Summability of series in the root functions of boundary-value problems of Bitsadze-Samarskii type
We investigate the Abel summability of a system of eigenfunctions and associated functions of Bitsadze-Samarskii-type boundary-value problems for elliptic equations in a rectangle. These problems are reduced to a boundary-value problem for elliptic operator differential equations with an operator in...
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| Datum: | 2008 |
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| Sprache: | Russisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2008
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509210629898240 |
|---|---|
| author | Aliev, I. V. Алиев, И. В. Алиев, И. В. |
| author_facet | Aliev, I. V. Алиев, И. В. Алиев, И. В. |
| author_sort | Aliev, I. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:47:27Z |
| description | We investigate the Abel summability of a system of eigenfunctions and associated functions of Bitsadze-Samarskii-type boundary-value problems for elliptic equations in a rectangle. These problems are reduced to a boundary-value problem for elliptic operator differential equations with an operator in boundary conditions in the corresponding spaces and are studied by the method of operator differential equations. |
| first_indexed | 2026-03-24T02:37:29Z |
| format | Article |
| fulltext |
UDK 517.947
Y. V. Alyev (Yn-t matematyky y mexanyky NAN AzerbajdΩana, Baku)
SUMMYRUEMOST| RQDOV PO KORNEVÁM FUNKCYQM
KRAEVÁX ZADAÇ TYPA BYCADZE – SAMARSKOHO
We investigate the summability of a system of eigenfunctions and associated functions of the Bitsadze –
Samarskii-type boundary-value problems for elliptic equations in a rectangular by the Abel method.
These problems can be reduced to the boundary-value problem for elliptic operator differential equations
with an operator under boundary conditions in the corresponding spaces. We investigate these problems
by the method of operator differential equations.
DoslidΩu[t\sq sumovnist\ za metodom Abelq systemy vlasnyx i pry[dnanyx funkcij krajovyx
zadaç typu Bicadze – Samars\koho dlq eliptyçnyx rivnqn\ u prqmokutnyku. Taki zadaçi zvodqt\-
sq do krajovo] zadaçi dlq eliptyçnyx dyferencial\no-operatornyx rivnqn\ z operatorom u
krajovyx umovax u vidpovidnyx prostorax i doslidΩugt\sq metodom dyferencial\no-operator-
nyx rivnqn\.
V rabote [1] dokazano suwestvovanye y edynstvennost\ klassyçeskoho reßenyq
πllyptyçeskoho uravnenyq vtoroho porqdka v prqmouhol\nyke s nelokal\n¥my
kraev¥my uslovyqmy. Polnota kraev¥x funkcyj zadaçy Bycadze – Samarskoho
v n-mernom sluçae yzuçena v rabote [2], a v rabote [3] dokazana neterovost\ πl-
lyptyçeskyx uravnenyj 2m-ho porqdka s kraev¥my uslovyqmy typa Bycadze –
Samarskoho.
Razreßymost\, a takΩe nekotor¥e spektral\n¥e vopros¥ kraevoj zadaçy
dlq dyfferencyal\no-operatorn¥x uravnenyj vtoroho porqdka v sluçae, kohda
koπffycyent¥ kraev¥x uslovyj — lynejn¥e operator¥, yzuçen¥ v rabotax
[4 – 10].
Pust\ H — separabel\noe hyl\bertovo prostranstvo. Çerez L Hp ( , ),0 1( )
oboznaçym banaxovo prostranstvo funkcyj x → u x( ) : ( , )0 1 → H, syl\no yzme-
rym¥x y summyruem¥x v p-j stepeny s normoj
u L H
p
p ( , ),0 1( ) = u x dxH
p( )
0
1
∫ < ∞ .
Pust\ H, H1 — dva hyl\bertov¥x prostranstva. Çerez B H H( , )1 y σ∞ (H ,
H1) oboznaçym sootvetstvenno klass ohranyçenn¥x y vpolne neprer¥vn¥x ope-
ratorov, dejstvugwyx yz H v H1:
B H H( , ) = B H( ), σ∞( , )H H = σ∞( )H .
© Y. V. ALYEV, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4 443
444 Y. V. ALYEV
Pust\ A ∈ ∞σ ( , )H H1 . Tohda A∗
∞∈σ ( , )H H1 . Znaçyt, operator T =
= A A∗( )
1
2 prynadleΩyt σ∞( )H y neotrycatelen. Sobstvenn¥e çysla operato-
ra T naz¥vagtsq s-çyslamy operatora A, t.De. s A H Hj ( , , )1 = λ j T( ), j =
= 1 ÷ ∞ , hde numeracyq v¥polnena po nevozrastanyg.
