Cloaking via Change of Variables in Electric Impedance - PDF document

CloakingviaChangeofVariablesinElectricImpedanceTomographyR.V.Kohn,H.Shen,M.S.Vogelius,andM.I.WeinsteinRevisedversion:December3,2007SubmittedtoInverseProblemsAbstractArecentpaperbyPendry,Schurig,andSmith[Science312,2006,1780-1782]usedthecoordinate-invarianceofMaxwell'sequationstoshowhowaregionofspacecanbe\cloaked"{inotherwords,madeinaccessibletoelectromagneticsensing{bysurroundingitwithasuitable(anisotropicandheterogenous)dielectricshield.EssentiallythesameobservationwasmadeseveralyearsearlierbyGreenleaf,Lassas,andUhlmann[MathematicalResearchLetters10,2003,685-693andPhysiologicalMeasurement24,2003,413-419]inthecloselyrelatedsettingofelectricimpedancetomography.Thesepapers,thoughbrilliant,havetwoshortcomings:(a)thecloakstheyconsiderarerathersingular;and(b)theanalysisbyGreenleaf,Lassas,andUhlmanndoesnotapplyinspacedimension=2.Thepresentpaperprovidesafreshtreatmentthatremediestheseshortcomingsinthecontextofelectricimpedancetomography.Inparticular,weshowhowaregularnear-cloakcanbeobtainedusinganonsingularchangeofvariables,andweprovethatthechange-of-variable-basedschemeachievesperfectcloakinginanydimensionContents1Introduction2Themainideas2.1Electricimpedancetomography..............................32.2Invariancebychangeofvariables.............................52.3Cloakingviachangeofvariables.............................62.4Relationtoknownuniquenessresults...........................82.5Commentsoncloakingatnonzerofrequency.......................9 CourantInstitute,NewYorkUniversity,kohn@cims.nyu.eduCourantInstitute,NewYorkUniversity,haiping@cims.nyu.eduDepartmentofMathematics,RutgersUniversity,vogelius@math.rutgers.eduDepartmentofAppliedPhysicsandAppliedMathematics,ColumbiaUniversity,miw2103@columbia.edu 3Analysisoftheregularnear-cloak103.1TheDirichlet-to-Neumannmap..............................113.2Dielectricinclusions....................................123.3Theregularnear-cloakisalmostinvisible........................144Analysisofthesingularcloak154.1Explicitformofthecloak.................................164.2Thepotentialoutsidethecloakedregion.........................174.3Thepotentialinsidethecloakedregion..........................194.4Thesingularcloakisinvisible...............................201IntroductionWesayaregionofspaceis\cloaked"withrespecttoelectromagneticsensingifitscontents{andeventheexistenceofthecloak{areinaccessibletosuchmeasurements.Iscloakingpossible?Theanswerisyes,atleastinprinciple.Acloakingschemebasedonchange-of-variableswasdiscussedforelectricimpedancetomographybyGreenleaf,Lassas,andUhlmannin2003[19,20],andforthetime-harmonicMaxwell'sequationbyPendry,Schurig,andSmithin2006[35,38].Otherschemeshavealsobeendiscussed,includingonebasedonopticalconformalmapping[28,29],anotherbasedonanomalouslocalizedresonances[32],andathirdbasedonuseofsensorsandactivesources[30,36].Recentdevelopmentsincludenumerical[6,12,45]andexperimental[39]implementationsofchange-of-variable-basedcloaking;adaptationsofthechange-of-variable-basedschemetoacousticorelasticsensing[13,31];andtheintroductionofrelatedschemesforcloakingactiveobjectssuchaslightsources[17].Iscloakinginteresting?Theanswerisclearlyyes.Onereasonistheoretical:theexistenceofcloaksrevealsintrinsiclimitationsofelectromagnetic-basedschemesforremotesensing,suchasinversescatteringandimpedancetomography.Asecondreasonispractical:cloakingprovidesaneasymethodformakinganyobjectinvisible{bysimplysurroundingitwithacloak.Theappealofthisideahasattractedalotofattention,e.g.[8,44].Iscloakingpractical?Theanswerisnotyetclear.Allapproachestocloakingrequirethedesignofmaterialswithexoticdielectricproperties.Onehopesthatthedesiredpropertiescanbeachieved(oratleastapproximated)bymeansof\metamaterials"[40];fortheschemesbasedonchange-of-variablesthisseemstobethecase[39].Foracloakingschemetobepracticalitmustbereasonablyinsensitivetoimperfection;therobustnessofthechange-of-variable-basedschemehasjustbeguntobeaddressed[10,18,37](seeSection2.3forcommentsonthiswork.)Thepresentpaperisrelatedtotherstandlastoftheprecedingquestions.Weask:(i)Doesthechange-of-variables-basedschemereallyachieveaperfectcloak?