Solvability and Spectral Properties of Boundary Value Problems for Equations of Even Order

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1 Malaysia Joral o Mathematical Scieces 3(): 7-48 (9) Solvability ad Spectral Properties o Bodary Vale Problems or Eqatios o Eve Order Dj.Amaov ad A.V.Yldasheva Istitte o Mathematics ad Iormatioal echologies o Academy Scieces o Uzbeista, 9, F. Khojaev str., ashet, Uzbeista, Natioal Uiversity o Uzbeista amed by M. Ulghbe. ashet, Beri str. amaov_d@rambler.r, asal-yldasheva@mail.r. ABSRAC We stdy bodary vale problems or a eqatio o the order ad prove reglar ad strog solvability o it, ivestigate spectrm o the problem. I case o eve we obtai a priori estimate or the soltio i the orm o the Sobolev space ad prove solvability almost everywhere. Keywords: solvability, bodary vale problem, spectrm, a priori estimate, reglar solvability, strog solvability, the Forier series, the Cachy-Schwarz ieqality, the Bessel ieqality, the Perceval eqality, the Lipchitz coditio, eve, odd, almost everywhere soltio. INRODUCION Bodary vale problems or the eqatios o the 3 rd ad 4 th order irst were ivestigated by Hadamard,(933) ad Sjöstrad,(937), ad developed by Davis,(954), Bitsadze,(96), Salahitdiov,(974), Dzhraev,(979), Wolersdor,(969) ad others. Bodary vale problems or the eqatios o the order 4 were stdied by Dzhraev ad Sopev,(), Salahitdiov ad Amaov,(5), Nicolesc,(954), Roitma,(97) ad Sobolev,(988). I preset paper we stdy bodary vale problems or a eqatio o the order.

2 Dj.Amaov & A.V.Yldasheva Statemet o the Problems We cosider the eqatio = ( x, t), () t i the domai { ( x,t ) : x p, t } Ω = < < < <, where is ixed positive iteger. Problem Fid the soltio ( x,t ) o the eqatio () i the domai Ω satisyig coditios m m (,t ) = ( p,t ) =, m =,,...,, t, () m m ( x, ) =, ( x, ) =, x p. (3) Problem Fid the soltio ( x,t ) o the eqatio () i the domai Ω satisyig coditios (3) ad (,t ) = ( p,t ) =, m =,,...,, t, (4) m+ m+ m+ m+ Problem 3 Fid the soltio ( x,t ) o the eqatio () i the domai Ω satisyig coditios () ad ( x, ) =, ( x, ) =, x p. (5) t 8 Malaysia Joral o Mathematical Scieces

3 Solvability ad Spectral Properties o Bodary Vale Problems or Eqatios o Eve Order Problem 4 Fid the soltio ( x,t ) o the eqatio () i the domai Ω satisyig coditios () ad ( x, ) ( x, ), = ( x, ) ( x, ) t =, x p. (6) We ivestigate Problem i detail ad other problems ca be similarly examied. Let t,, { x, t x,t V ( Ω ) = : C ( Ω ) C ( Ω ), ad coditios (), (3) are tre }, W { : C x,t,,t p,t, Lip α, p, ( Ω ) = ( Ω ) = = [ ] is iormly i t, < α }, + m, Ω = {} x,t Ω Ω = = + m { W : C, L,, with m,,..., We deie the operator L L mappig the domai V ( Ω ) ito C ( Ω ). t Deiitio A ctio ( x,t ) V problem with ( x,t ) C Ω. Ω is called the reglar soltio o the Ω i it satisies the eqatio () i the domai Malaysia Joral o Mathematical Scieces 9

4 Dj.Amaov & A.V.Yldasheva Deiitio A ctio ( x,t ) L problem with L ( Ω ) i there exists a seqece V Ω is called the strog soltio o the Ω, N, sch that, L as. L L, Deote by W the closre o the set V Ω i the orm W,, = m m Ω m= t t m= t m + m ad by W ( Ω ) the closre o the set V Ω i the orm dxdt, W = + m dxdt m. Ω m= t,, It is clear that W ( Ω ) ad W ( Ω ) are sbspaces o the Sobolev,, spaces W ( Ω ) ad W ( Ω ) respectively. I we complete the set V( Ω ), the operator L is also completed. Let L be the closre o operator L i, both cases with D( L) = W i is eve, ad D( L), = W i is odd. А Priori Estimate It is tre the ollowig Lemma. Let ( x,t ) be a reglar soltio o Problem havig cotios derivatives m+ (,t m ),,,,, t t t m =,,...,, 3 Malaysia Joral o Mathematical Scieces

