On the Basis Property of Eigenfunction. of the Frankl Problem with Nonlocal Parity Conditions. of the Third Kind

Transcription

1 It.. Cotemp. Math. Scieces Vol. 9 o HIKARI Lt O the Basis Popety o Eigeuctio o the Fakl Poblem with Nolocal Paity Coitios o the Thi Ki A. Sameipou a A. Ghaeahmati Math. Depatmet Loesta Uivesity Khoam Aba Loesta Ia Copyight A. Sameipou a A. Ghaeahmati. This is a ope access aticle istibute ue the Ceative Commos Attibutio Licese which pemits uesticte use istibutio a epouctio i ay meium povie the oigial wok is popely cite. Abstact I the peset pape we obtai the eigevalues a eigeuctios o the Fakl poblem with a olocal paity coitio o the thi ki.we pove the miimalist the completeess a Riesz basis o the eigeuctios coespoig to the eigevalues o the poblem i the space L D.. INTRODUCTION The pimay Fakl poblem was iquie i []. The poblem with a olocal bouay coitio o the seco ki was stai i []. I the peset pape we assume bouay caitios o the thi ki that whe y is limite to zeo a also i x = o x=o the uctio values ae liealy epeet i the elliptic a Hypebolic egios. I the poo o picipal theoem we ivestigate the miimalist the completeess a Riesz basis o a speciie system o cosies. Deiitio. System Deiitio. System { x X is calle complete i X i L[{ x } ] = X. } N N { x X is calle miimal i X i x L[{ x } ] k N. } N k k

2 3 A. Sameipou a A. Ghaeahmati Remak. I the system { x } is miimal i I the it is also miimal i N L p o I ; a i it is complete i L p I i L p ; o I. L p the it is also complete KIND. THE FRANKL PROBLEM WITH NONLOCAL CONDITION OF THE THIRD The Fakl poblem is to seek a solutio o equatio u sg y u sg x y u = xx yy D D i D with the bouay coitios u = [ ] u y = x y 3 u u x = x y y κ u y = u y y [] 5 κ u x = u x 6 The uctio u x y C D D D C D C D. which aeas D D a D ae eie as ollows: D = : < < < < D = x y : y < x < y < y < D = x y : x < y < x < x < Theoem.The eigevalues a eigeuctios o poblem -6 show by two seies.i the ist seies the eigevalues λ ae ou om the equatio k = k = k such that =... k =... a the α z ae the Bessel uctios [3 p. ] a the eigeuctios ae povie by the egulatios uk = Ak kcos i D

3 Fakl poblem with olocal paity coitios 35 uk ρ ψ = κak kρcosh ψ i D uk R ϕ = κak krcosh ϕ i D that we use o pola cooiate system = x y x = cos y = si o a i D o catesia cooiate system ρ = x y x = ρ coshψ y = ρ sihψ o <ψ < a < ρ < i D a R = y x x = Rsihϕ y = R coshϕ o < ϕ < a < R < i D. I the seco seies the eigevalues λ k = k ae esulte om the equatio Δ k = =... k =... a the eigeuctios ae etemie by the elatios u k Ak k = cosα id α κ cosh κ uk ρ ψ = Ak α kρ κ α sih id ψ α ψ κ κ u k R ϕ = κak α krcoshαϕ id whee; κ α = Δ Δ = acsi Δ κ Theoem.The system o uctios cos cos Δ = = is complete a a Riesz basis i the space L o Δ. o < 3 Δ the system is ot complete but is miimal o Δ > is complete but is ot miimal a i Δ = is complete a miimal. Poo. The poo o this theoem we use the covegece uctio = A cos cos Δ = B =

4 36 A. Sameipou a A. Ghaeahmati i L Riesz basis the system 3 o Δ a [3]. { si Δ } = Thoem 3.The system eigeuctio uk = Ak kcos cos u = α k Akα k is complete a basis i the space L theeoe uk = D u = D k a L D the = i D. D Poo. Usig obii theoem a Lebesgue s itegal o ay k =... we have = uk D k = agai sice L D so; cos < Isomuch system { k} k= i L is othogoal a complete it is eough to pove; cos L Usig the Hole iequality cos

5 Fakl poblem with olocal paity coitios 37 cos < cos = cos8 < < = = with the itegatio iteval < cos Thus < cos This iequality is equivalet to < cos Also system } { k is othogal a complete o... = k i L o elatio = cos k

6 38 A. Sameipou a A. Ghaeahmati imply that cos = Accoig to theoem we coclue that = i L. Similaly i we cosie the above calculatios o sequece { cos[ Δ ] } o =... we have; cos[ Δ] = Because completeess []-_=^ = i L.The poo o the theoem is complete. Theoem 3. The system o eigeuctios u k a u k o the poblem -6 is a Riesz basis i the space L D whee Ak = k Ak = Δ k Poo.Theoem 3 esults om Theoem a the completeess a othogality o the system A } = o a { A Δ k k} k= o i L. { k k k REFERENCES [] Fakl F.I. O the Poblems o Chaplygi o Mixe Sub- a Supesoic Flows Izv. Aka. Nauk SSSR 95 vol. 9 o. pp.. [] Moiseev E.I. a AmbatsumyaV.E. O the Basis Popety o Eigeuctios o the Fakl Poblem with a Nolocal Paity Coitio o the Seco Ki ISSN -66 Dieetial Equatios 9 Vol. 5 No. pp [3] A.Eelyi W.MagusF.Obehettigea F.G.TicomiHighe Tasceetal FuctiosBatema Mauscipt PojectMcGaw-HillNew YokTaslate ue the title Vysshie tastseetye FuktsiiVol.Moscow [] Abbasi.NSpectal Aspects o the Fakl Poblem o a Equatio o the Mixe Typeca.Sciphys-MathDissctatioMoscow9 [5] Moiseev.E.IO the Basis popety o a System o sies Die.uav987Vol.3. o [6] Moiseev E.I. The Basis Popety o Systems o Sies a Cosies Dokl. Aka. Nauk SSSR 98 vol. 75 o. pp Receive: Decembe 3