(ii)Whataboutaregularizedversionofthethisscheme?Howclosedoesitcometoachievingcloaking?Ouranalysisisrestrictedtoelectricimpedancetomography.Thisamountstoconsideringelectro-magneticsensinginthelow-frequencylimit[26];itissimplerthanthenite-frequencysetting,due tothereadyavailabilityofvariationalprinciples.Butwedodiscussthenite-frequencysetting,inSection2.5.Concerning(i):thereiscauseforconcern,becausetheunderlyingchangeofvariablesishighlysingular(seeSection2.3).Singularitiesaresometimessignicant;forexample,thefundamentalsolutionofLaplace'sequationisharmonicexceptatapoint.Thephysicsliteraturerecognizesthisissue;forexample,Cummeretal.writein[12]that\whetherperfectcloakingisachievable,evenintheory,is...anopenquestion."Theyalsosuggest,usinganargumentbasedongeometricaloptics,thatthepresenceofasingularity\maydegradecloakingperformancetoanunknowndegree."Actually,(i)wassettledforelectricimpedancetomographyby[19]inspacedimension3,usingamethodthatdoesnotworkinspacedimensiontwo.Onegoalofthepresentpaperistoshowthatthesituationisnotsignicantlydierentwhen=2:perfectcloakingisalsopossibleinspacedimensiontwo.Ourdiscussionofperfectcloaking,presentedinSection4,isnotfundamentallydierentfromthatin[19];inparticular,ourmaintool(like[19])isaresultabouttheremovabilityofsingularitiesforharmonicfunctions.Howeverourdiscussiondiersfrom[19]bytreatingalldimensions2simultaneously,andbyworkingdirectlywiththedivergence-formPDEofelectrostaticsratherthanrewritingitastheLaplace-BeltramiequationofanassociatedRiemannianmetric.Inaddition,ourexpositionisperhapsmoreelementary(thusmoreaccessibletonon-expertreaders).Concerning(ii):thequestionisasimportantastheanswer.Wesuggestthatthe\perfectcloak,"obtainedusingasingularchangeofvariables,notbetakenliterally.Instead,itshouldbeusedtodesignamoreregular\near-cloak,"basedonalesssingularchangeofvariable.Thenear-cloakisphysicallymoreplausible(forexample,itsdielectrictensorisstrictlypositiveandnite).Moreover,themathematicalanalysisofthenear-cloakisactuallyeasier,sincenothingissingular.Basically,theproblemreducestounderstandinghowboundarymeasurementsarein uencedbydielectricinclusions(seeSection2.3forfurtherexplanation).Thepaperisorganizedasfollows.Webegin,inSection2,byintroducingelectricimpedancetomographyandgivingabrief,nontechnicalexplanationofthechange-of-variable-basedcloakingscheme.Thatsectionalsoputsourworkincontext,discussingitsrelationtoknownuniquenessresultsandexplainingwhythenite-frequencycaseissimilartobutdierentfromtheoneconsid-eredhere.Then,inSections3and4,wegivearigorousanalysisofthechange-of-variable-basedcloakingscheme.InSection3weusearegularchangeofvariablesandprovethattheinclusionisalmostcloaked.InSection4weuseasingularchangeofvariablesandprovethattheinclusionisperfectlycloaked.2Themainideas2.1ElectricimpedancetomographyInelectricimpedancetomography,oneusesstaticvoltageandcurrentmeasurementsatthebound-aryofanobjecttogaininformationaboutitsinternalstructure.Mathematically,wesupposetheobjectoccupiesa(known)boundedregion 2.Its(unknown)electricalconductivity)isanon-negativesymmetric-matrix-valuedfunctionon . ThePDEofelectrostaticsisi;j ij(x) =0in ;(1)itrelatesthevoltageandtheassociatedelectriceldtotheresultingcurrent(seeSection2.5).ThePDE(1)determinesa\DirichlettoNeumannmap";bydenition,ittakesanarbitraryboundaryvoltagetotheassociatedcurrent ux:whereistheoutwardunitnormalto .Electricimpedancetomographyseeksinformationongivenknowledgeofthemapping.InthemathematicsliteraturethisproblemwasrstproposedandpartiallyaddressedbyCalderon[7].Doesdetermine?Ingeneral,theanswerisno:thePDEisinvariantunderchangeofvariables,socanatbestbedetermined\uptochangeofvariables."WeshallexplainthisstatementinSection2.2.If,however,isscalar-valued,positive,andnite,thentheanswerisbasicallyyes:undersomemodest(apparentlytechnical)conditionsontheregularityof,knowledgeoftheDirichlet-to-Neumannmapdeterminesaninternalisotropicconductivity)uniquely.WeshallreviewtheseresultsinSection2.4. Figure1:Theregioniscloakedbyif,regardlessoftheconductivitydistributiontheboundarymeasurementsatareidenticaltothoseofauniformregionwithconductivityWhatdoesitmeaninthiscontextforasubsetof tobecloaked?Inprinciple,itmeansthatthecontentsof{andeventheexistenceofthecloak{areinvisibletoelectrostaticboundarymeasurements.Tokeepthingssimple,however,weshalluseaslightlymorerestrictivedenition:wesay iscloakedbyaconductivitydistribution)denedoutsideiftheassociatedboundarymeasurementsat areidenticaltothoseofahomogeneous,isotropicregionwithconductivity1{regardlessoftheconductivityin(seeFigure1).Moreprecisely: Whenwesayis\positiveandnite"wemeanitisabounded,measurablefunctionwitha.e.in forsome Denition1Letbexed,andletbeanon-negative,matrix-valuedconductivitydenedonthecomplementof.Wesaycloakstheregionifitsextensionsacrossforforproducethesameboundarymeasurementsasauniformregionwithconductivity,regardlessofthechoiceoftheconductivityThenameisappropriate:acloakmakestheassociatedregioninvisiblewithrespecttoelectricimpedancetomography.Indeed,suppose inthesenseofofDenition1,andletbeanydomaincontaining .ThentheDirichlet-to-Neumannmapof)for)for1forisindependentof,andidenticaltothatofthedomain withconstantconductivity1.Thisholdsbecause communicateswithitsexterioronlythroughitsDirichlet-to-Neumannmap.Noticethatfromasingleexampleofcloaking,thisextensionargumentproducesmanyotherexamples.Indeed,accordingto(4),if