5 Solvability ad Spectral Properties o Bodary Vale Problems or Eqatios o Eve Order i Ω ad belogig to L, ( x,t) C L, where is odd. he there exists a costat C > that depeds oly o sizes o the domai x,t sch that ad the mber ad does t deped o the ctio C. (7) L, W Proo. We mltiply by ( x,t ) both sides o the eqatio () ad itegrate it over the regio Ω to obtai Usig the ormlas dxdt dxdt = t. (8) Ω Ω = t t t t, m m m = x + m m m x x x, = x ad coditios (), (3), the eqatio (8) becomes + = dxdt. (9) x t L L Ω Ω Applyig the ollowig evidet ieqality ε ab a + b ε with arbitrary ε > to the right-had side o (9) we obtai ε + + L L ( Ω x t ) L L Ω Ω ε. () Malaysia Joral o Mathematical Scieces 3

6 Dj.Amaov & A.V.Yldasheva It is obvios that t t t ( x,t ) = ( ( x, τ )) dτ = ( x, ) d ( x, ) d τ τ τ τ. τ τ τ Itegratig it with respect to х rom to р gives p p x,t dx x,t dtdx. Applyig the Cachy-Schwarz ieqality to the right-had side we have t p ( x,t ) dx. L t L Itegratig it with respect to t rom to yields L L t L. Dividig by rom () L ( Ω ) both parts o this ieqality ad sqarig it we obtai ε + L L L ε. () I we add the ieqalities () ad () by choosig ε = ad 4 + mltiply by both sides o it ad replace coeiciets by o the let-had side, the we obtai L L Ω t L L. () 3 Malaysia Joral o Mathematical Scieces

7 Solvability ad Spectral Properties o Bodary Vale Problems or Eqatios o Eve Order I we sqare both parts o () ad itegrate over Ω, the we have dxdt + = L ( Ω x t x t ) L Ω L. (3) Let s rearrage the itegrad by the ollowig way + + = ( ) ( + ) = t t t = ( ) ( + + ) = t t t = ( ) ( + + ) + 3 t t t ( ) ( ) 3 3 ( + ) = t t t + m m m = ( ) m m + + m= t t t t I m is odd, the m is eve ad accordig to () we have m m+ = at x = ad x = p, i case o eve m we have = at m m t x = ad x = p. Moreover = at t = ad t =. Coseqetly, + dxdt = t t. Ω Sbstittg it ito (3) ad droppig the coeiciet we get L t t L L L L. (4) Malaysia Joral o Mathematical Scieces 33

8 Addig () ad (4) yields Dj.Amaov & A.V.Yldasheva L t t t L ( Ω ) L ( Ω ) L ( Ω ) L ( Ω ) 4 x L ( Ω ) + ( + ) +. L ( Ω ) o obtai estimates or the orms o the orm we se ieqality m m L (5), m =,..., L L L. (6) that ca easily be checed. I we sm ieqalities (6) over п rom to ad se (5), we get L L ( ). L (7 ) Now smmig p ieqalities (6) over п rom to accordig to (7 ) we have + (4 + ) + L Ω Ω L Ω L (7 ) Proceedig i this way we obtai x L L ( ).... L (73) 34 Malaysia Joral o Mathematical Scieces

9 Solvability ad Spectral Properties o Bodary Vale Problems or Eqatios o Eve Order L ( Ω ) L L ( ). (7 -) Addig ieqalities (7), (7 ),..., (7 -) yields m m= L m m ( ) (4 + ) +. (8) L x ( Ω ) Addig ieqalities (5) ad (8) we obtai m m m m L m= t L m= t Ω L L ( ). (9) Smmig p the ieqalities m m + t t m m L L L which proo is evidet, over т rom to accordig to (9) we have, m m= L m ( ) ( 4 ). L x t + + ( Ω ) () Addig (9) ad () we get m + m m m m m L m= t L m= t L m= Ω Ω L or, W L ( 4 ) C, () C where his proves Lemma. Malaysia Joral o Mathematical Scieces 35

