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J.Math.PuresAppl.81(2002)1191–1206Onthefundamentalsolutionsofaclassofellipticquarticoperatorsindimension3PeterWagnerInstitutfürTechnischeMathematik,Geometrie,undBauinformatik,UniversitätInnsbruck,Technikerstr.13,A-6020Innsbruck,AustriaAccepted9March2002AbstractThe(uniquelydetermined)evenandhomogeneousfundamentalsolutionsofthelinearellipticpartialdifferentialoperatorswithconstantcoefficientsoftheform3j=13k=1cjk∂2j∂2karerepresentedbyellipticintegralsofthefirstkind.TherebywegeneralizeFredholm’sexample∂41+∂42+∂43,which,uptonow,wastheonlyirreduciblehomogeneousquarticoperatorin3variablesthefundamentalsolutionofwhichwasknowntobeexpressiblebytabulatedfunctions.2002ÉditionsscientifiquesetmédicalesElsevierSAS.Allrightsreserved.Keywords:Fundamentalsolution;Ellipticoperator;Ellipticintegral1.IntroductionandnotationsIn1908,I.Fredholmshowedthat“l’intégralefondamentale”Pofthehomogeneousellipticoperatorf(∂∂x,∂∂y,∂∂z)ofthedegreen4canbewrittenintheform:P=n2ν=1ξνηνξ0η0(ξx+ηy+z)n−3f2(ξ,η)dξ(1)(cf.[3,(4),p.3]).Heref2(ξ,η):=∂f(ξ,η,1)/∂η,andtheintegrationpathsrunonthealgebraiccurve{(ξ,η)∈C2;f(ξ,η,1)=0}fromsomefixedpoint(ξ0,η0)tothoseintersectionpoints(ξν,ην)withthelineξx+ηy+z=0thatsatisfyImξν0.E-mailaddress:wagner@mat1.uibk.ac.at(P.Wagner).URLaddress:–seefrontmatter2002ÉditionsscientifiquesetmédicalesElsevierSAS.Allrightsreserved.PII:S0021-7824(02)01258-81192P.Wagner/J.Math.PuresAppl.81(2002)1191–1206AsFredholmobserves,PistherebyrepresentedasasumofAbelianintegralsofthefirstkindpertainingtothealgebraiccurvegivenabove.Hethenappliesformula(1)totheequation∂4u∂x4+∂4u∂y4+∂4u∂z4=0andobtainsassolution:P=xFζ2x2+yFζ2y2+zFζ2z2,F(u)=∞udu√4u3−u,(2)whereζisthelargestofthethreerealrootsoftheequationζ3−(x4+y4+z4)ζ−2x2y2z2=0(cf.[3,p.6]).Letusformulatehisresultabitmorepreciselyandinournotation.Wewritex1,x2,x3,P(∂),Einsteadofx,y,z,f(∂∂x,∂∂y,∂∂z),P,respectively.ThentheonlyevenandhomogeneousfundamentalsolutionEofP(∂)=∂41+∂42+∂43isgivenbyE(x)=−18π3j=1|xj|∞ζ/(2x2j)du√4u3−u,(2)ζbeingdefinedasabove.Thoughformula(1)looksquiteexplicit,itsapplicationtoreal-valuedoperatorsinvolvestheuseofadditiontheorems.Thisgetsquitecumbersomeevenfortherelativelysimpleoperator∂41+∂42+∂43tackledbyFredholm(cf.[3,p.5]).Inlateryears,G.Herglotz,F.Bureau,H.G.Garnirthusonlyconsideredthecaseofreducibleoperatorsoffourthdegree(cf.[1,4,7]).Inthisarticle,wefirstshowthat,inodddimensions,thefundamentalsolutionsofellip-ticoperatorsareuniquelydeterminedbytherequirementsofhomogeneityandevenparity(Proposition1).Wethenrestrictouranalysistofourthorderoperatorsinthreedimensions(n=4inFredholm’snotation)andweavoidtheabove-mentioneddifficultybyreexpress-ingthesumofAbelianintegralsin(1)throughjustoneAbelianintegral(Proposition2).Weapplythismethodtoellipticoperatorsofthetype3j=13k=1cjk∂2j∂2k.Althoughtherespectivecurveshavegenus3ingeneral,itisstillpossibletoexpressthefundamentalsolutionbyellipticintegrals(Proposition3).ThespecialcaseofreducibleoperatorsisconsideredinProposition4.Byanalyticcontinuation,Proposition3(combinedwithpart3inProposition1)alsoallowstoconstructthefundamentalsolutionsEPofellipticoperatorsoftheformP(∂)=3j=13k=1cjk∂2j∂2kandhavingcomplexcoefficientscjk.Fromthis,oneobtainsthefundamentalsolutionsofhyperbolicoperatorsofthistypebyalimitingprocess.Thisinvestigationwillbecarriedoutinasubsequentpaper.Letusestablishsomenotations.WedenotebyN,R,Cthenatural,real,andcomplexnumbers,respectively,andweputN0:={0}∪N.WeconsiderRnasaEuclideanspacewiththeinnerproductx,y:=x1y1+···+xnynandwrite|x|:=√x,x;Sn−1P.Wagner/J.Math.PuresAppl.81(2002)1191–12061193denotestheunitsphere{ω∈Rn;|ω|=1}inRnanddσtheEuclideanmeasureonSn−1.ForasymmetricmatrixC=(cjk),wedenotebyCad=(Cadjk)theadjointmatrix,i.e.Cad=detC·C−1ifdetC=0,thedefinitionofCadbeingextendedtoallquadraticmatricesbycontinuity.VectorsareunderstoodascolumnsandthesuperscriptTmeansmatrixtransposition.Inthisarticle,allpartialdifferentialoperatorsarelinearandcontainconstantcoefficients,andweadoptforthemtheusualmulti-indexnotation:∂j:=∂/∂xj,j=1,...,n,∂α:=∂α11···∂αnn,|α|:=α1+···+αn,α∈Nn0.Whatconsidersthetheoryofdistributions,wereferto[8,10,13].Inparticular,theHeavisidefunctionisabbreviatedbyY,andφ,TstandsforthevalueofthedistributionTonthetestfunctionφ.WeusetheFouriertransformFintheform(Fφ)(x):=e−iξ,xφ(ξ)dξ,φ∈S(Rn),thisbeingextendedtoS(Rn)bycontinuity.2.Fundamentalsolutionsofhomogeneousellipticoperatorsinodd-numbereddimensionsTheMalgrange–EhrenpreisTheorem(cf.[2,11],[8,Theorem7.3.10,p.189],[12])statesthateverynotidenticallyvanishinglinearpartialdifferentialoperatorwithconstantcoefficients,i.e.everyoperator:P(∂)=α∈Nn0,|α|mcα∂α,wherecα∈Carenotallzero,possessesafundamentalsolutionE∈D(Rn).IfdegP0,thenEisnotunique.Forhyperbolicoperators,uniquenessisguaranteedbyrequiringthesupportofEtobecontainedinaclosedhalf-space(cf.[9,Theorem12.5.1,p.120]).Analogously,forhomogeneousellipticoperators,thereexistsjustoneevenandhomogeneousfundamentalsolutionifthedimensionnisodd.Proposition1.Letn∈{1,3,5,...}a
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