inthesenseofDenition1,thentheextensionofby1cloaksinanylargerdomain WeshallexplaininSection2.3,following[20,35],howtheinvarianceofelectrostaticsunderchangeofvariablesleadstoexamplesofcloaks.2.2InvariancebychangeofvariablesTheinvarianceofthePDE(1)bychange-of-variablesiswellknown.Soisthefactthatdetermineatbest\uptochangeofvariables."Thisobservationisexplicite.g.in[22,25],withanattributiontoLucTartar.Itisconvenienttothinkvariationally.Recallthatif)isboundedandpositivedenite,thenthesolutionof(1)withDirichletdatasolvesthevariationalproblemminMoreovertheminimum\energy"isdeterminedby,sincewhensolves(1)wehaveThus,knowledgeofdeterminestheminimumenergy,viewedasaquadraticformonDirichletdata.Theconverseisalsotrue:knowledgeoftheminimumenergyforallDirichletdatadeterminestheboundarymap.Thisfollowsfromthewellknownpolarizationidentity:foranyand Therighthandsideistheminimumenergyforminusthatfor,whilethelefthandsideistheboundarymap,viewedasabilinearformonDirichletdata.Weturnnowtochangeofvariables.Suppose)isaninvertible,orientation-preservingchangeofvariableson .Thenwecanchangevariablesinthevariationalprinciple(5): @x @xZ Xij @y @y det dy:Wecanwritethismorecompactlyaswhere det(inwhichisthematrixwithi;jelement=@xandtherighthandsideisevaluatedat).Wecallthepush-forwardbythechangeofvariablesWecomenallytothemainpoint:,thentheboundarymeasurementsassociatedwithareidentical,inotherwords)=)forallIndeed,if thenthechangeofvariablesdoesnotaecttheDirichletdata.Soforany=min=minThusanddetermineidenticalquadraticforms,fromwhichitfollowsby(7)that2.3CloakingviachangeofvariablesWenowexplainhowchange-of-variables-basedcloakingworks.Forsimplicitywefocusontheradialcase: =isaballofradius2,andtheregiontobecloakedis,theconcentricballofradius1(seeFigure2).Itwillbeclear,however,thatthemethodismuchmoregeneral.Westartbyexplaininghowcanbenearlycloakedusingaregularchangeofvariables.Fixingasmallparameter0,considerthepiecewise-smoothchangeofvariables x2�2 2�+1 2�jxjx Itskeypropertiesarethat iscontinuousandpiecewisesmooth,expands,whilemappingthefulldomaintoitself,attheouterboundary=2. Figure2:Thechangeofvariablesleadingtoaregularnear-cloak:expandsasmallballtoaballofradiusTheassociatednear-cloakisthepush-forwardviaoftheconstantconductivity=1,re-strictedtotheannulus.(Abusingnotationabit,wewritethisas1.)Toexplainwhy,consideranyconductivityoftheform)for)forBythechange-of-variablesprinciple(9)itsboundarymeasurementsareidenticaltothoseof)for1forwheredenotesthepush-forwardoftheconductivitydistributionbythemap.Thus,theboundarymeasurementsassociatedwitharethesameasthoseofauniformballperturbedbyasmallinclusionatthecenter.Thecontentsoftheinclusionareuncontrolled,sinceisarbitrary.Buttheradiusoftheinclusionissmall,namely.AsweexplaininSection3,thisisenoughtoassurethattheboundarymeasurementsareclosetothoseofacompletelyuniformball.Thus:whensucientlysmall,thisschemecomesclosetocloakingtheunitball(seeTheorem1inSection3.3).Nowweshowhowcanbeperfectlycloakedusingasingularchangeofvariables.Theideaisobvious:justtake=0in(10).Theresultingchangeofvariables 2jxjx isthesameoneusedin[19,20]forelectrostaticsandin[35]forelectromagnetics.Itskeypropertiesarethat: issmoothexceptat0;blowsupthepoint0totheball,whilemappingthefulldomaintoitself;andattheouterboundary=2.Aheuristic\proof"that1givesaperfectcloakusesthesameargumentasbefore.Thistimeoccupiesapointratherthanaball.Changingtheconductivityatapointshouldhavenoeectontheboundarymeasurements.Thereforeweexpectthatwhenisgivenby(11)withgivenby(12),theboundarymeasurementsshouldbeidenticaltothoseobtainedforauniformballwithThisheuristicproofneedssomeclarication.Thevalidityofthechangeofvariablesformulaisopentoquestionwhenissosingular.Worse:ourcloak1isquitesingularnearitsinnerboundary=1;somecareisthereforeneededconcerningwhatwemeanbyasolutionofthePDE(1).ThesetopicswillbeaddressedinSection4.Wehavefocusedontheradialcasebecausethesimple,explicitformofthedieomorphismleadstoanequallysimple,explicitformulafortheassociatedcloak(seeSection4.1).Howeverthemethodisclearlynotlimitedtotheradialcase(seeTheorems2and4).Our\regularnear-cloak"isquitedierentfromtheapproximatecloakingschemeconsideredin[18,37].Thosepapersstartwithaperfectcylindricalcloak,obtainedusingthe2Dversionofthefamiliarconstruction(12).Thiscloakllstheannulus12withananisotropic,heterogeneousmedium,whosebehaviorisrathersingularneartheinnerboundary=1(seeSection4.1).Theapproximatecloakconsideredin[18,37]isobtainedbyrestrictingtheperfectcloaktoaslightlysmallerannulus1+2.Perfectcloaking(atanyfrequency)isobtainedashowevertheconvergenceisextremelyslow.Theconvergencecanbegreatlyimprovedbyintroducingalayerattheedgeofthecloakthatpermitssurfacecurrents[18].Insummary:our\regularnear-cloak"avoidssingularbehaviorbyusingaregularizedchangeofvariables,whereas[18,37]avoidsingularbehaviorbytruncation.Wealsonotetheinterestingarticle[10],whichexploresthesensitivityoftheidealcloaktovarioustypesofmaterialormanufacturingimperfections.Thefocusofthispaperisoncloaking.Butwenoteinpassingthatitmightbepossibletodesignotherinterestingdevicesusingsimilarchange-of-variable-basedtechniques.Arecentexampleofthistypeistheschemeof[9]forrotatingelectromagneticelds.2.4RelationtoknownuniquenessresultsTheuniquenessproblemforelectricimpedancetomographyaskswhetheritispossible,inprinciple,todetermine)usingboundarymeasurements.Inotherwords,doesdetermineIfitisknowninadvancethattheconductivityisscalar-valued,positive,andnite,thentheanswerisbasicallyyes.Theearliestuniquenessresults{intheclassofanalyticorpiecewiseanalyticconductivities{datefromtheearly80's[14,23,24].Afewyearslater,usingentirelydierentmethods,uniquenesswasprovedforconductivitiesthatareseveraltimesdierentiableindimension3[42]andindimension=2[33].Recently,usingyetanothermethod,uniquenesshasbeenshownintwospacedimensionswithnoregularityhypothesisatall,assumingonlythat)is scalar-valued,strictlypositive,andnite[4].Wehavegivenjustafewofthemostimportantreferences;formorecompletesurveyssee[11,21,43].WeobservedinSection2.2thatwhen)issymmetric-matrix-valued,boundarymeasurementscanatbestdetermineit\uptochangeofvariables."Isthistheonlyinvariance?Inotherwords,iftwoconductivitiesgivethesameboundarymeasurements,musttheyberelatedbychangeofvariables?Ifcloakingispossiblethentheanswershouldbeno,sincetheconductivitiesin(3)arenotrelated,asvaries,bychangeofvariables.Paradoxically,Sylvesterprovedthatintwospacedimensions,boundarymeasurementsdo deter-mineuptochangeofvariables[41]!Theheartofhisproofwastheintroductionofisothermalcoordinates{i.e.constructionofa(unique)map suchthatisisotropicand .Byuniquenessintheisotropicsetting,determines;thusboundarymeasurementsdetermineuptochangeofvariables.DoescloakingcontradictSylvester'sresult?Notatall.TheresolutionoftheparadoxisthattheintroductionofisothermalcoordinatesdependscruciallyonhavingupperandlowerboundsforIndeed,if isaballand1withgivenby(10),thentheassociatedisothermalcoordinatetransformationis.As0in(10)theisothermalcoordinatesbecomesingular.Whenispositivewedonotgetperfectcloaking(consistentwithSylvester'stheorem).When=0wedogetcloaking{buttheeigenvaluesofareunboundedbothaboveandbelownear=1(seeSection4.1),Sylvester'sargumentnolongerapplies,andindeedthereisnoisothermalcoordinatesystem.Doboundarymeasurementsdetermineuptochangeofvariablesinthreeormorespacedimensions?Ifweassumeonlythatisnonnegativethentheanswerisno,sincecloakingispossible.If,however,weassumethatisstrictlypositiveandnite,thensucharesultcouldstillbetrue.Aproofforreal-analyticconductivitiesisgivenin[27].2.5CommentsoncloakingatnonzerofrequencyThispaperfocusesonelectricimpedancetomography,becausewecanexplaintheessenceofchange-of-variable-basedcloakinginthiselectrostaticsettingwithaminimumofmathematicalcomplexity.Thepracticalapplicationsofcloakingare,however,mainlyatnonzerofrequencies{forexam-ple,makingobjectsinvisibleatopticalwavelengths,orundetectablebyelectromagneticscatteringmeasurements.Wethereforediscussbrie yhowthepositive-frequencyproblemissimilarto,yetdierentfrom,thestaticcase.Fortime-harmoniceldsinalinearmedium,Maxwell'sequationsbecomei!i!H:Hereandarecomplexvectoreldsrepresentingtheelectricandmagneticelds;,andarereal-valued,positive-denitesymmetrictensorsrepresentingtheelectricalconductivity,dielectricpermittivity,andmagneticpermeabilityofthemedium;and0isthefrequency.ThephysicalelectricandmagneticeldsareRei!tandRei!t Sylvester'spaperprovedonlyalocalresult,andrequiredtobe.Whencombinedwith[33],however,hisanalysisgivesaglobalresult.Therecentimprovementin[5]assumesonlythatisboundedandpositive-denite. When=0,(13)reducesformallyto(1).Indeed,Maxwell'sequationsbecomeand=0.Thelatterimpliesandtheformerimpliesthatisdivergence-free.TheanalogueoftheDirichlet-to-Neumannmapatnitefrequencyisthecorrespondencebetweenthetangentialcomponentofandthetangentialcomponentof .Whenisnotaneigenfrequencythiscanbeexpressedasamapfrom,sometimesknownastheadmittance.(Whenisaneigenfrequencythemapisnotwell-denedandoneshouldconsiderinsteadallpairs(;H).)Mathematically,theadmittancespeciesthesetofpossibleCauchydatafor(13)atfrequency.Physically,abodyinteractswithitsexterioronlythroughitsadmittance;thereforetwoobjectswiththesameadmittanceareindistinguishablebyelectromagneticmeasurementsatfrequency{forexample,byscatteringmeasurements.Digressingabit,weremarkthatmanyoftheuniquenessresultssketchedinSection2.4havebeenextendedtonitefrequency.Inparticular,theadmittanceofa3Dbodyatasinglefrequencydetermines;,andprovidedtheyareknowninadvancetobescalar-valued,sucientlysmooth,andconstantneartheboundary[34].Adierentconnectionbetweenthepositive-frequencyandelectrostaticcasesisprovidedby[26],whichshowsthattheadmittancedeterminestheelectrostaticDirichlet-to-NeumannmapinthelimitLetusfocusnowoncloaking.Thepositive-frequencyanalogueofourdenitionofcloakingisclear:threenonnegativematrix-valuedfunctions;,anddenedon cloakaregiontheassociatedadmittanceat doesnotdependonhow;,andareextendedacross.Thepositive-frequencyanalogueofourchange-of-variablesschemeisalsoclear:if =and 2jxj
