ON THE BOUNDEDNESS OF A CLASS OF THE FIRST ORDER LINEAR DIFFERENTIAL OPERATORS

Transcription

1 PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Pysical ad Mateatical Scieces,, 3 6 Mateatics ON THE BOUNDEDNESS OF A CLASS OF THE FIRST ORDER LINEAR DIFFERENTIAL OPERATORS V Z DUMANYAN Cair of Nuerical Aalysis ad Mateatical Modelig, YSU I te reset article te first order liear differetial oerators wit ubouded coefficiets are ivestigated Te boudedess of te oerators uder cosideratio was roved Keywords: bouded differetial oerator, first order differetial oerator Let Q R,, be a bouded doai wit soot boudary Cosider te first order differetial eressio Q C Tu ( b( ), u( )) div( c( ) u( )) + d( ) u( ), u W, () ( ) () ( ) wit coefficiets b ( ) = ( b ( ),, b ( )), c ( ) = ( c ( ),, c ( )) ad d( ) tat are easurable ad bouded o eac strog ier subdoai of te doai Q For a arbitrary uv, W defie Tu, v (( b( ), u( )) v( ) + ( c( ) u( ), v( )) + d( ) u( ) v( )) d Q Te ai of tis article is to obtai coditios to be iosed o te coefficiets b ( ), c ( ) ad d( ), for wic T is a liear bouded oerator actig fro W ito W Tis roerty as iortat alicatios i studyig te robles of ateatical ysics (see, for eale, [, ]) Te followig teore is roved Teore Let te followig coditios old b ( ) = O as r ( ), () r ( ) were r ( ) is te distace of a oit Q fro te boudary Q, tc () t dt <, were Ct () = su c ( ), () 3 r( ) t td() tdt<, were D() t = su d( ) (3) r( ) t E-ail: dua@ysua

2 4 Proc of te Yereva State Uiv Pys ad Mate Sci,,, 3 6 Te te oerator T is a bouded liear oerator fro W ito W Proof of Teore Let Q be a arbitrary oit of te boudary Q of te doai Q, ( ', ) be a local coordiate syste wit te origi ad te ais directed alog te ier oral ν ( ) to Q at te oit Sice, Q C, tere eists a ositive uber r > ad a fuctio ϕ C ( R ) wit roerties ϕ () =, ϕ () = ad ϕ ( ') for all ' R, ( r ) suc tat te itersectio of te doai Q wit te ball U = { : < r } of ( r ) ( r ) radius r ad te cetre as te for Q U = U {( ', ) : ( ')} >ϕ Te Q U ( r ) ( ) U r {( ', ) : ( r = =ϕ ')} Let l = Fro te coverig ( l ) ( l ) { U, Q} of te boudary Q select a fiite subcoverig U, =,, ( l ) Deote for silicity U by U, r by r, l by l, ϕ by ϕ, =,, Now set = i(,, ) 3 5 r r Te eac of te curviliear cyliders l, Π = {( ', ) : ' < l, ϕ( ') < < ϕ( ') + }, =,,, is cotaied i te corresodig ball U, as well as i Q (recall tat l ( ', ) are te coordiates of a oit i a local syste of coordiates wit origi at ) Let l < be a ositive uber suc tat te coleet of te doai Q = { Q: r( ) = dist(, Q) > l } i Q is cotaied i te uio of te cyli- l ders, l l, Π, =,,, ie Q = { Q: r( ) = dist(, Q) l } Π It is easily verified tat for all U = ( ', ) l, Π, =,,, 5 r ( ) ϕ( ') r ( ) Now fi soe uber,, ad take a local coordiate syste wit origi at We defie aigs L ad L of te sace R oto itself usig relatios L ( ) = ( ', ϕ( ')), were = ( ', ) ad L ( y) = ( y', y + ϕ( y')), l y = ( y', y ) Te iage of, l Π uder te aig L will be deoted by, Π : l, l, L Π =Π ( ) Now take arbitrary fuctios u W ad η C ad ake te otatios uy ( ', y + ϕ( y')) = uy ( ), η( y', y + ϕ( y')) = η( y) =

