et aL, Oscillation Theory for Functional Differential Equations (1995)

191 S Agaian et aL, Binary Polynomial Transforms and Nonlinear Digital Filters (1995)

192 M L Gil’, Norm Estimations for Operation-Valued Functions and Applications (1995)

193 P.A Gdllet, Semigroups: An Introduction to the Structure Theory (1995)

194 S Kichenassamy, Nonlinear Wave Equations (1996)

195 V.F Krotov, Global Methods in Optimal Control Theory (1996)

196 K L Beidaret al., Rings with Generalized Identities (1996)

197 V L Amautov et aL, Introduction to the Theory of Topological Rings and Modules

(1996)

198 G Sierksma, Linearand Integer Programming (1996)

199 R Lasser, Introduction to Fourier Series (1996)

200 V Sima, Algorithms for Linear-Quadratic Optimization (1996)

201 D Redmond, Number Theory (1996)

202 J.K Beem et al., Global Lorentzian Geometry: Second Edition (1996)

203 M Fontana et aL, Pr0fer Domains (1997)

204 H Tanabe, Functional Analytic Methods for PaPal Differential Equations (1997)

205 C Q Zhang, Integer Flows and Cycle Covers of Graphs (1997)

206 E Spiegel and C J O’Donnell, Incidence Algebras (1997)

207 B Jakubczyk and W Respondek, Geometry of Feedback and Optimal Control (1998)

208 T W Haynes et al., Fundamentals of Domination in Graphs (1998)

209 T W Haynes et al., Domination in Graphs: Advanced Topics (1998)

210 L A D’Alotto et al., A Unified Signal Algebra Approach to Two-Dimensional Parallel Digital Signal Processing (1998)

211 F Halter-Koch, Ideal Systems (1998)

212 N.K Govilet a/., Approximation Theory (1998)

213 R Cross, Multivalued Linear Operators (1998)

214 A A Martynyuk, Stability by Liapunov’s Matrix Function Method with Applications (1998)

215 A Favini andA Yagi, Degenerate Differential Equations in Banach Spaces (1999)

216 A I/lanes and S Nadler, Jr., Hyperspaces: Fundamentals and Recent Advances (1999)

217 G Kato and D Struppa, Fundamentals of Algebraic Microlocal Analysis (1999)

218 G.X.-Z Yuan, KKM Theory and Applications in Nonlinear Analysis (1999)

219 D Motreanu and N H Pave/, Tangency, Flow Invariance for Differential Equations, and Optimization Problems (1999)

220 K Hrbacek and T Jech, Introduction to Set Theory, Third Edition (1999)

221 G.E Kolosov, Optimal Design of Control Systems (1999)

222 N.L Johnson, Subplane Covered Nets (2000)

223 B Fine and G Rosenberger, Algebraic Generalizations of Discrete Groups (! 999)

224 M V~th, Volterra and Integral Equations of Vector Functions (2000)

225 S S Miller and P T Mocanu, Differential Subordinations (2000)

226 R Li et aL, Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods (2000)

227 H Li and F Van Oystaeyen, A Primer of Algebraic Geometry (2000)

228 R P Agarwal, Difference Equations and Inequalities: Theory, Methods, and tions, Second Edition (2000)

Applica-229 A.B Kharazishvili, Strange Functions in Real Analysis (2000)

230 J M Appell et aL, Partial Integral Operators and Integro-Differential Equations (2000)

231 A L Pdlepko et aL, Methods for Solving Inverse Problems in Mathematical Physics (2000)

Additional Volumes in Preparation

Chapter 1

1.1 Preliminaries

In this section we give the basic notations and notions and present also

a number of relevant topics from functional analysis and the general ory of partial differential equations of parabolic type For more detail werecommend the well-known monograph by Ladyzhenskaya et al (1968).The symbol ~ is used for a bounded domain in the Euclidean spaceR’~, x = (xl, ,x,~) denotes an arbitrary point in it Let us denote Q:~ a cylinder ftx (0, T) consisting of all points (x, t) ’~+1 with x Ef~and t ~ (0,T)

the-Let us agree to assume that the symbol 0f~ is used for the boundary

of the domain ~ and ST denotes the lateral area of QT More specifically,

ST is the set 0~ x [0, T] ~ R’~+1 consisting of all points (x, t) with z ~

and t G [0, T]

In a limited number of cases the boundary of the domain f~ is supposed

to have certain smoothness properties As a rule, we confine our attention

to domains ~2 possessing piecewise-smooth boundaries with nonzero interiorangles whose closure (~ can be represented in the form (~ = tA~n=l~ for

1 Inverse Problems for Equations of Parabolic Type

fti r3 flj = e, i ¢ j, and every ~ can homeomorphically be mapped onto

a unit ball (a unit cube) with the aid of functions ¢~(x)’, i = 1,2,

k = 1,2, ,m, with the Lipschitz property and the 3acobians of the

transformations

are bounded from below by a positive constant

We say that the boundary Oft is of class C~, I _> 1, if t.here exists a

number p > 0 such that the intersection of Oft and the ball Be of radius pwith center at an arbitrary point z° E Oft is a connected surface area whichcan be expressed in a local frame of reference ((1,(~, ¯ ¯ ,(n) with origin the point z° by the equation (,~ = ¢o(~, ,(,~-1), where w(~l,

is a function of class C~ in the region /) constituting the projection of onto the plane ~,~ = 0 We will speak below about the class C~(/))

We expound certain exploratory devices for investigating inverse lems by using several well-known inequalities In this branch of mathemat-ics common practice involves, for example, the Cauchy inequality

prob-E aij ~i

i,j=l

) 1/2which is valid for an arbitrary nonnegative quadratic form aij ~i vii with

aij = aji and arbitrary real numbers ~, ,~n and ql, ,qn This isespecially true of Young’s inequality

Throughout this section, we operate in certain functional spaces, the

elements of which are defined in fl and QT We list below some of them In

what follows all the functions and quantities will be real unless the contrary

is explicitly stated

The spaces Lp(fl), 1 _< p < oo, being the most familiar ones, come

first They are introduced as the Banach spaces consisting of all measurablefunctions in fl that.are p-integrable over that set The norm of the space

Lp(fl) is defined by

It is worth noting here that in this chapter the notions of measurabilityand integrability are understood in the sense of Lebesgue The elements ofLp(f~) are the classes of equivalent functions on

When p = cx~ the space L~(f~) comprises all measurable functions f~ that are essentially bounded having

II ~ IIo~,a = esssup I u(z)

We obtain for p = 2 the Hilbert space L2(f~) if the scalar product

in that space is defined by

(u,v) =i u(x)v(x)

The Sobolev spaces W~(f~), where 1 is a positive integer, 1 _< p consists of all functions from Lv(f~) having all generalized derivatives of thefirst l orders that are p-integrable over fL The norm of the space

Oz<~ ’ Oz7 ~ Oz~,

and ~-~1~1=~ denotes summation over all possible c~th derivatives of u.Generalized derivatives are understood in the sense of Sobolev (seethe definitions in Sobolev (1988)) For ~ = 1 and ~ = 2 we will write,

4 1, Inverse Problems for Equations of Parabolic Type

usual, u~ and u~, respectively, instead of Dr u and D~ u This should notcause any confusion

0

It is fairly common to define the space W~l(f~) as a subspace of W~ (f~)

in which the set of all functions in f~ that are infinite differentiable and

have compact support is dense The function u(x) has compact support in

a bounded domain f~ if u(x) is nonzero only in a bounded subdomain ~

of the domain ~ lying at a positive distance from the boundary of ~.When working in HSlder’s spaces ch(~) and cl+h(~), we will as-

sume that the boundary of ~ is smooth A function u(x) is said to satisfy

HSlder’s condition with exponent h, 0 < h < 1, and HSlder’s constant

H~(u) in ~ if

sup l u(x) u(x’)l ~ H~(u) <

By definition, ch(~) is a Banach space, the elements of which are

contin-uous on ~ functions u having bounded

[u[~) : sup [u [+ H~(u).

In turn, c~+h(~), where l is a positive integer, can be treated as a Banach

space consisting of all differentiable functions with continuous derivatives

of the first l orders and a bounded norm of the form

I(~+~)

l u ~ = ~ ~ sup [D~ul+ ~ H~(D~u).

The functions depending on both the space and time variables with similar differential properties on x and t are much involved in solving non-stationary problems of mathematical physics

dis-Furthermore, Lp, q(Qr), 1 ~ p, q < ~, is a Banach space consisting all measurable functions u having bounded

][ u llp, q, Q~ = l u F d~ dt

0 ~

The Sobolev space W~’ ~ (Qr), p ~ 1, with positive integers I i ~ O,

i : 1, 2, is defined as a Banach space of all functions u belonging to the

sp~ce Lp(QT ) along with their weak x-derivatives of the first l~ orders and

t-derivatives of the first l: orders The norm on that space is defined by

P,QT :

The symbol W~:~(Qr) is used for a subspace of W~"(Qr) in which

the set of all smooth functions in QT that vanish on ST is dense.