x asin(12)and[19,20,35],weshouldbeabletocloakbytaking,andeachtobethe\push-forward"oftheconstant1.Thecorrectnessofthisschemeisdemonstratedin[17],thoughitisnotthemainfocusofthatpaper.Theirargumentis,roughlyspeaking,anite-frequency(andmoregeneral)analogueoftheoneinpresentedhereinSection4.Whataboutourregularnear-cloak?ThediscussioninSection3hasanobviousextensiontothetime-harmonicMaxwellsetting.Toanalyzetheperformanceofthisnear-cloak,wewouldneedanestimatefortheeectofasmallinclusion(withuncontrolleddielectricproperties)upontheboundarymeasurements(admittance).Unfortunately,thisquestionistothebestofourknowledgeopen,thoughtheeectofauniforminclusionisverywellunderstood[3].Weanticipatearesultsimilartotheelectrostaticsetting{theeectofaninclusionshouldtendtozeroasitsradiustendstozero.Sucharesultwould,asanimmediateconsequence,extendtheanalysisofSection3tothetime-harmonicMaxwellsetting.Wereferto[17]forfurtherdiscussionofthetime-harmonicproblem.Thatpaperincludes,amongotherthings,anewchange-of-variable-basedschemeforcloakinganactivedevice(suchasalightsource).3Analysisoftheregularnear-cloakThissectionreviewssomewellknownfactsabouttheDirichlet-to-Neumannmap,thenanalyzesthenear-cloakobtainedusingthechangeofvariable(10). 3.1TheDirichlet-to-NeumannmapIndiscussingthePDE(1),weassumethroughoutthissectionthattheconductivityisstrictlypositiveandboundedinthesensethatforsomeconstants0;M;forall and.Ourdiscussionofcloakingfocusedonthecasewhen isaball,butinthissection canbeanyboundeddomaininwithsucientlyregularboundary.Wewillmakeessentialuseofthevariationalprinciple(5).ThereforewemustrestrictourattentiontoDirichletdataforwhichthereexistsa\niteenergy"solution.Whensatises(14)itiswellknownthatthisoccurspreciselywhen )=forsomesuchthatdxWhenisconstantthesolutionisalsoconstant{atrivialcase{soitisnaturaltorestrictattentiontothesubspace )=,withthenaturalnorm=minThisisafractionalSobolevspace,consistingoffunctionswith\one-halfderivativein )"(seee.g.[1]).Weshallnottrytoexplainwhatthismeansingeneral,butwenotethatwhen isaballtheinterpretationisquitesimple.Infact,ifcos()attheboundarythentheoptimalfor(15)istheharmonicfunctionr=Rcos()),anddirectcalculationgivesSometimesitisconvenienttospecifyNeumannratherthanDirichletdata.Notethatwhenisanisotropic,thephrase\Neumanndata"refersto.ItiswellknownthatthespaceofniteenergyNeumanndatais )=.Itconsistsofmean-value-zerofunctionswith\minusone-halfderivativein )".Ingeneral=supwhen isaballofradiusandcos()thisreducestoWedenedtheDirichlet-to-Neumannmapin(2)astheoperatorthattakesDirichlettoNeumanndata.Itisaboundedlinearmapfrom )to ).Moreoveritispositive andsymmetric(intheinnerproduct)andinvertible,soitdenesapositivedenitequadraticformon ).Thisformcanbewritten\explicitly"aswhereandsolvethePDE(1)withDirichletdataandrespectively.Thenaturalnormonsymmetriclinearmapsofthistypeis=supf;fThisisequivalenttotheoperatornormofviewedasamapfrom,asaconsequenceofthepolarizationidentity(7).Whentwoconductivitiesareordered,theassociatedDirichlet-to-Neumannmapsarealsoor-dered.Moreprecisely:ifandsatisfy;ih;forall andalltheninthesensethatihforall ).Thisfollowseasilyfromthevariationalprinciple(5),sinceifand)=0in with ,thenf;ff;f3.2DielectricinclusionsThesimplestspecialcaseofourPDE(1)iswhen1.Thenthesolutionisharmonic.WeunderstandalmosteverythingaboutharmonicfunctionsandtheassociatedDirichlet-to-Neumannmap.Anotherrelativelysimplecaseariseswhenisuniformexceptforaconstant-conductivitysphericalinclusionofradiuscenteredatsome;for1forInviewof(17),theeectoftheinclusiondependsmonotonicallyonitsconductivity.Itisthereforenaturaltoconsidertheextremelimitsas0and Wenowdiscusstheselimitsindetail,sincetheyareimportanttoouranalysis.Givenany )letdenotethesolutionto=0in with =0onandSimilarlyletletdenotethesolutionto=0in withandwheretheconstantis(uniquely)determinedby Usingverystandardenergyargumentsitiseasytoseethatandweaklyin )).Indeed,energyconsiderationsimmediatelyyieldthat isboundeduniformlyin,that0as,andthat0.Byextractionofsubsequenceswenowgetweak ))limits,and,thatsatisfy(19)and(20),respectively.Theboundaryconditionson)followfromthecontinuityof(acrossthis\interface."Thecondition(21),determining,followssince =Z(x0) andtherefore =lim Itisnothardtoseethatthissamemayalsobecharacterizedastheconstantthatgivesrisetothesmallestenergy(of).Thefactthatwegetsinglelimitsas0and,respectively,isaconsequenceoftheuniquenessofthesolutionto(19),andthesolutionto(20).WenowdeneandIntegrationbyparts,togetherwiththeweakconvergence,givesthatand0and,respectively.Inparticularf;ff;fandf;ff;f