Oboznaçym çerez σq H H( , )1 , q > 0, mnoΩestvo operatorov A, dejstvugwyx
vpolne neprer¥vno yz H v H1, dlq kotor¥x s A H Hj
q
j
( , , )11=
∞∑ < ∞.
Operator vloΩenyq yz odnoho hyl\bertova prostranstva v druhoe oboznaçym
çerez J.
Opredelenye 1. Lynejn¥j zamknut¥j operator A naz¥vaetsq syl\no po-
zytyvn¥m v H , esly oblast\ opredelenyq D H( ) plotna v H , pry nekoto-
rom δ π∈( , )0 vse toçky yz uhla arg λ ≥ δ prynadleΩat rezol\ventnomu
mnoΩestvu y rezol\venta udovletvorqet ocenke
( – )–A Iλ 1 ≤ c 1 1+( )λ –
.
Otmetym, çto yz syl\noj pozytyvnosty operatora A sleduet syl\naq pozy-
tyvnost\ operatora Aα
, α ∈( , )0 1 .
Pust\ A syl\no pozytyven v H. Tohda H Aa( ) = u u D Aa: ∈ ( ){ , u H Aa( ) =
= A ua
H
, a ≥ }0 — hyl\bertovo prostranstvo.
Opredelenye 2. Pust\ A — proyzvodqwyj operator analytyçeskoj pry
t > 0 poluhrupp¥ e t A–
, ub¥vagwej na beskoneçnosty. Tohda
H H An
p
, ( )
,( )θ = u u H u t A e u dt
H H A
p p n t A
H
p
n: ,
, ( )
( – ) – –∈ = < ∞
( )
∞
∫ 1 1
0
θ
,
hde 0 < θ < 1, 1 < p < ∞, n — natural\noe çyslo.
Çerez
W H A Hp
n n( , ), ( ),0 1( ) = u A u L H u L Hn
p
n
p: ( , ), , ( , ),( )∈ ( ) ∈ ( ){ }0 1 0 1
oboznaçym prostranstvo vektor-funkcyj s normoj
u W H A Hp
n n( , ), ( ),0 1( ) = A un
L Hp ( , ),0 1( ) + un
L Hp ( , ),0 1( ) .
Yzvestno [10, c. 46], çto esly u ∈ Wp
n ( , )0 1( , H An( ), H), to u j( )( )0 ∈ H( ,
H An
n j n p
( ))( )– – / / ,1 2
, j = 0 1÷ ( – )n .
Pust\ Ff = ( ) ( )
– –
–
2
1
2π σe f x dxix
∞
∞
∫ — preobrazovanye Fur\e.
Opredelenye 3. OtobraΩenye σ → T( )σ : R → B H( ) naz¥vaetsq mul\ty-
plykatorom Fur\e typa ( , )p q , esly
F TFf
L R Hq
–
( , )
1 ≤ c f L R Hp ( , ) , f L R Hp∈ ( , ) .
Yzvestno [10, c. 196], çto esly otobraΩenye σ → T( )σ neprer¥vno dyffe-
rencyruemo y
T C( )σ ≤ , ′T ( )σ ≤ C σ –1, σ ∈R , σ ≠ 0 ,
to T( )σ qvlqetsq mul\typlykatorom Fur\e typa ( , )p q (teorema Myxlyna –
Ívarca).
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
SUMMYRUEMOST| RQDOV PO KORNEVÁM FUNKCYQM KRAEVÁX ZADAÇ … 445
Pust\ f H∈ . Rassmotrym rqd
s s sc t e∑ ( ) . Zdes\ c ts( ) zavysyt ot t y opre-
delqetsq sledugwym obrazom. Esly es — sobstvenn¥j vektor operatora T,
sootvetstvugwyj xarakterystyçeskomu çyslu λ p , y operator T ne ymeet pry-
soedynennoho vektora, to
c ts( ) = e cpt
s
−λδ
,
hde λδ = λ δ δ λe i arg
, δ > 0, cs — koπffycyent normal\noho razloΩenyq vek-
tora f po systeme ei{ }, i = 1, 2, … .
