Asymptotic behavior of the eigenvalues of a boundary-value problem for a second-order elliptic operator-differential equation
We study the asymptotic behavior of the eigenvalues of a boundary-value problem with spectral parameter in the boundary conditions for a second-order elliptic operator-differential equation. The asymptotic formulas for the eigenvalues are obtained.
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| Datum: | 2006 |
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| Format: | Artikel |
| Sprache: | Russisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2006
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509623457415168 |
|---|---|
| author | Aliev, B. A. Алиев, Б. А. Алиев, Б. А. |
| author_facet | Aliev, B. A. Алиев, Б. А. Алиев, Б. А. |
| author_sort | Aliev, B. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:56:35Z |
| description | We study the asymptotic behavior of the eigenvalues of a boundary-value problem with spectral parameter in the boundary conditions for a second-order elliptic operator-differential equation. The asymptotic formulas for the eigenvalues are obtained. |
| first_indexed | 2026-03-24T02:44:03Z |
| format | Article |
| fulltext |
K�O�R�O�T�K�I���P�O�V�I�D�O�M�L�E�N�N�Q
UDK 517.9
B.�A.�Alyev (Yn-t matematyky y mexanyky NAN AzerbajdΩana, Baku)
ASYMPTOTYÇESKOE POVEDENYE
SOBSTVENNÁX ZNAÇENYJ ODNOJ KRAEVOJ ZADAÇY
DLQ ∏LLYPTYÇESKOHO DYFFERENCYAL|NO-
OPERATORNOHO URAVNENYQ VTOROHO PORQDKA
We study the asymptotic behavior of eigenvalues of a boundary-value problem with spectral parameter
under boundary conditions for an elliptic operator-differential equation of second order. We obtain
asymptotic formulas for eigenvalues.
Vyvça[t\sq asymptotyçna povedinka vlasnyx znaçen\ odni[] krajovo] zadaçi zi spektral\nym
parametrom u hranyçnyx umovax dlq eliptyçnoho dyferencial\no-operatornoho rivnqnnq dru-
hoho porqdku. OderΩano asymptotyçni formuly dlq vlasnyx znaçen\.
Pust\ H — separabel\noe hyl\bertovo prostranstvo. Çerez L2 ( ( 0 , b ) ; H ) , 0 <
< b < + ∞ , oboznaçym mnoΩestvo vsex vektor-funkcyj x → u ( x ) : ( 0 , b ) → H ,
syl\no yzmerym¥x y takyx, çto || ||∫ u x dxH
b
( ) 2
0
< + ∞ . Kak yzvestno, L2 ( ( 0 , b ) ; H )
qvlqetsq hyl\bertov¥m prostranstvom otnosytel\no skalqrnoho proyzvedenyq
( , ) ( )( , );
u
L b H
v
2 0
=
u x x dxH
b
( ), ( )v( )∫
0
.
Pust\ A — samosoprqΩenn¥j poloΩytel\no opredelenn¥j operator v H
( A = A∗ ≥ ω2
I , ω > 0, I — edynyçn¥j operator v H ) s oblast\g opredelenyq
D ( A ) .
Poskol\ku A
–1 ohranyçen v H , H ( A ) = u u D A u AuH A H: ( ); ( )∈ ={ }|| || || || qv-
lqetsq hyl\bertov¥m prostranstvom, norma v kotorom πkvyvalentna norme hra-
fyka operatora A . PoloΩym
W b H A H2
2 0( )( , ); ( ), = { u : A u , u″ ∈ L2 ( ( 0 , b ) ; H ) ,
|| || || || || ||= + ′′ }u Au u
W b H A H L b H L b H2
2
2 20
2
0
2
0
2
( ) ( ) ( )( , ); ( ), ( , ); ( , ); .
MnoΩestvo W b H A H2
2 0( )( , ); ( ), qvlqetsq hyl\bertov¥m prostranstvom [1,
s. 23].
