一类Helmholtz方程混合边值问题解的存在唯一性.docx

一类Helmholtz方程混合边值问题解的存在唯一性
Introduction:
Helmholtzequationisasecond-orderpartialdifferentialequation,whichhasawiderangeofapplicationsinvariousfields,suchasacoustics,electromagnetics,andfluiddynamics,etc.Theequationarisesinsituationswherewavespropagateinamediumandthemediumdependsonspace.Inthispaper,wewilldiscusstheexistenceanduniquenessofsolutionsforaclassofHelmholtzequationmixedboundaryvalueproblems.
MathematicalFormulation:
LetΩbeaboundeddomaininR^dwithasmoothboundary∂Ω.TheHelmholtzequationwithawavenumberk>0andaforcingtermf(x)inΩcanbewrittenas:
∆u+k^2u=f(x)inΩ,
whereuistheunknownfunction,and∆denotestheLaplaceoperator.Themixedboundaryvalueproblemweconsideristofindu(x)suchthat:
(1)∆u+k^2u=f(x)inΩ,
(2)∂u/∂n+iku=g(x)on∂Ω,
wherendenotestheunitoutwardnormalto∂Ω.Theboundarycondition(2)isamixedboundarycondition,whichinvolvesboththeNeumannandDirichletconditions.Here,g(x)isagivencomplex-valuedfunctionontheboundary∂Ω,whichdeterminestheboundarydata.
ExistenceResult:
Webeginbyprovingtheexistenceofasolutiontotheabovemixedboundaryvalueproblem.Forthis,weconsidertheweakformulation:
(3)∫Ω(∇u·∇v+k^2uv)dx=∫Ωfvdx+∫∂Ω(g-∂u/∂n)vdSforallv∈H^1(Ω),
whereH^1(Ω)denotestheSobolevspaceoffunctionsinL^2(Ω)withweakderivativesinL^2(Ω).Notethatthesecondtermontheright-handsideof(3)involvesthetraceofuandtheboundarydatag.Hence,weneedtoensurethatthetraceoperatoriswell-definedandcontinuous.
Tothisend,weassumethatg∈H^{1/2}(∂Ω),thespaceoffunctionsinL^2(∂Ω)withasquare-integrableboundarytrace.Then,itcanbeshownthatthereexistsaboundedlinearoperatorT:H^{1/2}(∂Ω)→H^{-1/2}(∂Ω)suchthat:
(4)Tu=u|∂Ωforu∈H^1(Ω)∩C(¯Ω),
withnorm||Tu||_{H^{-1/2}(∂Ω)}≤C||u||_{H^1(Ω)},whereCisapositiveconstantindependentofu.Moreover,Tiscontinuousandinjective,andhence,wecanidentifyH^{1/2}(∂Ω)withitsimageinH^{-1/2}(∂Ω)viaT.
LetV={v∈H^1(Ω):∂v/∂n+ikv∈H^{-1/2}(∂Ω)}bethespaceofadmissibletestfunctionsin(3).Then,(3)definesaboundedlinearfunctionalonV′,thedualspaceofV,andbytheRieszrepresentationtheorem,thereexistsauniquesolutionu∈V⊂H^1(Ω)to(3).
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