Esly Ωe es, … , es j+ obrazugt Ωordanovu cepoçku, to c ts i+ ( ) , i = 0, 1, … , j,
— koπffycyent¥ razloΩenyq v¥çeta pod¥ntehral\noj funkcyy
1
2
1
π
λ λλ
γ
δ
i
e T I T f dt
p
– –( – )∫
po vektoram es, … , es j+ , hde γ p — zamknut¥j kontur, oxvat¥vagwyj tol\ko
λ p .
Opredelenye 4. Rqd, sootvetstvugwyj f, summyruetsq k f po metodu
Abelq porqdka σ, esly dlq πtoho rqda suwestvuet podposledovatel\nost\
çastyçn¥x summ SNν
, sxodqwaqsq v H pry vsex t, takaq, çto
u t( ) = c t es s
s N
N
( )
==
∞
+
+
∑∑
ν
ν
ν 1
1
1
, lim ( )
t
u t f
→ +
=
0
.
V prostranstve L Hp ( , ),0 1( ) rassmotrym sledugwug kraevug zadaçu:
( – )L I uλ = – ( )′′u x + ( – ) ( )A I u xλ = 0, (1)
L u1 = u( )0 + B u1 1( )ξ = f1,
(2)
L u2 = u( )1 + B u2 2( )ξ = f 2,
hde x ∈( , )0 1 , ξ1, ξ2 0 1∈( , ) .
Teorema 1. Pust\ v¥polnen¥ sledugwye uslovyq:
1) A — syl\no pozytyvn¥j operator v H;
2) operator Bi , i = 1, 2, ohranyçenno dejstvuet yz H v H y yz H A( )
v H A( );
3) f H H Ai p
p
p
∈( ), ( ) –
,
2 1
2
, i = 1, 2.
Tohda zadaça (1), (2) pry dostatoçno bol\ßyx λ , udovletvorqgwyx us-
lovyg argλ ≥ δ, ymeet edynstvennoe reßenye
u W H A Hp∈ ( )2 0 1( , ), ( ), .
Dokazatel\stvo. Yzvestno (sm. [11, s. 72] ), çto reßenye uravnenyq (1),
prynadleΩawee Wp
2 0 1( , )( , H A( ), H), ymeet vyd
u x( ) = e gx A– λ
1
2
1 + e gx A– ( – )1
2
1
2
λ . (3)
Podstavlqq (3) v (2), poluçaem sledugwug systemu dlq opredelenyq πlementov
g1 y g2:
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
446 Y. V. ALYEV
g1 D+D e gA– λ
1
2
2 + B e g e gA A
1 1
1
2
1
1
2
1
1
2– –( – )ξ ξλ λ+
= f1,
(4)
e gA– λ
1
2
1 + g2 + B e g e gA A
2 1
1
2
2
1
2
2
1
2– –( – )ξ ξλ λ+
= f2.
Yz (4) naxodym
g1 = I S I S e B eA A+ + +( ) +
1 1 1( ) ( ) – –( – )λ λ λ λξ
1
2
1
1
21 ×
× I S e B e I S fA A+( ) +
+( )
( ) ( )– –λ λλ λξ
1
2
2
1
2
12 1 +
+ I S I S e B e I S fA A+ +( ) +
+( )
( ) – ( ) ( )– –( – )λ λ λλ λξ
1 1
1
2
1
1
21
2 , (5)
g2 = – ( ) ( )– –I S e B e I S fA A+( ) +
+( )λ λλ λξ
1
2
2
1
2
1 12 + I S f+( )( )λ 2, (6)
hde
I S+( )1( )λ = I B e A+
1
–
–
ξ λ1
1
2
1
,
I S+ ( )λ = I B e e B eA A A+ +
2 2
–( – ) – ––1 2
1
2
1
2
2
1
2ξ ξλ λ λ ×
× I S e B eA A+( ) +
1
1
11
2
1
1
2
( ) – –( – )
–
λ λ λξ
1 .
Yzvestno, çto esly A syl\no pozytyven, to
e x A– λ
1
2
≤ ce x–β λ , argλ δ≥ , β > 0, x > 0 . (7)
Poskol\ku pry dostatoçno bol\ßyx λ yz uhla argλ δ≥ v sylu uslovyq 2 y
ocenky (7)
B ei
A–ξ λ
1
2
≤ ce x–β λ → 0, 0 < ξ < 1,
to
S1( )λ = B ei
A
k
k
–ξ λ1
1
2
1
=
∞
∑ ,
(8)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
SUMMYRUEMOST| RQDOV PO KORNEVÁM FUNKCYQM KRAEVÁX ZADAÇ … 447
S( )λ = B e e B eA A A
k
2 2
–( – ) – ––1
1
2
1
2
1
2
2
1
2ξ ξλ λ λ+
=
∞
∑ ×
× I S e B eA A
k
+( ) +
1
1
1
2
1
1
2
( ) – –( – )λ λ λξ
1 .