V prostranstve L2 ( ( 0 , b ) ; H ) rassmotrym kraevug zadaçu
– u″ ( x ) + A u ( x ) + q ( x ) u ( x ) = λ u ( x ) , x ∈ ( 0 , b ) , (1)
© B.8A.8ALYEV, 2006
1146 ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8
ASYMPTOTYÇESKOE POVEDENYE SOBSTVENNÁX ZNAÇENYJ … 1147
u′ ( 0 ) + λ u ( 0 ) = 0,
(2)
u′ ( b ) – λ u ( b ) = 0,
hde, kak y ranee, A = A∗ ≥ ω2I v H , operator A–1 vpolne neprer¥ven, q ( x ) —
syl\no neprer¥vnaq operator-funkcyq, znaçenyqmy kotoroj qvlqgtsq samoso-
prqΩenn¥e ohranyçenn¥e operator¥ v H , λ > 0 — spektral\n¥j parametr.
V prostranstve H = L2 ( ( 0 , b ) ; H ) ⊕ H ⊕ H opredelym operator¥ L0 y L
ravenstvamy
D ( L0 ) = D ( L ) = { v = ( u ( x ) , – u ( 0 ) , u ( b ) ), u ∈ W b H A H2
2 0( )( , ); ( ), },
( L0 v ) ( x ) = ( – u″ ( x ) + A u ( x ) , u′ ( 0 ) , u′ ( b ) ),
( L v ) ( x ) = ( – u″ ( x ) + A u ( x ) + q ( x ) u ( x ) , u′ ( 0 ) , u′ ( b ) ).
Netrudno vydet\, çto sobstvenn¥e znaçenyq zadaçy (1), (2) y operatora L sov-
padagt.
Cel\ nastoqwej rabot¥ — yzuçyt\ asymptotyçeskoe povedenye sobstvenn¥x
znaçenyj zadaçy (1), (2), znaq asymptotyçeskoe raspredelenye sobstvenn¥x çy-
sel operatora A .
Otmetym, çto asymptotyka sobstvenn¥x znaçenyj kraev¥x zadaç dlq dyffe-
rencyal\no-operatornoho uravnenyq Íturma – Lyuvyllq na koneçnom otrezke s
odnym y tem Ωe spektral\n¥m parametrom v uravnenyy y v odnom yz hranyçn¥x
uslovyj yzuçena v rabotax [2, 3]. Toçnee, v ukazann¥x rabotax, v çastnosty, yzu-
çeno asymptotyçeskoe raspredelenye sobstvenn¥x znaçenyj zadaçy (1) s krae-
v¥my uslovyqmy
u′ ( 0 ) + λ u ( 0 ) = 0,
(3)
u ( b ) = 0
v L2 ( ( 0 , b ) ; H ) ⊕ H . Dokazano, çto esly spektr operatora A dyskreten, to
spektr operatora, poroΩdennoho kraevoj zadaçej (1), (3), takΩe dyskretn¥j.
Sobstvenn¥e znaçenyq zadaçy (1), (3) ( pry q ( x ) = 0 ) obrazugt beskoneçn¥e po-
sledovatel\nosty λk � µk y λn, k � µk + ( π2
/ b2
) n2, n , k ∈ N , hde µ k = µk ( A )
— sobstvenn¥e znaçenyq operatora A .
Asymptotyka sobstvenn¥x znaçenyj samosoprqΩenn¥x hranyçn¥x zadaç dlq
uravnenyq (1) v sluçae, kohda hranyçn¥e uslovyq soderΩat ohranyçenn¥e samo-
soprqΩenn¥e operator¥, yzuçalas\ ranee (sm., naprymer, [4, 5]).
V dannoj rabote s yspol\zovanyem ydej y texnyky rabot [2, 3] dokaz¥vaetsq,
çto zadaça (1), (2) takΩe ymeet dve seryy sobstvenn¥x znaçenyj λk � µk y
λn, k � µk + ( π2
/ b2
) n2.
