Optimal stability estimate of the inverse boundary

arXiv:0708.3289v1[math.AP]24Aug2007OPTIMALSTABILITYESTIMATEOFTHEINVERSEBOUNDARYVALUEPROBLEMBYPARTIALMEASUREMENTSHORSTHECKANDJENN-NANWANGAbstract.InthisworkweestablishlogtypestabilityestimatesfortheinversepotentialandconductivityproblemswithpartialDirichlet-to-Neumannmap,wheretheDirichletdataishomoge-neousontheinaccessiblepart.Thisresult,tosomeextent,im-provesourformerresultonthepartialdataproblem[HW06]inwhichlog-logtypeestimateswerederived.1.IntroductionInthispaperwestudythestabilityquestionoftheinversebound-aryvalueproblemfortheSchr¨odingerequationwithapotentialandtheconductivityequationbypartialCauchydata.Thistypeofin-verseproblemwithfulldata,i.e.,Dirichlet-to-Neumannmap,werefirstproposedbyCalder´on[Ca80].Forthreeorhigherdimensions,theuniquenessissuewassettledbySylvesterandUhlmann[SU87]andareconstructionprocedurewasgivenbyNachman[Na88].Fortwodi-mensions,Calder´on’sproblemwassolvedbyNachman[Na96]forW2,pconductivitiesandbyAstalaandP¨aiv¨arinta[AP06]forL∞conductivi-ties.Thisinverseproblemisknowntobeill-posed.Alog-typestabilityestimatewasderivedbyAlessandrini[Al88].Ontheotherhand,itwasshownbyMandache[Ma01]thatthelog-typeestimateisoptimal.Allresultsmentionedaboveareconcernedwiththefulldata.Re-cently,theinverseproblemwithpartialdatahasreceivedlotsofat-tentions[GU01],[IU04],[BU02],[KSU05],[FKSU07],[Is07].Alog-logtypestabilityestimatefortheinverseproblemwithpartialdatawasderivedbytheauthorsin[HW06].Themethodin[HW06]wasbasedon[BU02]andastabilityestimatefortheanalyticcontinuationprovedin[Ve99].Webelievethatthelogtypeestimateshouldbetherightestimatefortheinverseboundaryproblem,evenwithpartialdata.Inthispaper,motivatedbytheuniquenessproofinIsakov’swork[Is07],weprovealogtypeestimatefortheinverseboundaryvalueproblemThesecondauthorwassupportedinpartbytheNationalScienceCouncilofTaiwan(NSC95-2115-M-002-003).12HECKANDWANGunderthesameaprioriassumptionontheboundaryasgivenin[Is07].Precisely,theinaccessiblepartoftheboundaryiseitherapartofasphereoraplane.Also,oneisabletousezerodataontheinaccessiblepartoftheboundary.Thestrategyoftheproofin[Is07]followstheframeworkin[SU87]wherecomplexgeometricalopticssolutionsarekeyelements.Akeyobservationin[Is07]isthatwhenΓ0isapartofasphereoraplane,weareabletouseareflectionargumenttoguaranteethatcomplexgeometricalopticssolutionshavehomogeneousdataonΓ0.Letn≥3andΩ⊂Rnbeanopendomainwithsmoothboundary∂Ω.Givenq∈L∞(Ω),weconsidertheboundaryvalueproblem:(Δ−q)u=0inΩu=fon∂Ω,(1.1)wheref∈H1/2(∂Ω).Assumethat0isnotaDirichleteigenvalueofΔ−qonΩ.Then(1.1)hasauniquesolutionu∈H1(Ω).TheusualdefinitionoftheDirichlet-to-NeumannmapisgivenbyΛqf=∂νu|∂Ωwhere∂νu=∇u·νandνistheunitouternormalof∂Ω.LetΓ0⊂∂ΩbeanopenpartoftheboundaryofΩ.WesetΓ=∂Ω\Γ0.WefurthersetH1/20(Γ):={f∈H1/2(∂Ω):suppf⊂Γ}andH−1/2(Γ)thedualspaceofH1/20(Γ).ThenthepartialDirichlet-to-NeumannmapΛq,ΓisdefinedasΛq,Γf:=∂νu|Γ∈H−1/2(Γ)whereuistheuniqueweaksolutionof(1.1)withDirichletDataf∈H1/20(Γ).Inwhatfollows,wedenotetheoperatornormbykΛq,Γk∗:=kΛq,ΓkH1/20(Γ)→H−1/2(Γ)Weconsidertwotypesofdomainsinthispaper:(a)Ωisaboundeddomainin{xn0}andΓ0=∂Ω∩{xn0};(b)ΩisasubdomainofB(a,R)andΓ0=∂B(a,R)∩∂ΩwithΓ06=∂B(a,R),whereB(a,R)isaballcenteredatawithradiusR.DenotebyˆqthezeroextensionofthefunctionqdefinedonΩtoRn.Themainresultofthepaperreadsasfollows:Theorem1.1.AssumethatΩisgivenasineither(a)or(b).LetN0,sn2andqj∈Hs(Ω)suchthatkqjkHs(Ω)≤N(1.2)OPTIMALSTABILITYESTIMATE3forj=1,2,and0isnotaDirichleteigenvalueofΔ−qjforj=1,2.ThenthereexistconstantsC0andσ0suchthatkq1−q2kL∞(Ω)≤ClogkΛq1,Γ−Λq2,Γk∗−σ(1.3)whereCdependsonΩ,N,n,sandσdependsonnands.Theorem1.1canbegeneralizedtotheconductivityequation.Letγ∈Hs(Ω)withs3+n2beastrictlypositivefunctiononΩ.Theequationfortheelectricalpotentialintheinteriorwithoutsinksorsourcesisdiv(γ∇u)=0inΩu=fon∂Ω.Asabove,wetakef∈H1/20(Γ).ThepartialDirichlet-to-NeumannmapdefinedinthiscaseisΛγ,Γ:f7→γ∂νu|Γ.Corollary1.2.LetthedomainΩsatisfy(a)or(b).Assumethatγj≥N−10,sn2,andkγjkHs+3(Ω)≤N(1.4)forj=1,2,and∂βνγ1|Γ=∂βνγ2|Γon∂Ω,∀0≤β≤1.(1.5)ThenthereexistconstantsC0andσ0suchthatkγ1−γ2kL∞(Ω)≤ClogkΛγ1,Γ−Λγ2,Γk∗−σ(1.6)whereCdependonΩ,N,n,sandσdependonn,s.Remark1.3.Forthesakeofsimplicity,weimposetheboundaryiden-tificationcondition(1.5)onconductivities.However,usingtheargu-mentsin[Al88](alsosee[HW06]),thisconditioncanberemoved.Theresultingestimateisstillintheformof(1.6)withpossibledifferentconstantCandσ.2.PreliminariesWefirstproveanestimateoftheRiemann-Lebesguelemmaforacertainclassoffunctions.Letusdefineg(y)=kf(·−y)−f(·)kL1(Rn)foranyf∈L1(Rn).Itisknownthatlim|y|→0g(y)=0.4HECKANDWANGLemma2.1.Assumethatf∈L1(Rn)andthereexistδ0,C00,andα∈(0,1)suchthatg(y)≤C0|y|α(2.1)whenever|y|δ.ThenthereexistsaconstantC0andε00suchthatforany0εε0theinequality|Ff(ξ)|≤C(exp(−πε2|ξ|2)+εα)(2.2)holdswithC=C(C0,kfkL1,n,δ,α).Proof.LetG(x):=exp(−π|x|2)andsetGε(x):=ε−nG(xε).Thenwedefinefε:=f∗Gε.Nextwewrite|Ff(ξ)|≤|Ffε(ξ)|+|F(fε−f)(ξ)|.Forthefirsttermontherighthandsideweget|Ffε(ξ)|≤|Ff(ξ)|·|FGε(ξ)|≤kfk1|ε−nεnFG(εξ)|≤kfk1exp(−πε2|ξ|2).(2.3)Toestimatethesecondterm,weusetheassumption(2.1)andderive|F(fε−f)(ξ)|≤kfε−