Inverse Problems for Partial Differential Equations

358 Pages·1998·1.912 MB·English

by  Victor Isakov (auth.)

Applied Mathematical Sciences
Volume 127
Editors
S.S.Antman J.E.Marsden L.Sirovich
Advisors
J.K.Hale P.Holmes J.Keener
J.Keller B.J.Matkowsky A.Mielke
C.S.Peskin K.R.Sreenivasan
Applied Mathematical Sciences
1. John:PartialDifferentialEquations,4thed. 33. Grenander:RegularStructures:LecturesinPattern
2. Sirovich:TechniquesofAsymptoticAnalysis. Theory,Vol.III.
3. Hale:TheoryofFunctionalDifferentialEquations, 34. Kevorkian/Cole:PerturbationMethodsinApplied
2nded. Mathematics.
4. Percus:CombinatorialMethods. 35. Carr:ApplicationsofCentreManifoldTheory.
5. vonMises/Friedrichs:FluidDynamics. 36. Bengtsson/Ghil/Ka¨lle´n:DynamicMeteorology:
6. Freiberger/Grenander:AShortCoursein DataAssimilationMethods.
ComputationalProbabilityandStatistics. 37. Saperstone:SemidynamicalSystemsinInfinite
7. Pipkin:LecturesonViscoelasticityTheory. DimensionalSpaces.
8. Giacaglia:PerturbationMethodsinNon-linear 38. Lichtenberg/Lieberman:RegularandChaotic
Systems. Dynamics,2nded.
9. Friedrichs:SpectralTheoryofOperatorsinHilbert 39. Piccini/Stampacchia/Vidossich:Ordinary
Space. DifferentialEquationsinRn.
10. Stroud:NumericalQuadratureandSolutionof 40. Naylor/Sell:LinearOperatorTheoryin
OrdinaryDifferentialEquations. EngineeringandScience.
11. Wolovich:LinearMultivariableSystems. 41. Sparrow:TheLorenzEquations:Bifurcations,
12. Berkovitz:OptimalControlTheory. Chaos,andStrangeAttractors.
13. Bluman/Cole:SimilarityMethodsforDifferential 42. Guckenheimer/Holmes:NonlinearOscillations,
Equations. DynamicalSystems,andBifurcationsofVector
14. Yoshizawa:StabilityTheoryandtheExistenceof Fields.
PeriodicSolutionandAlmostPeriodicSolutions. 43. Ockendon/Taylor:InviscidFluidFlows.
15. Braun:DifferentialEquationsandTheir 44. Pazy:SemigroupsofLinearOperatorsand
Applications,3rded. ApplicationstoPartialDifferentialEquations.
16. Lefschetz:ApplicationsofAlgebraicTopology. 45. Glashoff/Gustafson:LinearOperationsand
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Theory,Vol.I. Programs.
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Applications. 47. Hale/Magalha˜es/Oliva:DynamicsinInfinite
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Waves. 49. Ladyzhenskaya:TheBoundary-ValueProblemsof
22. Rouche/Habets/Laloy:StabilityTheoryby MathematicalPhysics.
Liapunov’sDirectMethod. 50. Wilcox:SoundPropagationinStratifiedFluids.
23. Lamperti:StochasticProcesses:ASurveyofthe 51. Golubitsky/Schaeffer:BifurcationandGroupsin
MathematicalTheory. BifurcationTheory,Vol.I.
24. Grenander:PatternAnalysis:LecturesinPattern 52. Chipot:VariationalInequalitiesandFlowinPorous
Theory,Vol.II. Media.
25. Davies:IntegralTransformsandTheir 53. Majda:CompressibleFluidFlowandSystemsof
Applications,2nded. ConservationLawsinSeveralSpaceVariables.
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forConstrainedandUnconstrainedSystems. 55. Yosida:OperationalCalculus:ATheoryof
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Edition. 56. Chang/Howes:NonlinearSingularPerturbation
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Equations. DynamicalSystems.