Snaçala yssleduem zadaçu (1), (2) pry q ( x ) ≡ 0. Uslovye A = A∗ ≥ ω2I v H
vleçet symmetryçnost\ y poloΩytel\nug opredelennost\ operatora L0 v H .
Dejstvytel\no, esly v1 = ( u1 ( x ) , – u1 ( 0 ) , u1 ( b ) ), v2 = ( u2 ( x ) , – u2 ( 0 ) , u2 ( b ) )
— πlement¥ yz D ( L0 ) , to
( L0 v1 , v2 )H =
− ′′ +( ) − ′( ) + ′( )∫ u x Au x u x dx u u u b u b
b
H H1 1 2
0
1 2 1 20 0( ) ( ), ( ) ( ), ( ) ( ), ( )H =
= ′ ′( ) − ′( ) + ′( ) + ( )∫ ∫u x u x dx u b u b u u Au x u x dxH
b
H H H
b
1 2
0
1 2 1 2 1 2
0
0 0( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) –
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8
1148 B.8A.8ALYEV
– ′( ) + ′( ) = ′( ) − ′′( )∫u u u b u b u x u x u x u x dxH H H
b
H
b
1 2 1 2 1 2 0 1 2
0
0 0( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) +
+ u x Au x dx u x u x Au x dxH
b
H
b
1 2
0
1 2 2
0
( ), ( ) ( ), ( ) ( )( ) = − ′′ +( )∫ ∫ +
+ u b u b u uH H1 2 1 20 0( ), ( ) ( ), ( )′( ) − ′( ) = ( v1 , L0 v2 )H ,
t.8e. L0 — symmetryçeskyj operator.
S druhoj storon¥, dlq lgboho v = ( u ( x ) , – u ( 0 ) , u ( b ) ) ∈ D ( L0 ) ymeem
( L0 v , v )H = − ′′ +( ) − ′( ) + ′( )∫ u x Au x u x dx u u u b u bH
b
H H( ) ( ), ( ) ( ), ( ) ( ), ( )
0
0 0 =
= ′ + ( ) ≥ ′ +∫ ∫ ∫ ∫u x dx Au x u x dx u x dx u x dxH
b
H
b
H
b
H
b
( ) ( ), ( ) ( ) ( )2
0 0
2
0
2 2
0
ω .
Poskol\ku vloΩenye W b H2
1 0( )( , ); ⊂ C ( [ 0 , b ] ; H ) neprer¥vno, to (sm. [6], teo-
rema 1.7.7, [1, s. 48])
u c u xH W b H( ) ( ) ( )( , );0 1 02
1≤ ,
u b c u xH W b H( ) ( ) ( )( , );≤ 2 02
1 ,
hde c1 , c2 > 0 — nekotor¥e konstant¥. Sledovatel\no,
( L0 v , v )H ≥ c u x dx u u b cH
b
H H( ) ( ) ( )2
0
2 2 20∫ + +
= v H ,
t.8e. operator L0 poloΩytel\no opredelen.
MoΩno takΩe pokazat\, çto esly A–1 vpolne neprer¥ven v H , to operator
L0
1− vpolne neprer¥ven v H .
Teorema 1. Pust\ A = A∗ ≥ ω2I v H y A–1 vpolne neprer¥ven. Tohda dlq
sobstvenn¥x znaçenyj zadaçy (1), (2) pry q ( x ) = 0 ( operatora L 0 ) spraved-
lyv¥ sledugwye asymptotyçeskye formul¥: λk � µk ; λn, k � µk + ( π2
/ b2
) n2,
n , k = 1, 2, … , hde µk = µk ( A ) — sobstvenn¥e znaçenyq operatora A .
Dokazatel\stvo. Sobstvenn¥e vektor¥ operatora A , sootvetstvugwye sob-
stvenn¥m znaçenyqm µk ( A ) , oboznaçym çerez ϕk . Yzvestno, çto { ϕk } obrazuet
ortonormyrovann¥j bazys v H . Tohda, uçyt¥vaq spektral\noe razloΩenye, dlq
koπffycyentov uk = ( u , ϕk ) poluçaem sledugwug zadaçu:
– ′′u xk ( ) + ( µk – λ ) uk ( x ) = 0, x ∈ ( 0 , b ) , (4)
′uk ( )0 + λ uk ( 0 ) = 0,
(5)
′u bk ( ) – λ uk ( b ) = 0.