Finallywenotethatif isaballofradiusandtheinclusionliesatitscenter,thentheaboveconvergenceoftheDirichlet-to-Neumannmapscaneasilybederivedbyexplicitsolutionof(19)and(20),usingseparationofvariables.Inthesmall-particlelimit0,theperturbationintroducedbythepresenceofasmallinclusion(extremeornot)iswellunderstood.Weshallnotuseitsexactform;ratherwhatmatterstousisitsmagnitude,whichisproportionaltothevolumeoftheinclusion:Proposition1LetbetheDirichlet-to-Neumannmapwhen,andletbetheDirichlet-to-Neumannmapsassociatedwiththeproblems(19)and(20)respectively.Thenwhenissucientlysmall.Hereisthespatialdimensionandwemeantheoperatornorm(16)onthelefthandsideofeachinequality.AproofoftheestimateforisgiveninSection2of[16]andthesameargumentcanbeusedfor.Theconstantdependsofcourseonthelocationofandtheshapeof .Muchmoredetailedresultsareknown,includingafullasymptoticexpansionforthedependenceoftheDirichlet-to-Neumannmapon;seee.g.[2]forarecentreview.Wehavefocusedonsphericalinclusionsonlyforthesakeofsimplicity.Theprecedingdiscussionextendsstraightforwardlytoinclusionsofanyxedshape,i.e.tothesituationwhen)isreplacedbywhereisany\inclusionshape"(aboundeddomainin,containingtheorigin,withsucientlyregularboundary).3.3Theregularnear-cloakisalmostinvisibleNowconsiderthe\regularnear-cloak"discussedinSection2.3: =isaballabouttheoriginofradius2,andhastheform)for)forwhereisgivenby(10).Thesymbolstandsfor\arbitrary:")isthe(scalarormatrix-valued)conductivityintheregionbeingcloaked.Weassumeitispositivedeniteandnite,;forsothesolutionofthePDE(1)iswell-denedandunique.HoweverourestimateswillnotdependonthelowerandupperboundsandAsweexplainedinSection2.3,theDirichlet-to-Neumannmapofisidenticaltothatof)for1forBytheorderingrelation(17),andtheconvergenceresultsdescribedintheprevioussection,weconcludethat=lim whenceItfollowsusingProposition1thattheboundarymeasurementsobtainedusingthisnear-cloakarealmostidenticaltothoseofauniformballwithconductivity1:wherethelefthandsideistheoperatornorm(16).Theconstantisindependentof;infactitdoesnotevendependonthevaluesofandin(22).Wehaveproved:Theorem1Supposetheshellhasconductivity,whereisgivenby(10).Ifsucientlysmallthenisnearlycloaked,inthesensemadepreciseby(23).Wehavefocusedonthesphericallysymmetricsettingduetoitssimple,explicitcharacter.Howeverourargumentdidnotusethissymmetryinanyessentialway.Indeed,thesameargumentproves(seeFigure3):Theorem2LetbeaLipschitzcontinuousmapwithaLipschitzcontinuousinverse,andlet.ThenispiecewiseLipschitz;moreoverexpands,andIftheshellhasconductivityisnearlycloakedwhenissmall.Moreprecisely:whentheconductivityofhastheformforfortheDirichlet-to-Neumannmapisnearlyindependentofinthesensethat4AnalysisofthesingularcloakThissectiondiscussestheperfectcloakobtainedusingthesingularchangeofvariables(12).Wefocusontheradialcaseforsimplicity,butourargumentextendsstraightforwardlytoabroadclassofnon-radialexamples(seeTheorem4).AsweexplainedinSection2.3,thebasicassertionofcloakingisthatforconductivitiesoftheform(11)withgivenby(12),theDirichlet-to-Neumannmapisidenticaltothatoftheuniformballwithconductivity1.Thus,iftheshellhasconductivity1thentheballiscloaked.ThisassertionfollowsfromTheorem1bypassingtothelimit0(seeRemark1inSection4.2).Butitcanalsobeproveddirectly,andthedirectargument{beingverydierent{gives Figure3:Themapblowsupwhileactingastheidentityonadditionalinsight.Inparticular,itrevealsthemechanismofcloaking:thepotentialinconstant,renderingtheconductivityinthisregionirrelevant.Theessenceoftheargumentpresentedinthissectionissimilartothatof[19].Inparticular,ourmaintoolisawell-knownresultontheremovabilityofisolatedsingularitiesforsolutionsofLaplace'sequation(seetheproofofProposition2).4.1ExplicitformofthecloakRecallthat1isdenedby(8).Whenisgivenby(12)itiseasytomakeexplicit.Indeed,theJacobianmatrix=@x)is 2+1 jxjI�1 for=0,whereistheidentitymatrixand^.Thusissymmetric;^isaneigenvec-torwitheigenvalue12,and(inspacedimensionisan1-dimensionaleigenspacewitheigenvalue 2+1 .Thedeterminantisevidentlydet( 21 2+1 +2) Itfollowsbyabriefcalculationthatintheshell1 (2+ 4jxjn�1+jxjn�2+jxjn�3�^x^xT