The space C2+~’1+~/2(QT), 0 < a < 1, is a Banach space of all functions u in QT that are continuous on (~T and that possess smooth x-

derivatives up to and including the second order and t-derivatives of thefirst order In so doing, the functions themselves and their derivativesdepend continuously on x and t with exponents a and a/2, respectively.The norm on that space is defined by

In specific cases the function u depending on x and t will be treated as

an element of the space V~’°(Qr) comprising all the elements of W~’°(Qr)that are continuous with respect to t in the L~(fl)-norm having finite

T u T~r= sup II~(,~)ll~,a+ll~ll~,~,

[o, T]

where u = (u~,, , u.~) and ~ =~u~ [~ Th e meaning of the continuity

of the function u(-,t) with respect to t in the L~(fl)-norm is

°l 0For later use, the symbol V~’ (@) will appear once we agree to con-sider only those elements of V~’°(@) that vanish on ST

In a number of problems a function depending on x and t can beviewed as a function of the argument.t with values from a Banach spaceover ~ For example, Le(O,T; W~(~)) is a set of all functions u( ,t)

(0, T) with wlues in Wtp(a) and norm

T

6 1 Inverse Problems for Equations of Parabolic Type

Obviously, the spaces Lq(O, T; Lp(~)) and Lp, q(Qw) can be identified in a

natural way In a similar line, the space

holds-true for all the functions u from W~(~), where ~ is a bounded domain

in the space R’~ The constant c1(~) depending only on the domain ~ bounded by the value 4 (diamg)

~

Theorem 1.1.1 Let ~ be a bounded domain in the space Rn with the

pieccwise smooth boundary 0~ and let Sr be an intersection of ~ with any r-dimensional hypersurface, r <_ n (in particular, if r = n then Sr =- ~,"

if r = n - 1 we agree to consider O~ as St) Then for any function

u 6 W~p(~), where l > 1 is a positive integer and p > 1, the following

assertions are valid:

(a) for n > pl and r > n-pl there exists a trace of u on ST belonging

to the space Lq(Sr) with any finite q < pr/(n-pl) and the estimate

(b) for n = pl the assertion of item (a) holds with any q < c~;

(c) for n < pl the function u is Hb’Ider’s continuous and belongs to the

class ck+h((~), where k : l - 1 - In/p] and h : 1 + [n/p] -

if niP is not integer and Vh < 1 if nip is integer In that case the estimate

Notice that the constants c arising from (1.1.4)-(1.1.5) depend

on n, p, l, r, q, Sr and ~ and do not depend on the function u The proof

of Theorem 1.1.1 can be found in Sobolev (1988)

In establishing some subsequent results we will rely on Rellich’s orem, whose precise formulation is due to Courant and Hilber-t (1962)

the-o

Theorem 1.1.2 If ~ is a bounded domain, then W~(~) is compactly

em-bedded into the space L~(~), that is, a set of elements {us) of the space o

W~(Q) with uniformly bounded norms is compact in the space L2(~).

Much progress in solving inverse boundary value problems has beenachieved by serious developments in the general theory of elliptic and par-abolic partial differential equations The reader can find deep and diverseresults of this theory in Ladyzhenskaya (1973), Ladyzhenskaya and Uralt-seva (1968), Friedman (1964), Gilbarg and Trudinger (1983), Berezanskij(1968) Several facts are known earlier and quoted here without proofs,the others are accompanied by explanations or proofs Some of them werediscovered and proven in recent years in connection with the investigation

of the series of questions that we now answer Being of independent valuealthough, they are used in the present book only as part of the auxiliarymathematical apparatus The theorems concerned will be formulated here

in a common setting capable of describing inverse problems of interest thatmake it possible to draw fairly accurate outlines of advanced theory.Let ~ be a bounded domain in the space R~ with boundary c~ of class

C2 In the domain ~ of such a kind we consider the Dirichlet boundary

value (direct) problem for the elliptic equation of the second order

with certain positive constants # and u and arbitrary real numbers ~, ,

~,~ The left inequality (1.1.9) reflects the ellipticity property and the rightone means that the coefficients aij are bounded

8 1 Inverse Problems for Equations of Parabolic Type

In trying to solve the direct problem posed above we look for thefunction u by regarding the coefficients of the operator L, the source term

f and the domain f~ to be known in advance

Theorem 1.1.3 Let the operator L satisfy (1.1.8)-(1.1.9), aij ¯ C(~),

~ ¯ C(~), i ¯ L~and c < 0 al most ever ywhere (a.e ) in ~ I

where ~he constant c* is independent of u.