Rqd¥ v prav¥x çastqx ravenstv (8) sxodqtsq po norme prostranstva B H( ).
DokaΩem, çto πlement¥ g1 y g2, opredelenn¥e formulamy (5), (6), prynad-
leΩat H H A p
p
p, ( ) –
,( )2 1
2
. DokaΩem πto, naprymer, dlq g2. Dlq πtoho dostatoçno
pokazat\, çto operator¥ S( )λ y S1( )λ ohranyçenno dejstvugt yz H( ,
H A p
p
p
( ) –
,
)2 1
2
v H H A p
p
p, ( ) –
,( )2 1
2
.
Operator S1( )λ ohranyçenno dejstvuet yz H v H . S druhoj storon¥, ope-
rator S1( )λ ohranyçenno dejstvuet yz H A( ) v H A( ), çto sleduet yz uslo-
vyqD2 y ocenky (7). Tohda v sylu ynterpolqcyonnoj teorem¥ [12, s. 24] operator
S1( )λ dejstvuet ohranyçenno v prostranstve H H A p
p
p, ( ) –
,( )2 1
2
y ymeet mesto
ocenka
S B H H A p
p
p
1 2 1
2
( ) , ( ) –
,
λ ( ) ≤ C S SB H A B H
p
p
p
p
1 1
2 1
2
2 1
2( ) ( )( ) ( )
––
λ λ( ) ≤ q < 1, (9)
argλ δ≥ , λ → ∞.
Analohyçno moΩno pokazat\, çto operator S( )λ ohranyçenno dejstvuet v pros-
transtve H H A p
p
p, ( ) –
,( )2 1
2
y ocenka (9) ymeet mesto y dlq operatora S( )λ .
Yz (6) v sylu uslovyq 2 y ocenok (7), (9) poluçaem
g H H A p
p
p
2 2 1
2
, ( ) –
,( ) ≤ C f fH H A H H Ap
p
p
p
p
p
1 22 1
2
2 1
2
, ( ) , ( )–
,
–
,( ) ( )+
,
(10)
λ → ∞, argλ δ≥ .
Analohyçno moΩno pokazat\, çto
g H p
p
1 2 1
2
–
≤ C f fH Hp
p
p
p
1 22 1
2
2 1
2
– –
+
, λ → ∞, argλ δ≥ . (11)
TeoremaD1 dokazana.
V prostranstve L Hp ( , ),0 1( ) rassmotrym operator L, opredelenn¥j ravens-
tvamy
D L( ) = W H A H L up
2
1
20 1 0( , ), ( ), , ν ν = =( ) ,
Lu = − ′′u x( ) + Au x( ) ,
hde
L u1 = u( )0 + B u1 1( )ξ , L u2 = u( )1 + B u2 2( )ξ , ξi ∈( , )0 1 , i = 0, 1.
Teorema 2. Pust\ v¥polnen¥ sledugwye uslovyq:
1) A — syl\no pozytyvn¥j operator v H;
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
448 Y. V. ALYEV
2) operator Bi, i = 1, 2, ohranyçenno dejstvuet yz H v H y yz H A( ) v
H A( ).
Tohda pry nekotorom r > 0 vse toçky kompleksnoj ploskosty yz uhla
argλ ≥ δ, po modulg bol\ße r , prynadleΩat rezol\ventnomu mnoΩestvu
operatora L y spravedlyva ocenka
R L( , )λ = ( – )–L Iλ 1 ≤ c λ –1
.
Dokazatel\stvo. Uravnenye ( – )L I uλ = f πkvyvalentno zadaçe
– ( )′′u x + ( – ) ( )A I u xλ = f x( ), (12)
L uν = 0, ν = 1 2, . (13)
Reßenye zadaçy (12), (13) predstavlqetsq v vyde summ¥ u x( ) = u x1( ) + u x2( ) ,
hde u x1( ) — suΩenye na 0 1,[ ] reßenyq uravnenyq
– ˜ ( )′′u x + ( – ) ˜ ( )A I u xλ 1 = f x( ) (14)
na vsej osy,
˜( )f x =
f x x
x
( ), , ,
, , ,
∈[ ]
∈[ ]
0 1
0 0 1
a u x2( ) — reßenye zadaçy
– ( )′′u x2 + ( – ) ( )A I u xλ 2 = 0, (15)
L uν 2 = –L uν 1, ν = 1 2, . (16)
PokaΩem, çto reßenye ˜ ( )u x1 uravnenyq (14) prynadleΩyt W R H A Hp
2 , ( ),( ).