+1 wheretherighthandsideisevaluatedat)=2( jyj:16 Sinceissingularat=0weexpect1tobeabitstrangeneartheinnerboundaryoftheshell.Thedetailsdependonthespatialdimensionwhen=2,oneeigenvalueof1tendsto0andtheotherto;(28)when=3,oneeigenvaluetendsto0whiletheothersremainnite;(29)when4,alleigenvaluestendto0.(30)Infact:writing=2(1),when=2theeigenvaluesbehavelikeandwhen=3oneeigenvaluebehaveslikeandtwolike;when4oneeigenvaluebehaveslikeandtheremaining1like.Noticethatfor3,theconductivity1dependssmoothlyonneartheinnerboundaryoftheshell.The\strangeness"wementionedaboveisnotalackofsmoothnessbutratheradegeneracy(lackofauniformlowerbound).Inspacedimension=2thesituationisalittledierent:1becomesdegeneratebutalsolackssmoothnesssincethecircumferentialeigenvaluebecomesinnite.Thisdierencebetween=2and3willplaynoessentialroleinouranalysis.4.2ThepotentialoutsidethecloakedregionLetbethepotentialassociatedwithDirichletdata)=0in,with,(31)whereisgivenby(11)usingthesingularchangeofvariable(12).Weassume,asinSection3,thatisboundedaboveandbelowinthesensethat(22)holds.DoesthisPDEhaveauniquesolution?Theanswerisnotimmediatelyobvious,duetothedegeneracyof1near=1.Weshallshow,hereandinSection4.3,thattheonlyreasonablesolutionof(31)is)for(0)forwhereistheharmonicfunctionwiththesameDirichletdata=0in,withandWhatcanweassumeaboutthesolutionof(31)?Later,inSection4.3,wewillaskthatandbothbesquare-integrable.Forthemoment,however,weaskonlythatbeboundednear=1.Moreprecisely,weaskthatforforsomeconstantsand1.(WedonotassumeisboundedintheentireballbecausetheDirichletdatacanbeunbounded{anfunctionneednotbe.)Thisisaverymodesthypothesis.Indeed,since1issmoothfor1,ellipticregularityassuresusthatuniformlyboundedinanycompactsubsetof.Theessentialcontentof(34)isthusthat doesnotdivergeas1.Iftheconductivitywerepositiveandnitesuchgrowthwouldberuledoutbythevariationalprinciple(5)andaneasytruncationargument.Withthismodesthypothesison,wecanidentifyitsvaluesinbychangingvariablesthenusingastandardtheoremabouttheremovabilityofpointsingularitiesforharmonicfunctions.Proposition2solves(31)andsatises(34)thenfor2(35)whereistheharmonicfunctiononwiththesameDirichletdataasProof.Since)issmoothandboundedawayfromzeroforstrictlylargerthan1,ellipticregularityappliesandisaclassicalsolutionofthePDEin .Whenissupported ,thePDEcombineswiththedenitionofandthechangeofvariablesformulatogiveSince)issupportedon ,thetestfunction))vanishesat0andbutisotherwisearbitrary.So(36)tellsusthat))isaweaksolutionof
=0inthepuncturedball.Byellipticregularity,itisalsoaclassicalsolution.Wenowusethefollowingwellknownresultaboutremovablesingularitiesforharmonicfunctions:if
=0inapuncturedballabout0andifindimension3,orindimension0,thenhasaremovablesingularityat0(seee.g.[15]).Inotherwords,(0)isdeterminedbycontinuityand(soextended)isharmonicintheentireball.))satises(37){indeed,itisuniformlyboundednear0asaconsequenceof(34).Soisharmonicon.MoreoverhasthesameDirichletdataas,since.Thusispreciselythefunctionthatappearsin(35),andtheproofiscomplete.Remark1Wehaveshownusingelliptictheorythatforthecloakconstructedusingthesingularchangeofvariable(12),thepotentialoutsidethecloakedregionisgivenby(35).Analternative,morephysicaljusticationof(35)isthis:itgivesthelimitingvalueofthepotentialassociatedwithourregularnear-cloak(10)inthesingularlimitTojustifytheRemark,letbetheregularizedchangeofvariable(10),andletbethepotentialinthenear-cloakforagivenchoiceoftheDirichletdata.Then))isharmonicoutside.Itisalsouniformlybounded(awayfromtheouterboundary=2),withaboundindependentof.Sobyastandardcompactnessargument,thelimitas0existsandisharmonic.Sincethelimitisbounded,0isaremovablesingularityand)=lim)istheuniqueharmonicfunctioninwiththegivenDirichletdata.Nowforanyxed1wecanpasstothelimit0intherelation))togetconrming(35). 4.3ThepotentialinsidethecloakedregionWehaveassertedthatthesolutionof(31)isgivenby(32).Proposition2justiesthisassertionoutside;thissectioncompletesthejusticationbyshowingthat(i)theproposedisindeedasolution,and(ii)itistheonlyreasonablesolution.Toshowthatisasolution,wemustdemonstratethatisdivergence-free.ThisisthemaingoalofthefollowingProposition.Proposition3,letbedenedby(32).ThenisLipschitzcontinuousawayfrom,i.e.isuniformlyboundedinforeveryisalsouniformlyboundedawayfromuniformlyas,whereisthenormalto,andisweaklydivergence-freeintheentiredomainProof.Weobserverstthat(d)followsimmediatelyfrom(b),(c),and(36).Indeed,aboundedvector-eldisweaklydivergence-freeonifandonlyifitisweaklydivergence-freeonthesubdomainsand anditsnormal uxiscontinuousacrosstheinterface.(Thenormal uxiswell-denedfromeitherside,asaconsequenceofbeingdivergencefreeinanditscomplement.)Weapplythisto,whichisclearlyclearlydivergence-freein(whereitvanishes)andin (byequation(36)).If(c)holdsthenthenormal ux=0vanishesonbothsidesof.Inparticularitiscontinuous,so(d)holds.Theproofsof(a)-(c)arestraightforwardcalculationsbasedonthechangeofvariableformulaandthesmoothnessof)),togetherwithourexplicitformulasfor(24)and(26).Toseethatisboundedawayfromweobservethat,bychainruleandthesymmetry,wehavefor12.Thematrix(isuniformlybounded,by(24);andisbounded(exceptperhapsnear)sinceisharmonicin.ThusisboundedandisLipschitzcontinuouson1forany2.Itismoreoverconstanton,andcontinuousacross.ThereforeisLipschitzcontinuousontheentireballforeveryIndimensions3(b)followsimmediatelyfrom(a),since1isuniformlybounded.Indimension=2howeverwemustbemorecareful,since1becomesunboundedas1.Usingthedenitionof,chainrule,andthesymmetryofwehavefor12.Thesymmetricmatrices1and(havethesameeigenvectors,namely^and^.Taking=2in(24)and(26)weseethattheeigenvalueof1indirection^behaveslike,whilethatof(behaveslike.Theeigenvaluesofbothmatricesindirection^ arebounded.Thustheproductisbounded.Thisyields(b),sinceisboundedawayfromand=0forTheproofof(c)issimilartothatof(b).Since1correspondsto0and,wemustshowthatthe^componentof(38)tendstozeroas0.Sinceissymmetricand^isaneigenvector,itsucestoshowthatthecorrespondingeigenvaluetendsto0.Infact,itsvalueaccordingto(24)and(26)is (2+whichtendstozerolinearly(if=2)orbetter(if3).Theproofisnowcomplete.Wehaveshownthatthefunctiondenedby(32)solvesthePDE(31).Isittheonlysolution?werestrictlypositiveandnite,uniquenesswouldbestandard.Whenisdegenerate,however,uniquenesscansometimesfail.Forexample,ifwereidentically0inthenthesolutionwouldnotbeunique:wouldbearbitraryin.Oursituation,however,ismuchmorecontrolled:thedegeneracyoccursonlyat,andithasaveryspecicform.Uniquenessshouldbeprovedinaspecicclass.WeassumedinSection4.2thatwasuniformlyboundednear.Hereweassumefurtherthat)andProposition4isaweaksolutionofthePDE(31)whichalsosatises(34)and(39)thenmustbegivenbytheformula(32).Proof.WeknowfromProposition2that)outside.Whatremainstobeprovedisthat(0)inRecallthathasaremovablesingularityat0.Inparticularitiscontinuousthere.Sincemapsto0,itfollowsthat(0)asapproachesfromoutside.Since)byhypothesis,therestrictionofmakessense,anditisthesamefromoutsideorinside.Evidentlythisrestrictionisconstant,identicallyequalto(0).Itfollows,byuniquenessforthePDE)=0in,that(0)throughout,asasserted.Theprecedingargumentactuallyusessomewhatlessthan(39).Anyconditionthatmakescontinuousacrosswouldbesucient.Howeverwealsoneedahypothesison(forexamplethatitbeintegrable)forthePDE(31)tomakesense.4.4ThesingularcloakisinvisibleOurmainpointisthatiftheshellhasconductivity1thentheballiscloaked.Thisisaneasyconsequenceoftheprecedingresults:Theorem3Supposeisgivenby(11),whereisgivenby(12)andisuniformlypositiveandnite(22).ThentheassociatedDirichlet-to-Neumannmapisthesameasthatofauniformballwithconductivity Proof.ItsucestoprovethatanddeterminethesamequadraticformonDirichletdata,whereistheDirichlet-to-Neumannmapoftheuniformball.Butby(32)wehavedy;andthedenitionofcombinedwiththechangeofvariablesformulagiveswhereisharmonicwiththesameDirichletdataas.Thusf;ff;fforall,whenceasasserted.Wehavefocusedontheradialsettingforthesakeofsimplicity.HowevertheanalysisinthissectionextendsstraightforwardlytothenonradialcloaksdiscussedattheendofSection3.Theorem4LetbeaLipschitzcontinuousmapwithLipschitzcontinuousinverse,and.Thenactsastheidentityon,while\blowingup"thepoint.(ThisisthelimitofFigure3).Consideraconductivitydenedonoftheformforforwhereissymmetric,positive,andnitebutotherwisearbitrary.TheassociatedDirichlet-to-Neumannmapisindependentof;infact,istheDirichlet-to-NeumannmapassociatedwithconductivityProof.Weclaimthat)for)forwhere)andsolves
=0in withthesameDirichletdataas.Theproofisparalleltoourargumentintheradialcase,soweshallberelativelybrief.TheproofofProposition2madenouseofradialsymmetry;itappliesequallyinthepresentsetting.Wemustassumeofcoursethatisboundedawayfrom ,andweconcludethat(40)iscorrectoutsideTheanalogueofProposition3(a)isthestatementthatisuniformlyLipschitzin exceptperhapsnear .Withtheconventions),and),wehavebychainrule.Byhypothesis,andareuniformlybounded.Therefore(isuni-formlyboundedtoo.Since
=0,isasmoothfunctionofexceptperhapsnear .Itfollowsthat))isuniformlyLipschitzcontinuousawayfrom TheanalogueofProposition3(b)isthestatementthatisuniformlyboundedawayfrom .Recallingthedenition detDHDHandusingthat,weseethat detSinceisharmonic,itissmoothawayfrom .Asfordet():ithasthesamebehaviorDF=det(),sinceandarebounded.Oneveriesusingtheexplicitformula(24)thatDF=det()staysboundedasTheanalogueofProposition3(b)isthestatementthatthenormal ux(0asapproachesfromoutside,whereistheunitnormalat.Weusethefactthatisparallelto(),ifistheunitnormaltoatthecorrespondingpoint).(Toseethis,notethatifistangenttothenDGistangentto,andDG;;=0.)ItfollowsthatNow,=(det=(detDGDFDGSotheinnerproductontherightsideof(41)isequalto(detDGDFDG=(detDFDGu;Sinceandarebounded,thisisboundedbyaconstanttimes(detDFDGu;Butrecallthatisaneigenvectorofthesymmetricmatrix(det,withaneigenvaluethattendsto0as0.Therefore0asasasserted.TheargumentsusedforProposition3(d),Proposition4andTheorem3didnotuseradialsymmetryortheexplicitformofthecloak,sotheyextendimmediatelytothepresentsetting.Wenotethatfor3theresultsinTheorem3andTheorem4coincidewiththosealreadyestablishedin[19].AcknowledgementsThisworkwassupportedbyNSFthroughgrantsDMS-0313744andDMS-0313890(RVKandHS),DMS-0412305andDMS-0707850(MIW),andDMS-0604999(MSV).Wethanktheanonymousrefereeforhisconstructivecriticism,whichsignicantlyimprovedthepaper. 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