A similar result concerning the unique solvability can be obtained gardless of the sign of the coefficient c However, in this case the coefficients

re-of the operator L should satisfy some additional restrictions such as, forexample, the inequality

(1.1.11)

where b = ~i=~ b~(z) and c~(a) is the same constant as in (1.1.g).For further motivations we cite here the weak principle of m~imumfor elliptic equations following ~he monograph of Gilbarg and Trudinger(198~), p 170-17g To facilitate understanding, it will be convenientintroduce some terminology which will be needed in subsequent reasonings

A function ~ ~ W~(~) is said to satisfy the inequality ~ ~ 0 on 0~ if itspositive part u+ = max{u,0} belongs to W~(~) This definition permits

us go involve inequalities of other types on 0~ Namely, u ~ 0 on 0~ if-u ~ 0 on 0~; functions u and v from W~(~) satisfy the inequality u ~

Theorem 1.1.4 /the weak principle of maximum) Let the conditions

of Theorem 1.1.3 hold for the operator L and let a function u

satisfy the inequality Lu > 0 in f2 in a weak sensẹ Then

sup u _< sup ự

Corollary 1.1.1 Let the operator L be in line with the premises of Theorem

o

1.1.3 and let a function ~o ~ W~(Q) [1 W~(Q) comply with the conditions

~(x) > O ạẹ in Q and ~(x) ~ const Then there exists a measurable set Q’ C f~ with

mes,~Q’ > 0

such that L~ < 0 in

Proof On the contrary, let L~ > 0 in ~2 If so, the theorem yields either

~ < 0 in Q or ~o = const in fL But this contradicts the hypotheses ofCorollary 1.1.1 and proves the current corollarỵ ¯

Corollary 1.1.2 Let the operator L meet the requirements of Theorem

o

1.1.3 and let a function ~ 6 W~(QT) f] W~(Q) follow the conditions

~(x) > 0 ạẹ in a and L~o(:~) co nst in ạ

Then there exists a measurable set Q’ C Q with

mes~Q’ > 0

such that LW < 0 in

Proof Since LT ~ O, we have T ~ O, giving either T const > 0 or

T ~ const If ~_ const > O, then

=-and the above assertion is simple to follow For ~ ~ const applying lary 1.1.1 leads to the desired assertion ¯

Corol-lO 1 Inverse Problems for Equations of Parabolic Type

For the purposes of the present chapter we refer to the parabolicequation

(1.1.12) ut(x,t ) - (Lu)(x,t) = F(x,t), (x,t) QT= Qx (O,T ),

supplied by the initial and boundary conditions

The direct problem for equation (1.1.12) consists of finding a solution

u of the governing equation subject to the initial condition (1.1.13) and theboundary condition (1.1.14) when operating with the functions F and the coefficients of the operator L and the domain ~ × (0, T)

Definition 1.1.1 A function u is said to be a solution of the direct problem

2 1

(1.1.12)-(1.1.14) from the class w2’l(I~2 ,,~., ~ if u ¯ W2:o(Qr) and relations

(1.1.12)-(1.1.14) sati sfied almo st ever ywhere in t he corresponding

2,1(1.1.12)-(1.1.14) has a solution u ¯ W~,o(Qr), this solution is unique the indicated class of functions and the following estimate is valid:

(1.1.17) Ilull~)~, <c* Ilfll~,o~÷lla u,~ ,

where the constant c* does not depend on u.