Prymenyv k uravnenyg (14) preobrazovanye Fur\e, poluçym
Fũ1 = A I Ff– –
–
λ σ2 1( )[ ] .
Qsno, çto esly λ naxodytsq v uhle argλ > δ, to λ – σ2
takΩe naxodytsq v
πtom uhle. Poπtomu v uhle argλ ≥ δ ymeet mesto ocenka
A I– –
–
λ σ2 1( )[ ] ≤ C 1 2 1
+ ( )( )λ σ–
–
≤ C 1 2 1
+ +( )λ σ
–
.
Tohda
A u L Hpλ ˜
( , ),1 0 1( ) ≤ A u L R Hpλ ˜
,1 ( ) = F A A I Ff
L R Hp
– –
,
– – ˜1 2 1
λ λ σ( )[ ]
( )
. (17)
Oçevydno, çto T1( )σ = A A Iλ λ σ– –
–2 1( )[ ] — mul\typlykator Fur\e typa
( , )p p . Poπtomu k T1( )σ prymenyma teorema Myxlyna – Ívarca. Tohda yz (17)
ymeem
A u L Hpλ 1 0 1( , ),( ) ≤ C f
L R Hp
˜
,( ) = C f L Hp ( , ),0 1( ). (18)
S druhoj storon¥,
′′ ( )u L Hp1 0 1( , ), ≤ ˜
,
′′
( )u
L R Hp
1 = F A I Ff
L R Hp
– –
,
– –1 2 2 1
σ λ σ( )[ ]
( )
.
MoΩno lehko pokazat\, çto T2( )σ = σ λ σ2 2 1
A –
–( )[ ]I takΩe qvlqetsq mul\ty-
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
SUMMYRUEMOST| RQDOV PO KORNEVÁM FUNKCYQM KRAEVÁX ZADAÇ … 449
plykatorom Fur\e typa ( , )p p , t.De.
′′ ( )u L Hp1 0 1( , ), ≤ C f L Hp ( , ),0 1( ). (19)
Yz (18) y (19) sleduet, çto
u W H A Hp1 0 12 ( , ), ( ),( ) ≤ C f L Hp ( , ),0 1( ). (20)
Ytak, suΩenye ˜ ( )u x1 na 0 1,[ ] prynadleΩyt W H A Hp
2 0 1( , ), ( ),( ) y u x1( ) udov-
letvorqet uravnenyg
– ˜ ( )′′u x1 + ( – ) ( )A J u xλ 1 = f x( ), x ∈[ ]0 1, .
Teper\ ocenym reßenye zadaçy (15), (16). Yz teorem¥ o sledax [12, s. 46]
sleduet, çto L u1 1, L u2 1 ∈ H H A p
p
p
, ( ) –
,
( )2 1
2
. Tohda v sylu teorem¥D1 πta zadaça
pry dostatoçno bol\ßyx λ yz uhla argλ δ≥ ymeet edynstvennoe reßenye
u2 ∈ W H A Hp
2 0 1( , ), ( ),( ) y spravedlyva ocenka
u L Hp2 0 1( , ),( ) ≤ C g gH H
p
p
p
p
λ –
– –
1
1 2
2 1
2
2 1
2
+
. (21)
S druhoj storon¥, v sylu (10), (11)
g H
p
p
1
2 1
2
–
+ g H
p
p
2
2 1
2
–
≤ C L u L u
H H
p
p
p
p
1 1 2 1
2 1
2
2 1
2
– –
+
. (22)
Yspol\zuq ocenku (20), v sylu [12, c. 46] poluçaem
L u
H
p
p
1 1
2 1
2
–
≤ C f L Hp ( , ),0 1( ), (23)
L u
H
p
p
2 1
2 1
2
–
≤ C f L Hp ( , ),0 1( ), (24)
Hθ = H H A p, ( ) ,( )θ .