10 Dj.Amaov & A.V.Yldasheva he Reglar Solvability o the Problem It is tre the ollowig heorem. Let ( x,t) W ( Ω ) i is eve ad ( x,t) W odd ad mbers Р ad satisy the coditio Ω i is π si δ >, N. () p he there exists a reglar soltio o Problem. We search a reglar soltio o Problem i the orm o Forier series x,t = t X x, (3) = expaded i ll orthoormal system i L (, p ). π X ( x ) = si λx, λ =, N, p p It is clear that ( x,t ) satisies coditios (). We expad the ctio ( x,t ) ito the Forier series i ctios X ( x ) = x,t = t X x, (4) where p t = x,t X x dx. (5) Sbstittig (3) ad (4) ito the eqatio () we obtai the ollowig eqatio = t λ t t. (6) '' 36 Malaysia Joral o Mathematical Scieces

11 Solvability ad Spectral Properties o Bodary Vale Problems or Eqatios o Eve Order or ow ctio t. Coditios (3) tae the orm =, =. (7) he soltio o the eqatio (6) satisyig coditios (7) has the orm i is eve, ad has the orm = ( τ ) ( τ ) τ = x,t X x K t, d, λ = λ τ τ τ = x,t X x K t, d, (8) (9) i is odd, where ( ) shλ ( τ ) = shλ K t, shλ τ shλ t ( ), τ t, shλ t shλ τ, t τ, with ( ) siλ ( τ ) = siλ K t, siλ τ si λ t ( ), τ t, si λ t si λ τ, t τ, ( i ) ( i ) K t, τ = K τ,t, i =,, ( ) C K ( t, τ ), C cost, t e = > (3) λ τ ( ) ( τ ) K t, δ. (3) Malaysia Joral o Mathematical Scieces 37

12 Dj.Amaov & A.V.Yldasheva Let be a eve mber. We have to prove iormly covergece o the series (8) ad = = λ ( ) λ X x K t, τ τ d τ, = + t = = λ ( ) X x t λ X x K t, τ τ d τ, (3) (33) I we show iormly covergece o the series = λ X x K t, τ τ d τ, (34) the which implies iormly covergece o the series (8), (3), (33). I the eqality (5) we itegrate the itegral by parts, where Sice Lip [, p] t = t λ p ( t) = cos λ xdx. x p α is iormly with respect to t, the [ 5] So t α C, C = cost >, < α <. λ C ( τ ) + α λ. (35) 38 Malaysia Joral o Mathematical Scieces

13 Solvability ad Spectral Properties o Bodary Vale Problems or Eqatios o Eve Order We ext tr to estimatig the itegral i (34). Accordig to (3) ad (35) we have ( ) ( (, ) ) (, ) K t τ τ dτ K t τ τ dτ t CC dτ C C λ ( t τ ) λ ( τ t) e d e d + = + = t + τ τ α λ τ α λ e λ o t λ t λ ( t) ( e ) ( e ) CC CC =. + α + α λ λ λ λ λ (36) he estimate (36) implies iormly covergece o the series (34), (33), (3), (8). his iishes the proo o heorem or eve. We ow tr to the case where is odd. It has to be show iormly covergece o the series (9) ad = t = = ( ) t X x λ X x K t, τ τ d τ, (37) = = λ X x K t, τ τ d τ, (38) It sices to show covergece o the series (38). Let W ( Ω ). We itegrate the itegral (5) by parts + times where t = t, (39) λ p + +. x t = X x dx + Malaysia Joral o Mathematical Scieces 39

14 Dj.Amaov & A.V.Yldasheva We proceed to estimate the itegral. Accordig to (3) ad (39) we obtai K t τ τ dτ K t τ τ dτ τ dτ ( ) (, ), + δλ + τ ( τ ) τ + d d = δλ δ λ L (, ) (4) Here we have sed the Cachy-Schwartz ieqality. aig ito accot (4) yields As ( ) λ X x K ( t, τ ) ( τ ) dτ λ + = δ p = λ = L (, ) = ( ) + <, δ p L, ( ) L, = λ δ p = λ = + =, L (, + ) = x L the the series (38) coverges iormly. By the estimate (4) the series (9) ad (37) are also coverget iormly, ad the proo o heorem is completed. Remar. As to the coditio (3) x, = x, =, x p, (3) the coditio is ecessary at t=. I we do t impose ay coditio at t= ad chage it to = =, t t the the problem is ot correct or eve. 4 Malaysia Joral o Mathematical Scieces