In subsequent studies of inverse problems some propositions on ability of the direct problem (1.1.12)-(1.1.14) in the "energy" space

solv-~l’°(Qr) will serve as a necessary background for important conclusions

Definition 1.1.2 A function u is said to be a weak solution of the direct

°10

problem (1.1.12)-(1.1.14) from the c]ass ~’°(QT) ifu ¯ V~’ (QT) the system (1.1.12)-(1.1.14) sat isfied in thesense of t he foll owing integral identity:

element of W~’l(Qr) such that O(x,t) = 0 for

is an excellent start in this direction

Theorem 1.1.6 Let the coefficients of the operator L satisfy

(1.1.15)-(1.1.16) and let F ¯ L2,1(QT) and a ¯ L2(~) Then the direct problem

~l’°t~) ~ this solution is unique

(1.1.11)-(1.1.14) has a weak solution u ¯ ~ ~’~TZ,

in the indicated class of functions and the energy balance equation is valid:

t

(1.1.19) ~ II~( t

12 1 Inverse Problems for Equations of Parabolic Type

Differential properties of a solution u ensured by Theorem 1.1.6 arerevealed in the following proposition

Lemma 1.1.1 If all the conditions of Theorem 1.1.6 are put together with

o

(1.1.12)-(1.1.14) from V2’°(QT) admits for 0 < t < T the estimate

(1.1.20) tlu( ,t)112,~ -< exp {-oct)H a I1~,

/.z,= max{esssupa IC(¢)1, esssup [a i=1 ~ Bf(*)]I/=}

and c~(Q) is the constant from the Poincare-Friedrichs inequality (1.1.3).Observe that we imposed no restriction on the sign of the constant a

At the next stage, holding a number ¢ from the interval (0, T) fixedand taking t = ¢, we appeal to identity (1.1.18) After subtracting theresulting expression from (1.1.18) we get

(l.1.21)

t

where ¯ is an arbitrary element of W~’I(QT) that vanish on ST Due to

the differential properties of the function u established in Lemma 1.1.1 wecan rewrite (1.1.21) for 0 < e < t < T

Obvi-into (1.1.22) we arrive

(1.1.23)

dr, 0<¢<t<T

14 1 Inverse Problems for Equations of Parabolic Type

o

It is worth noting here that C~([¢, T]) is dense in the space L2([¢, T]) By

minor manipulations with relation (1.1.23) we are led

#1 =max esssup Ic(x)l, esssup a~(x

The estimation of the first term on the right-hand side of (1.1.~5) can done relying on Young’s inequality with p : q = 2 and ~ = u/#l, whoseuse permits us to establish the relation

and c~(~) is the same constant as in (1.1.3)

Let us multiply both sides of (1.1.27) by exp {at} and integrate thenthe resulting expression from e to t Further passage to the limit as e + 0-4-leads to the desired estimate (1.1.20)

T~l,0The second estimate for u E v2 (QT) in question follows directlyfrom (1.1.20):

16 1 Inverse Problems for Equations of Parabolic Type

u e ~’°(QT) of the direct problem (1.1.12)-(1.1.14)satisfies the estimate

in Lemma 1.1.1, under some additional restrictions on the input data anysolution u of the direct problem (1.1.12)-(1.1.14) from I}~’°(QT) belongs the space W~:~(a x (¢, T)) for any ~ ¢ (0, T) This, in p~rticu[ar, that the derivative ~,( ,t) belongs to the space L~(~) for any t ~ (¢,T)and is really continuous with respect to t in the L2(Q)-norm on the segmentLet t be an arbitrary number from the half-open interval (0, T] Hold-ing a number e from the interval (0, t) fixed we deduce that there exists moment r* ~ [e, t], at which the following relation occurs:

if v* and t were suitably chosen in conformity with (1.1.31).

readily see that (1.1.33) yields the inequality

i,j=l Aij(x) u~(x,t) u~,(x,t)

t

r ° f~

i,j=l Aij(x) %~(x, v*)%,(x, 7") dx -t-2

where

#1 ~ max ess sup I c(x) I,

and (5 is an arbitrary positive number

By merely setting (5 = 1/(4#1) we derive from (1.1.34)-(1.1.35)

18 1 Inverse Problems for Equations of Parabolic Type

and ~ is an arbitrary positive number from the interval (0, t)

The first term on the right-hand side of the preceding inequality can

be estimated on the basis of (1.1.30) as follows:

Differential properties of a solution u ensured by Theorem 1.1.5 areestablished in the following assertion

Lemma 1.1.2 If, in addition to the premises of Theorem 1.1.5, Ft ~ L2(QT) and a E W~(12), then the solution u(x,t) belongs to C([O,T]; W~(~)), its derivative ut(x , t) belongs to

in the context of Lemma 1.1.2 a solution u of the system (1.1.12)-(1.1.14)has the estimates