Uçyt¥vaq ocenky (23), (24), yz (22) naxodym
g H
p
p
1
2 1
2
–
+ g H
p
p
2
2 1
2
–
≤ C f L Hp ( , ),0 1( ). (25)
Tohda yz (21) y (25) ymeem
u L Hp2 0 1( , ),( ) ≤ C f L Hp
λ–
( , ),
1
0 1( ) .
Otsgda s uçetom (18) sleduet, çto
u L Hp ( , ),0 1( ) ≤ C f L Hp
λ–
( , ),
1
0 1( ) .
TeoremaD2 dokazana.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
450 Y. V. ALYEV
Sledstvye. Pry uslovyqx teorem¥D1 zadaça (12), (13) koπrcytyvno razre-
ßyma v prostranstve L Hp ( , ),0 1( ), t .De. pry dostatoçno bol\ßyx λ yz
uhla argλ δ≥ y pry lgbom f ∈ L Hp ( , ),0 1( ), fi ∈ H
p
p
2 1
2
–
, i = 1, 2, zadaça
(12), (13) ymeet edynstvennoe reßenye u ∈ W H A Hp
2 0 1( , ), ( ),( ) y spravedlyva
ocenka
′′ ( )u L Hp ( , ),0 1 + Au L Hp ( , ),0 1( ) + λ u L Hp ( , ),0 1( ) ≤
≤ C f f fL H H Hp p
p
p
p
( , ),
– –
0 1 1 2
2 1
2
2 1
2
( ) + +
.
Teorema 3. Pust\:
1) A — syl\no pozytyvn¥j operator v H y pry nekotorom p 0 > 0 A–1 ∈
∈ σ p H
0
( );
2) operator Bi ohranyçenno dejstvuet yz H v H y yz H A( ) v H A( ).
Tohda R L( , )λ ∈ σq L H2 0 1( , ),( ) pry q >
1 2
2
0+ p
.
Dokazatel\stvo. Oboznaçym çerez S 1 takoj samosoprqΩenn¥j poloΩy-
tel\no opredelenn¥j operator v L2 0 1( , ), dlq kotoroho D Si( ) = W2
2 0 1( , ) . Ta-
koj operator suwestvuet (sm., naprymer, [12, s. 468] ), y dlq sobstvenn¥x çysel
verna ocenka c k1
2 ≤ λk S( )1 ≤ c k2
2
.
Çerez S2 oboznaçym v H takoj samosoprqΩenn¥j poloΩytel\no oprede-
lenn¥j operator, dlq kotoroho D S( )2 = H A( ). Poskol\ku A–1
∈ σ p H
0
( ), to y
S2
1– ∈ σ p H
0
( ). Otsgda sleduet, çto dlq sobstvenn¥x çysel v¥polnqetsq nera-
venstvo
µn S( )2 > cn p
1
0
.
Sobstvenn¥e πlement¥ operatora S1 oboznaçym çerez ϕk x( ), a sobstvenn¥e
πlement¥ operatora S2 — çerez en:
( )( )S xk1 ϕ = λ ϕk k x( ), S en2 = µn ne .
Tohda ϕk nx e( ) obrazuet bazys L H2 0 1(( , ), ). V prostranstve L H2 0 1(( , ), ) opre-
delym operator
∧u = ( ) ( )
,
,1
1
+ +
=
∞
∑ λ µ ϕk n
k n
k n k na x e , u =
k n
k n k na x e
,
, ( )
=
∞
∑
1
ϕ .
Operator ∧ ohranyçenno dejstvuet yz W H A H2
2 0 1( , ), ( ),( ) v L H2 0 1( , ),( ).
Tohda operator F = ∧ R L( , )λ qvlqetsq ohranyçenn¥m operatorom v L2 0 1( , )( ,
H) :
R L( , )λ = ∧–1F .
Poskol\ku ∧–1 ∈ σq L H2 0 1( )( , ),( ), to R L( , )λ ∈ σq L H2 0 1( )( , ),( ) pry q >
>
2 1
2
0p +
(sm. [13, s. 476] ).
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
SUMMYRUEMOST| RQDOV PO KORNEVÁM FUNKCYQM KRAEVÁX ZADAÇ … 451
Teorema dokazana.
V prostranstve L H2 0 1( , ),( ) rassmotrym spektral\nug zadaçu
− ′′u x( ) + Au x( ) = λu x( ), x ∈( , )0 1 , (26)
u( )0 + B u1 1( )ξ = 0, ξ1 0 1∈( , ), (27)
u( )1 + B u2 2( )ξ = 0, ξ2 0 1∈( , ) .