Obwee reßenye ob¥knovennoho dyfferencyal\noho uravnenyq (4) ymeet vyd
uk ( x ) = c e c ex b xk k
1 2
− − − − −+µ λ µ λ( ) , (6)
hde ci , i = 1, 2, — proyzvol\n¥e postoqnn¥e. Podstavyv (6) v (5), poluçym sys-
temu otnosytel\no ci , i = 1, 2, opredelytel\ kotoroj ymeet vyd
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8
ASYMPTOTYÇESKOE POVEDENYE SOBSTVENNÁX ZNAÇENYJ … 1149
K ( λ ) =
λ µ λ λ µ λ
λ µ λ λ µ λ
µ λ
µ λ
− − + −( )
− + −( ) − − −( )
− −
− −
k k
b
k
b
k
e
e
k
k
=
= − − −( ) + + −( ) − −λ µ λ λ µ λ µ λ
k k
be k2 2 2 .
Sledovatel\no, sobstvenn¥e znaçenyq operatora L0 — πto nuly uravnenyq
e b
k k
k2 2 2µ λ λ µ λ λ µ λ− − −( ) − + −( ) = 0, (7)
a znaçyt, nuly uravnenyj
eb
k k
kµ λ λ µ λ λ µ λ− − −( ) − + −( ) = 0 (8)
y
eb
k k
kµ λ λ µ λ λ µ λ− − −( ) + + −( ) = 0. (9)
Takym obrazom, spektr operatora L0 sostoyt yz tex vewestvenn¥x λ ≠ µk ,
kotor¥e xotq b¥ pry odnom k udovletvorqgt, po krajnej mere, odnomu yz urav-
nenyj (8) yly (9).
Najdem sobstvenn¥e znaçenyq operatora L 0 , men\ßye µk . PoloΩym
µ λk − = y. Uravnenyq (8) y (9) v πtom sluçae πkvyvalentn¥ sootvetstvenno
uravnenyqm
y
by
cth
2
+ y2 – µk = 0, 0 < y < µk , (10)
y
y
by
th
2
+ y2 – µk = 0, 0 < y < µk . (11)
Uravnenye (10) yssledovano v rabotax [2, 3], hde pokazano, çto v promeΩutke
( 0, µk ) ono, naçynaq s nekotoroho k , ymeet toçno odyn koren\ yk , kotor¥j
asymptotyçesky vedet sebq kak µk – 1 / 2 . Otsgda dlq sobstvenn¥x znaçenyj
poluçaem asymptotyçeskug formulu λk � µk pry k → ∞ .
Analohyçn¥m obrazom yssleduetsq uravnenye (11). Oboznaçym fk ( y ) =
= y
by
th
2
+ y2 – µk , y ∈ ( 0, µk ) . Proyzvodnaq ′f yk ( ) =
sh
ch
by by
by
+
2 22 ( / )
+ 2y > 0 pry
y ∈ (0, µk ) , t.8e. funkcyq fk ( y ) monotonno vozrastaet na (0, µk ) . Poskol\-
ku fk ( 0 ) = – µk < 0 y fk ( µk ) = µ µ
k
kb
th
2
> 0, oçevydno, çto v promeΩutke
( 0, µk ) uravnenye (11), naçynaq s nekotoroho k , ymeet toçno odyn koren\ yk .
PokaΩem, çto πtot koren\ yk takΩe vedet sebq kak µk – 1 / 2 . Dejstvytel\no,
pry ε > 0
f bk k
k
k
( / )
/
µ ε
µ ε
µ ε− −
− −
= − −
1 2
1 2 2
1
2
th +
+
− + + +
− −
2 1 2 1 2
1 2
2µ ε ε
µ ε
k
k
( / ) ( / )
/
→ – 2ε pry k → + ∞ .
