ebook img

Inverse Problems for Partial Differential Equations PDF

358 Pages·1998·1.912 MB·English
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Inverse Problems for Partial Differential Equations

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 17. Collatz/Wetterling:OptimizationProblems. Approximation:AnIntroductiontotheTheoretical 18. Grenander:PatternSynthesis:LecturesinPattern AnalysisandNumericalTreatmentofSemi-Infinite Theory,Vol.I. Programs. 19. Marsden/McCracken:HopfBifurcationandIts 46. Wilcox:ScatteringTheoryforDiffractionGratings. Applications. 47. Hale/Magalha˜es/Oliva:DynamicsinInfinite 20. Driver:OrdinaryandDelayDifferentialEquations. Dimensions,2nded. 21. Courant/Friedrichs:SupersonicFlowandShock 48. Murray:AsymptoticAnalysis. 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. 26. Kushner/Clark:StochasticApproximationMethods 54. Wasow:LinearTurningPointTheory. forConstrainedandUnconstrainedSystems. 55. Yosida:OperationalCalculus:ATheoryof 27. deBoor:APracticalGuidetoSplines:Revised Hyperfunctions. Edition. 56. Chang/Howes:NonlinearSingularPerturbation 28. Keilson:MarkovChainModels—Rarityand Phenomena:TheoryandApplications. Exponentiality. 57. Reinhardt:AnalysisofApproximationMethodsfor 29. deVeubeke:ACourseinElasticity. DifferentialandIntegralEquations. 30. Sniatycki:GeometricQuantizationandQuantum 58. Dwoyer/Hussaini/Voigt(eds):Theoretical Mechanics. ApproachestoTurbulence. 31. Reid:SturmianTheoryforOrdinaryDifferential 59. Sanders/Verhulst:AveragingMethodsinNonlinear 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. Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(SpringerScience+BusinessMedia,Inc.,233SpringStreet,NewYork, NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Use in connection with any form of information storage and retrieval, electronic adaptation, computer software,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimliarterms,evenifthey arenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyare subjecttoproprietaryrights. PrintedintheUnitedStatesofAmerica. (SBA) 9 8 7 6 5 4 3 2 1 springeronline.com 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

See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.