Yz teoremD2, 3 v sylu [14, s. 345] sleduet takaq teorema.
Teorema 4. Pust\:
1) A — syl\no pozytyvn¥j operator v H y pry nekotorom p 0 > 0 A–1
∈
∈ σ p H
0
( ), 0 < δ < π
1 2 0+ p
;
2) operator Bi ohranyçenno dejstvuet yz H v H y yz H A( ) v H A( ).
Tohda systema sobstvenn¥x y prysoedynenn¥x vektor-funkcyj zadaçy (26),
(27) obrazuet bazys dlq metoda summyrovanyq Abelq porqdka γ v L2 0 1( , )( ,
H), hde γ ∈D
2 1
2 2
0p +
, π
δ
.
Rassmotrym v L2 0 1 0, ,[ ] × [ ]( )l spektral\nug zadaçu dlq uravnenyq Laplasa
s kraev¥my uslovyqmy typa Bycadze – Samarskoho:
Lu = ∂
∂
2
2
u
x
+ ∂
∂
2
2
u
y
= λu x y( , ) , (28)
L u1 = u y( , )0 +
b y t u t dt1 1( , ) ( , )ξ
0
l
∫ = 0,
(29)
L u2 = u y( , )1 +
b y t u t dt2 2( , ) ( , )ξ
0
l
∫ = 0,
Q uν = αν
νu xy
k( ) ( , )0 + βν
νu xy
k( ) ( , )l = 0, ν = 1 2, , (30)
hde 0 ≤ k1 ≤ k2 ≤ 1, αν , βν — kompleksn¥e çysla, αν + βν ≠ 0.
Teorema 5. Pust\ v¥polnen¥ sledugwye uslovyq:
1) b y ti( , ),
∂
∂
2
2
b y t
t
i( , )
∈ L2 0 0, ,l l[ ] × [ ][ ], i = 1, 2;
2) θ =
α β
α β
1 1
2 2
1
1
1
2
( )
( )
−
−
k
k
≠ 0.
Tohda system¥ sobstvenn¥x y prysoedynenn¥x funkcyj zadaçy (28) – (30)
poln¥ v L2 0 1 0( , ) ( , )×[ ]l y obrazugt bazys dlq metoda summyrovanyq Abelq
porqdka γ v L2 0 1 0( , ) ( , )×[ ]l , hde γ > 1.
Dokazatel\stvo. V prostranstve L2 0( , )l opredelym operator¥ A , Bi
sledugwym obrazom:
D A( ) = u u W L u: ( , ), , , ,∈ = ={ }2
2 0 0 1 2l ν ν , ( )( )Au y = –
( )d u y
dy
2
2 , (31)
D Bi( ) = L2 0( , )l , ( )( )B u yi =
b y t u t dti i( , ) ( , )ξ
0
l
∫ , (32)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
452 Y. V. ALYEV
i = 1, 2, 0 < ξ1, ξ2 < 1.
Tohda oçevydno, çto dokazatel\stvo teorem¥D5 svodytsq k proverke uslovyj
teorem¥D4. Syl\naq pozytyvnost\ operatora A, opredelennaq formuloj (31),
sleduet yz [11, s. 19]. V sylu [12, s. 317] ymegt mesto kompaktn¥e vloΩenyq
W2
2 0( , )l � L2 0( , )l , a v sylu [12, s. 437]
S J W Ln , ( , ), ( , )2
2
20 0l l( ) ∼ n–2
.
S druhoj storon¥, pry j = 1, 2, …
Sj =
J W Q u L, ( , ), , ( , )2
2
1
2
20 0 0l lν ν=
=
≤ S J W Lj , ( , ), ( , )2
2
20 0l l( ).
Tohda v sylu [12, s. 494] pry lgbom p0 > 1
2
A–1 ∈
σp L
0 2 0( , )l( ) .
Netrudno pokazat\, çto operator Bi, opredelenn¥j ravenstvom (32), udov-
letvorqet vtoromu uslovyg teorem¥D4.
1. Bycadze A. V., Samarskyj A. A. O nekotor¥x reßenyqx prostejßyx obobwenn¥x lynejn¥x
πllyptyçeskyx kraev¥x zadaç // Dokl. AN SSSR. – 1969. – 185, # 4. – S. 739 – 740.
2. Eroßenkov E. P., Kal\menov T. Í. O polnote kornev¥x vektorov πllyptyçeskoj zadaçy
Bycadze – Samarskoho // Tam Ωe. – 1987. – 296, # 3. – S. 528 – 531.