Sledovatel\no, fk k( / )µ ε− −1 2 < 0. Analohyçno
fk k
k
( / )
/
µ ε
µ ε
− +
− +
1 2
1 2
→ 2ε
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8
1150 B.8A.8ALYEV
pry k → + ∞ , t.8e. fk k( / )µ ε− −1 2 > 0. Takym obrazom, yk leΩyt meΩdu µk –
– 1 / 2 – ε y µk – 1 / 2 + ε , y v sylu proyzvol\nosty ε yk – ( µk – 1 / 2 ) → 0
pry k → ∞ , otkuda sleduet, çto λk � µk pry k → ∞ , hde λk = µk – yk
2 .
Dlq sobstvenn¥x znaçenyj operatora L0 , bol\ßyx µk , uravnenyq (8) y (9)
prynymagt vyd
ctg bz z
z
k
2
2
= + µ
, (12)
tg bz z
z
k
2
2
= − + µ
, z ∈ ( 0 , + ∞ ) , (13)
sootvetstvenno, hde z = λ µ− k .
Uravnenye (12) yssledovano v [2, 3], hde pokazano, çto v kaΩdom promeΩutke
2 2 1n
b
n
b
π π
,
( )+
πto uravnenye ymeet tol\ko odyn koren\ zn, k :
2n
b
π < zn, k <
2 1( )n
b
+ π
.
Otsgda dlq sobstvenn¥x znaçenyj poluçaem sledugwug asymptotyçeskug
formulu:
λn k,
1
� µk + π
2
2
22
b
n( ) . (14)
Analohyçno yssleduetsq uravnenye (13). A ymenno, rassmotrym funkcyg
uk ( z ) =
z bz z
z
ktg( / )2 2+ + µ
=
ϕk z
z
( )
, hde ϕk ( z ) = z bztg
2
+ z2 + µk . Nuly funk-
cyj ϕk ( z ) y uk ( z ) sovpadagt. Funkcyq ϕk ( z ) opredelena na ( 0 , + ∞ ) vsgdu,
za ysklgçenyem toçek zn = ( π / b ) ( 2n + 1) , n = 0, 1, 2, … . Poskol\ku v kaΩdom
promeΩutke ((π / b ) (2n + 1) , (π / b ) (2n + 3)) ϕk ( z ) yzmenqetsq ot – ∞ do + ∞ , a
′ϕk z( ) =
sin
cos ( / )
bz bz
bz
+
2 22 + 2z > 0,
v nem pry kaΩdom k funkcyq ϕk ( z ) ymeet tol\ko odyn nul\ zn, k :
π
b
n( )2 1+ < zn, k < π
b
n( )2 3+ .
Otsgda dlq sobstvenn¥x znaçenyj poluçaem asymptotyçeskug formulu
λn k,
2
� µk + π
2
2
22 1
b
n( )+ . (15)
Yz (14) y (15) v¥tekagt asymptotyçeskye formul¥ dlq sobstvenn¥x znaçenyj
λ > µk operatora L0 :
λn, k � µk + π
2
2
2
b
n .
Teorema81 dokazana.
Sledstvye 1. Pust\ v¥polnen¥ uslovyq teorem¥81. PredpoloΩym takΩe,
çto sobstvenn¥e znaçenyq operatora A , raspoloΩenn¥e v porqdke vozrasta-
nyq, udovletvorqgt uslovyg µk ( A ) � a kα ( lim ( )k k A→∞ µ ⋅ k–
α = a, 0 < a, α =
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8
ASYMPTOTYÇESKOE POVEDENYE SOBSTVENNÁX ZNAÇENYJ … 1151
= const ) . Tohda sobstvenn¥e znaçenyq operatora L0 ymegt asymptotyku
λm ( L0 ) � dmδ,
hde
δ =
2
2
2
2
2
1 2
α α
α α
α
a +
>
<
=
pry
pry
pry
,
,
,
d > 0 — nekotoraq konstanta.