3 Proc of te Yereva State Uiv Pys ad Mate Sci,,, I view of (), () ad (3) we ave u ( ) η( ) Tu, η K d+ Cr ( ( )) u ( ) η( ) d+ Dr ( ( )) u ( ) η( ) d r ( ), Q Q Q were K is a costat Let us estiate u ( ) η( ) u ( ) η( ) I = d d+ u( ) η( ) d r ( ) l r ( ) l Q Q Q l u ( ) η( ) d + u r ( ) l = Π l, For =,, te followig estiate olds: η ( ) ( ) W Q W Q / / u () η() 5 uy () () y 5 ( y) d dy u( y) dy dy l, r () η η l, y l, l, Π Π Π y Π / / η ( y) 5 u ( ) d dy cost u y l, l, Π Π η W W We used ere te Hardy iequality (see, for eale, [3]), i virtue of wic η ( y) η ( y', y ) ' cost ( ) dy = dy dy η y dy l, y y l, Π Tus, y' < l W W were te costat does ot deed o u ad η Net, Π I cost u η, (4) I = Cr ( ( )) u ( ) η( ) d Cr ( ( )) u ( ) η( ) d+ Cl ( ) u ( ) η( ) d Q l Q Q l Cr ( ( )) u ( ) η( ) d+ Cl ( ) u = Π l, For =,, we ave l, l, Π Π η W W Cr (()) u () η() d C(()) r u() d η W / / y C y u () y dy η C y (',) W y u y τ dτdy η W l, 5 l, Π 5 Π / dyc y y dy ' dτ u( y ', τ) η W 5 ' y < l / C y y dy u η 5 / W W

4 6 Proc of te Yereva State Uiv Pys ad Mate Sci,,, 3 6 Tus, we obtai I cost u η, (5) W W were te costat does ot deed o u ad η Siilarly we obtai I = Dr ( ( )) u ( ) η( ) d Dr ( ( )) u ( ) η( ) d+ Dl ( ) u ( ) η( ) d 3 Q l Q Q l Dr ( ( )) u ( ) η( ) d+ Dl ( ) u = Π l, Fially, for =,, we ave η W W D( r ( )) u ( ) η( ) d D y uy ( ) η( y) dy 5 l, l, Π Π / / η ( y) D y yu ( y) dy dy, 5 l l, Π y Π / y 3 τ τ η W 5 cost D y y u( y', ) d dy l, Π / 3 W W cost D y y dy u η 5 Tus, I3 cost u η, (6) W W were te costat is ideedet of u ad η Terefore, i view of (4) (6) te followig estiate olds Tu, η cost u η, were te costat is ideedet of u ad η W W Sice te fuctios η ( ) fro C are dese everywere i W, te roof of te Teore iediately follows fro te establised estiate Te Teore is roved Received 5 REFERENCES Ladyzeskaya OA, Uraltseva NN Liear ad Quasiliear Ellitic Equatios Acadeic Press, 968 Mikailov VP Partial Differetial Equatios st ed M: Nauka, 976; Eglis trasl of st ed M: Mir, Maz'ya VG Sobolev Saces Leigrad: Leigrad Uiv Press, 985 (i Russia); Eglis traslatio fro te Russia by TO Saosikova Berli New York: Sriger-Verlag, Mikailov VP, Gusci AK Advaced Toics of Equatios of Mateatical Pysics Lecture courses of SEC M: Steklov Mateatical Istitute of RAS, 7

5 Proc of te Yereva State Uiv Pys ad Mate Sci,,, Վ Ժ Դումանյան Առաջին կարգի գծային դիֆերենցիալ օպերատորների սահմանափակության մասին Հոդվածում ուսումնասիրվում են առաջին կարգի անսահմանափակ գործակիցներով գծային դիֆերենցիալ օպերատորներ: Ապացուցվում է այդ օպերատորների սահմանափակությունը: В Ж Думанян Об ограниченности линейных дифференциальных операторов первого порядка В настоящей работе исследуются линейные дифференциальные операторы первого порядка с неограниченными коэффициентами Установлена ограниченность рассматриваемых операторов