15 Solvability ad Spectral Properties o Bodary Vale Problems or Eqatios o Eve Order Ideed, i this is the case, the we have the ollowig eqatio or λ = t t t '' t. he geeral soltio o this eqatio has the orm t t a e b e τ shλ t τ dτ λ t λ t = + ( ) λ We reqire the obtaied soltio to satisy the ollowig coditios he we get ', = =. = a + b =, b = a λ λ λ ' ( ) = a b = a = a =, b =. It is clear that the seqece t λ = ( ) t t shλ t τ dτ does t coverge. hs the problem is icorrect. Lemma. Let is odd mber. he the soltio (9) satisies the estimate C, (4) L, W where С positive costat depedig oly o sizes o the domai ad ot depedig o the ctio ( x,t ). Proo. We rewrite the soltio (9) i the orm x,t = t X x, (4) = Malaysia Joral o Mathematical Scieces 4

16 Dj.Amaov & A.V.Yldasheva where λ t = K t, τ τ dτ (43) We evalate the orm. By (3) ad Cachy-Schwartz ieqality we get t = K t, τ τ dτ ( ) λ λ Itegratig the ieqality K t, d d. ( ) δ λ ( τ ) τ ( τ ) τ L (, ) t δ λ L (, ) with respect to t rom to we obtai. (44) δ λ L (, ) L (, ) By sig (44) we estimate L ( Ω ). = ( t) X ( x), ( t) X ( x) = L Ω p m = m= L = t t X ( x) X ( x) dxdt = m m = m= p = t t dt X x X x dx = t dt = m m = m= m= p = L (, ) L (, ) = δ π = p = δ π. L (, ) L = δ π 4 Malaysia Joral o Mathematical Scieces p

17 Solvability ad Spectral Properties o Bodary Vale Problems or Eqatios o Eve Order So p. (45) δ π L L Now we estimate the orm t L [ Ω ]. o this ed we irst estimate ' L. Ω ( ) ( t) t ' si λτ cosλ ( t) = ( τ ) dτ + si λ cosλ t si λ τ + si λ t ( τ ) dτ. t ' δ δ t δ t τ dτ + τ dτ = τ dτ dτ ( τ ) dτ. L (, ) δ = δ Sqarig this ieqality ad itegratig with respect to t rom to Т we obtai. δ ' L, (, ) L Usig this ieqality ad the Parceval idetity yields From here we get ' ' t =, L t X x m t X m x Ω = = m=. = =. ' L, L Ω L Ω = δ = δ. (46) δ t L L Malaysia Joral o Mathematical Scieces 43

18 Dj.Amaov & A.V.Yldasheva We estimate x. Combiig (44) ad the Bessel ieqality gives L ( Ω ) ' ' x =, L t X x m t X m x Ω = = m= = = ' λ L (, ), L = δ = λ L (, ) L (, ) = = p p = δ π δπ ; x p L L δπ. (47 ) For L we have the ollowig estimatio p. (47 L x ) L δπ... L δ L. (47 ) Addig the ieqalities (45), (46), (47 ),..., (47 ) yields or where ( δ π ) m + C Ω t m m= L L C, L, W C = C p,,,, = cost >. he proo o Lemma is completed. L 44 Malaysia Joral o Mathematical Scieces

19 Solvability ad Spectral Properties o Bodary Vale Problems or Eqatios o Eve Order he Strog Solvability It is tre the ollowig heorem. For ay Ω there exists a iqe strog soltio o L Problem ad it satisies estimatio (7), i is eve, ad estimatio (4) i is odd. Proo. Let be a arbitrary ctio i L Accordig to the act that W ( Ω ) is dese i L { } W ( Ω ), N sch that L Ω Ω ad be a eve mber. Ω there exists a seqece as.coseqetly, { } is Cachy seqece i L ( Ω ). We deote by ( x, t) V soltio o the eqatio () with the right part ( x, t ). By (7) we have L m, W m Ω the C,, m, (48) that is { } is a Cachy seqece i o the space,, (, ) lim (, ), W ( Ω ). Accordig to completeess W ( Ω ) there exists a iqe limit x t = x t W Ω which is the strog soltio o Problem. Passig to limit i ieqality C as we coclde L, W that estimatio (7) is also tre or the strog soltio ( x, t ) Passig to L =, V ( Ω ), W ( Ω ), as we get limit i eqatio, L =, W ( Ω ), L ( Ω ). Coseqetly, the strog soltio is a soltio almost everywhere. I a similar way oe ca prove that Problem, is strog solvable i the space W ( Ω ) i case o odd. Spectrm o Problem he spectrm o a problem is the set o eigevales o the operator o the problem. We examie spectrm o the problem i case o eve. he ivestigatio o the spectrm or odd is similar. Malaysia Joral o Mathematical Scieces 45