32. Meis/Markowitz:NumericalSolutionofPartial
DifferentialEquations. (continuedfollowingindex)
Victor Isakov
Inverse Problems
for Partial
Differential Equations
Second Edition
VictorIsakov
DepartmentofMathematicsandStatistics
TheWichitaStateUniversity
Wichita,KS67260-0033
USA
[email protected]
SeriesEditors:
S.S.Antman J.E.Marsden L.Sirovich
DepartmentofMathematics ControlandDynamical LaboratoryofApplied
and Systems,107-81 Mathematics
InstituteforPhysicalScience CaliforniaInstituteof Departmentof
andTechnology Technology BiomathematicalSciences
UniversityofMaryland Pasadena,CA91125 MountSinaiSchool
CollegePark,MD20742–4015 USA ofMedicine
USA [email protected] NewYork,NY10029-6574
[email protected] USA
[email protected]
MathematicsSubjectClassification(2000):35R30,86A22,80A23,78A46,65M32,31B20,
35B60,91B26
LibraryofCongressControlNumber:2005924713
ISBN-10:0-387-25364-5
ISBN-13:978-0387-25364-0
Printedonacid-freepaper.
(cid:1)C 2006SpringerScience+BusinessMedia,Inc.
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To my wife Julie
Mostpeople,ifyoudescribeatrainofeventstothem,willtellyouwhattheresult
wouldbe.Theycanputthoseeventstogetherintheirminds,andarguefromthem
thatsomethingwillcometopass.Therearefewpeople,however,who,ifyoutold
themaresult,wouldbeabletoevolvefromtheirowninnerconsciousnesswhat
thestepswerewhichleduptothatresult.ThispoweriswhatImeanwhenItalk
ofreasoningbackward,oranalytically.
—ArthurConanDoyle,AStudyinScarlet
Preface to the Second Edition
In8yearsafterpublicationofthefirstversionofthisbook,therapidlyprogress-
ingfieldofinverseproblemswitnessedchangesandnewdevelopments.Partsof
the book were used at several universities, and many colleagues and students as
wellasmyselfobservedseveralmisprintsandimprecisions.Someoftheresearch
problemsfromthefirsteditionhavebeensolved.Thiseditionservesthepurposes
of reflecting these changes and making appropiate corrections. I hope that these
additionsandcorrectionsresultedinnottoomanynewerrorsandmisprints.
Chapters1and2containonly2–3pagesofnewmateriallikeinsections1.5,
2.5. Chapter 3 is considerably expanded. In particular we give more convenient
definition of pseudo-convexity for second order equations and included bound-
arytermsinCarlemanestimates(Theorem3.2.1(cid:2))andCounterexample3.2.6.We
giveanew,shorterproofofTheorem3.3.1andnewTheorems3.3.7,3.3.12,and
Counterexample3.3.9.Werevisedsection3.4,whereanewshortproofofexact
observability inequality in given: proof of Theorem 3.4.1 and Theorems 3.4.3,
3.4.4, 3.4.8, 3.4.9 are new. Section 3.5 is new and it exposes recent progress on
Carlemanestimates,uniquenessandstabilityofthecontinuationforsystems.In
Chapter4weaddedtosections4.5,4.6somenewmaterialonsizeevaluationof
inclusionsandonsmallinclusions.Chapter5containsnewresultsonidentification
ofanellipticequationfrommanylocalboundarymeasurements(Theorem5.2.2(cid:2),
Lemma 5.3.8), a counterexample to stability, a brief description of recent com-
pleteresultsonuniquenessofconductivityintheplanecase,somenewresultson
identificationofmanycoefficientsandofquasilinearequationsinsectiosn5.5,5.6,
andchangesandmostrecentresultsonuniquenessforsomeimportantsystems,
likeisotropicelasticitysystems.InChapter7weinformaboutnewdevelopments
inboundaryrigidityproblem.Section7.4nowexposesacompletesolutionofthe
uniquenessproblemintheattenuatedplanetomographyoverstraightlines(The-
orem7.4.1)andanoutlineofrelevantnewmethodsandideas.Insection8.2we
giveanewgeneralschemeofobtaininguniquenessresultsbasedonCarlemanes-
timatesandapplicabletoawideclassofpartialdifferentialequationsandsystems
(Theorem 8.2.2) and describe recent progress on uniqueness problem for linear
isotropicelasticitysystem.InChapter9weexpandedtheexpositioninsection9.1
vii
viii PrefacetotheSecondEdition
to reflect increasing importance of the final overdetermination (Theorems 9.1.1,
9.1.2).Insection9.2weexposenewstabilityestimatefortheheatequationtrans-
form (Theorem 9.2.1’ Lemma 9.2.2). New section 9.3 is dedicated to emerging
financialapplications:theinverseoptionpricingproblem.Wegivemoredetailed
proofsinsection9.5(Lemma9.5.5andproofofTheorem9.5.2).InChapter10we
addedabriefdescriptionofanewefficientsinglelayeralgorithmforanimporatnt
inverseprobleminacousticsinsection10.2andanewsection10.5onso-called
rangetestsfornumericalsolutionsofoverderminedinverseproblems.