3. Skubaçevskyj A. L. Razreßymost\ πllyptyçeskyx zadaç s kraev¥my uslovyqmy typa Bycad-
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4. Yl\yn V. A., Fylyppov V. S. O xaraktere spektra samosoprqΩennoho rasßyrenyq operatora
Laplasa v ohranyçennoj oblasty // Dokl. AN SSSR. – 1970. – 191, # 2. – S. 167 – 169.
5. Horbaçuk V. Y., Horbaçuk M. L. Nekotor¥e vopros¥ spektral\noj teoryy dyfferencyal\-
n¥x uravnenyj πllyptyçeskoho typa v prostranstve vektor-funkcyj // Ukr. mat. Ωurn. –
1976. – 28, # 3. – S. 313 – 324.
6. Myxajlec V. A. Hranyçn¥e zadaçy dlq operatornoho uravnenyq Íturma – Lyuvyllq s pol-
noj systemoj sobstvenn¥x y prysoedynenn¥x funkcyj // Dyfferenc. uravnenyq. – 1975. –
11, # 9. – S. 1595 – 1600.
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çeskoho dyfferencyal\no-operatornoho uravnenyq vtoroho porqdka // Dokl. AN AzSSR. –
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Poluçeno 14.07.06,
posle dorabotky — 10.04.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
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| id | umjimathkievua-article-3167 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:37:29Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/4a/ed989436fd04c32b8f0f6fdab0fb0d4a.pdf |
| spelling | umjimathkievua-article-31672020-03-18T19:47:27Z Summability of series in the root functions of boundary-value problems of Bitsadze-Samarskii type Суммируемость рядов по корневым функциям краевых задач типа Вицадзе - Самарского Aliev, I. V. Алиев, И. В. Алиев, И. В. We investigate the Abel summability of a system of eigenfunctions and associated functions of Bitsadze-Samarskii-type boundary-value problems for elliptic equations in a rectangle. These problems are reduced to a boundary-value problem for elliptic operator differential equations with an operator in boundary conditions in the corresponding spaces and are studied by the method of operator differential equations. Досліджується сумовність за методом Абеля системи власних i приєднаних функцій крайових задач типу Віцадзе - Самарського для еліптичних рівнянь у прямокутнику. Такі задачі зводяться до крайової задачі для еліптичних диференціально-операторних рівнянь з оператором у крайових умовах у відповідних просторах і досліджуються методом диференціально-операторних рівнянь. Institute of Mathematics, NAS of Ukraine 2008-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3167 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 4 (2008); 443–452 Український математичний журнал; Том 60 № 4 (2008); 443–452 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3167/3081 https://umj.imath.kiev.ua/index.php/umj/article/view/3167/3082 Copyright (c) 2008 Aliev I. V. |
| spellingShingle | Aliev, I. V. Алиев, И. В. Алиев, И. В. Summability of series in the root functions of boundary-value problems of Bitsadze-Samarskii type |
| title | Summability of series in the root functions of boundary-value problems of Bitsadze-Samarskii type |
| title_alt | Суммируемость рядов по корневым функциям краевых задач типа Вицадзе - Самарского |
| title_full | Summability of series in the root functions of boundary-value problems of Bitsadze-Samarskii type |
| title_fullStr | Summability of series in the root functions of boundary-value problems of Bitsadze-Samarskii type |
| title_full_unstemmed | Summability of series in the root functions of boundary-value problems of Bitsadze-Samarskii type |
| title_short | Summability of series in the root functions of boundary-value problems of Bitsadze-Samarskii type |
| title_sort | summability of series in the root functions of boundary-value problems of bitsadze-samarskii type |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3167 |
| work_keys_str_mv | AT alieviv summabilityofseriesintherootfunctionsofboundaryvalueproblemsofbitsadzesamarskiitype AT alieviv summabilityofseriesintherootfunctionsofboundaryvalueproblemsofbitsadzesamarskiitype AT alieviv summabilityofseriesintherootfunctionsofboundaryvalueproblemsofbitsadzesamarskiitype AT alieviv summiruemostʹrâdovpokornevymfunkciâmkraevyhzadačtipavicadzesamarskogo AT alieviv summiruemostʹrâdovpokornevymfunkciâmkraevyhzadačtipavicadzesamarskogo AT alieviv summiruemostʹrâdovpokornevymfunkciâmkraevyhzadačtipavicadzesamarskogo |