20 We rewrite the soltio (8) as Dj.Amaov & A.V.Yldasheva p (), (, ;, ) (, ) x t = K x t ξ τ ξ τ dξdτ, (49) where X ( x) X ( ξ ) K ( x, t;, ) = K ( t, τ ). (5) () () ξ τ = λ As K () ( t, τ ) is symmetric, the K () ( x, t; ξ, τ ) is symmetric. he estimatio (3) implies its bodedess, i.e. K ( x, t; ξ, τ ) C (5) () Combiig (49) with (5) we coclde that it is deied boded symmetric operator W Ω which is iverse o the operator L ad acts rom it by L o W Ω to V Ω by the rle p () L x, t = K ( x, t; ξ, τ ) ( ξ, τ ) dξdτ, (5) It ca be exteded to whole space L L, is the closre o L, D( L ) L Ω. his extesio, we deote = Ω. he operator symmetric, boded, ad deied o the whole space L Ω, so it is seladjoit. It ollows rom (5) that a compact operator i L () K x t ξ τ L L is (, ;, ) ( Ω Ω ) thereore Ω. he the spectrm o the operator discrete ad cosists o real eigevales o iite mltiplicity. he relatio betwee eigevales o the operators L i µ is a eigevale o the operator operator L. L L ad L is as ollows (Dezi,98): L, the is is µ is eigevale o the hs, i case o eve the spectrm o Problem cosists o real eigevales o iite mltiplicity. 46 Malaysia Joral o Mathematical Scieces

21 Solvability ad Spectral Properties o Bodary Vale Problems or Eqatios o Eve Order A similar assertio is also tre i case o odd. Corollary. Problem is sel adjoit or all. CONCLUSION I this article we have ivestigated or bodary vale problems or the eqatio o the eve order i a rectaglar domai. Oe o these problems is stdied i detail. Other problems ca be hadled i mch the same way. I case eve we have obtaied a priori estimate or the soltio, i the orm o the space W ( Ω ), proved its reglar ad strog solvability almost everywhere. I case o odd we have drive the estimate or the, reglar soltio i the orm o the space W ( Ω ). he spectrm o the problem has bee researched ad its discreteess has bee proved. he seladjoitess o problem has bee established. REFERENCES Bitsadze, A.V. 96. O the mixed-composite type eqatios, i: Some problems o mathematics ad mechaics. Novosibirs. (i Rssia). Bogoa, L. ad Moylay, M.S. 3. Geeralized soltios to parabolichyperbolic eqatios. Electroic Jor. o Di. Eqatios, 3(9): -6. Davis, R.B A bodary vale problem or third-order liear partial dieretial eqatios o composite type, Proceedigs Amer.Math.Soc.,5:7. Dezi, A.A. 98. Geeral qestios o bodary vale problems theory, Naa, Moscow (i Rssia). Dzhraev,.D Bodary vale problems or eqatios o mixed ad mixed-composite type. Fa. ashet, (i Rssia). Dzhraev,.D. ad Sopev, A.. O theory o the orth order partial dieretial eqatios. Fa. ashet (i Rssia). Malaysia Joral o Mathematical Scieces 47

22 Dj.Amaov & A.V.Yldasheva Hadamard, J Proprietes d'e eqatio lieaire ax derives partielles d qatrie ordre, he oho Math. J., 37:33-5. Nicolesc, M Eqatia iterata a calolri. St.Cers.Mat., 5:3. Roitma, E. 97. Some observatios abot o odd order Parabolic Eqatio, Joral o Di. Eqatios, 9:335. Salahitdiov, M.S Eqatios o mixed-composite type. Fa. ashet. Salahitdiov, M.S. ad Amaov, D. 5. Solvability ad spectral properties o sel-adjoit problems or the orth order eqatio, i: rasactios o the iter. Coerece:Moder problems o mathematical physics ad iormatio techologies, :5-55. (i Rssia). Sjöstrad, O Sr e eqatio ax derivees partielles d type composite, Ariv.M.A.o.F., Bd. 6A():. Sobolev, S.L Some applicatios o ctioal aalysis i mathematical physics, Naa, Moscow (i Rssia). Wolersdor, L Sjöstradsche probleme der Richtg-Sableierte aalytishe tioe, Math. Nachrichte, 4: Zygmd, A rigoometrical series, Naa, Moscow (i Rssia). 48 Malaysia Joral o Mathematical Scieces