Manyexerciseshavebeensolvedbystudents,whilemostoftheresearchprob-
lemsawaitsolutions.Chapter7ofthefinalversionofthemanuscripthavebeen
readbyAlexanderBukhgeim,whofoundseveralmisprintsandsuggestedmany
corrections. The author is grateful to him for attention and help. He also thanks
the National Science Foundation for long-term support of his research, which
stimulatedhisresearchandthewritingofthisrevision.
Wichita,Kansas VictorIsakov
Preface to the First Edition
This book describes the contemporary state of the theory and some numerical
aspectsofinverseproblemsinpartialdifferentialequations.Thetopicisofsub-
stantialandgrowinginterestformanyscientistsandengineers,andaccordinglyto
graduatestudentsintheseareas.Mathematically,theseproblemsarerelativelynew
andquitechallengingduetothelackofconventionalstabilityandtononlinearity
andnonconvexity.Applicationsincluderecoveryofinclusionsfromanomaliesof
their gravitational fields; reconstruction of the interior of the human body from
exterior electrical, ultrasonic, and magnetic measurements, recovery of interior
structuralparametersofdetailofmachinesandoftheundergroundfromsimilar
data (non-destructive evaluation); and locating flying or navigated objects from
theiracousticorelectromagneticfields.Currently,therearehundredsofpublica-
tions containing new and interesting results. A purpose of the book is to collect
and present many of them in a readable and informative form. Rigorous proofs
arepresentedwhenevertheyarerelativelyshortandcanbedemonstratedbyquite
generalmathematicaltechniques.Also,weprefertopresentresultsthatfromour
pointofviewcontainfreshandpromisingideas.Insomecasesthereisnocom-
plete mathematical theory, so we give only available results. We do not assume
thatareaderpossessesanenormousmathematicaltechnique.Infact,amoderate
knowledgeofpartialdifferentialequations,oftheFouriertransform,andofbasic
functionalanalysiswillsuffice.However,somedetailsofproofsneedquitespecial
andsophisticatedmethods,butwehopethatevenwithoutcompletelyunderstand-
ing these details a reader will find considerable useful and stimulating material.
Moreover,westartmanychapterswithgeneralinformationaboutthedirectprob-
lem, where we collect, in the form of theorems, known (but not simple and not
alwayseasytofind)resultsthatareneededinthetreatmentofinverseproblems.
Wehopethatthisbook(oratleastmostofit)canbeusedasagraduatetext.Not
onlydowepresentrecentachievements,butweformulatebasicinverseproblems,
discussregularization,giveashortreviewofuniquenessintheCauchyproblem,
andincludeseveralexercisesthatsometimessubstantiallycomplementthebook.
Allofthemcanbesolvedbyusingsomemodificationofthepresentedmethods.
ix
x PrefacetotheFirstEdition
Partsofthebookinapreliminaryformhavebeenpresentedasgraduatecoursesat
theJohannes-KeplerUniversityofLinz,attheUniversityofKyoto,andatWichita
StateUniversity.Manyexerciseshavebeensolvedbystudents,whilemostofthe
researchproblemsawaitsolutions.Partsofthefinalversionofthemanuscripthave
beenreadbyIlyaBushuyev,AlanElcrat,MatthiasEller,andPeterKuchment,who
foundseveralmisprintsandsuggestedmanycorrections.Theauthorisgratefulto
thesecolleaguesfortheirattentionandhelp.HealsothankstheNationalScience
Foundationforlong-termsupportofhisresearch,whichstimulatedthewritingof
thisbook.
Wichita,Kansas VictorIsakov