Dokazatel\stvo sledstvyq81 sleduet yz [3] (sm. takΩe [4]).
Pust\ teper\ q ( x ) ≠ 0. V8prostranstve H opredelym operator Q sledug-
wym obrazom:
D ( Q ) = H , Q v = ( q ( x ) u ( x ) , 0 , 0 ) .
Oçevydno, çto operator Q ohranyçenn¥j y samosoprqΩenn¥j. Tohda operator
L moΩno predstavyt\ kak L = L0 + Q .
Sledstvye 2. Pust\ v¥polnen¥ uslovyq teorem¥ 1. PredpoloΩym takΩe,
çto pry kaΩdom x ∈ ( 0 , b ) q ( x ) — samosoprqΩenn¥j ohranyçenn¥j operator
v H . Tohda sobstvenn¥e znaçenyq kraevoj zadaçy (1), (2) ( operatora L ) yme-
gt asymptotyku
λm ( L ) � λm ( L0 ) .
Dokazatel\stvo sledstvyq 2 provodytsq po toj Ωe sxeme, çto y v [3, 4].
Prymer. Rassmotrym v prqmouhol\nyke Ω = [ 0 , b ] × [ 0 , 1 ] zadaçu na sob-
stvenn¥e znaçenyq
− −∂ ϑ
∂
∂ ϑ
∂
2
2
2
2
( , ) ( , )x y
x
x y
y
+ q ( x , y ) ϑ ( x , y ) = λ ϑ ( x , y ) , (16)
∂ϑ
∂
( , )0 y
x
+ λ ϑ ( 0 , y ) = 0,
(17)
∂ϑ
∂
( , )b y
x
– λ ϑ ( b , y ) = 0,
ϑ ( x , 0 ) = ϑ ( x , 1 ) ,
∂ϑ
∂
∂ϑ
∂
( , ) ( , )x
y
x
y
0 1= , (18)
hde q ( x , y ) — neprer¥vnaq funkcyq na Ω .
Zapyßem zadaçu (16) – (18) v operatornoj forme
– u″ ( x ) + A u ( x ) + q ( x ) u ( x ) = λ u ( x ) , x ∈ ( 0 , b ) ,
u′ ( 0 ) + λ u ( 0 ) = 0,
u′ ( b ) – λ u ( b ) = 0,
hde u ( x ) = ϑ ( x , ⋅ ) — vektor-funkcyq so znaçenyqmy v hyl\bertovom prostran-
stve H = L2 ( 0 , 1 ) , a operator¥ A y q ( x ) opredelen¥ sledugwym obrazom:
D ( A ) = { u ∈ W2
2 0 1( , ) | u ( 0 ) = u ( 1 ) , u′ ( 0 ) = u′ ( 1 )}, A u = –
d u
dy
2
2 + ω u (19)
( ω > 0 — nekotoroe çyslo ) ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8
1152 B.8A.8ALYEV
D ( q ( x ) ) = L2 ( 0 , 1 ) , q ( x ) u = q ( x , y ) u – ω u . (20)
Oçevydno, çto operator A , opredelenn¥j ravenstvom (19), samosoprqΩen-
n¥j y pry dostatoçno bol\ßyx ω > 0 poloΩytel\no opredelenn¥j, a A–1
vpolne neprer¥ven v L2 ( 0 , 1 ) . Prost¥e v¥çyslenyq pokaz¥vagt, çto sobstven-
n¥e znaçenyq operatora A ymegt vyd
µk ( A ) = ω + 4π2
k2, k = 0, 1, 2, … .
Poskol\ku operator q ( x ) , opredelenn¥j v (20), pry kaΩdom x ∈ ( 0, b ) ohra-
nyçen y samosoprqΩen v L2 ( 0, 1) , to na osnovanyy sledstvyq 2 sobstvenn¥e zna-
çenyq zadaçy (16) – (18) vedut sebq kak λm � const ⋅ m .
Zametym, çto v rabotax [7, 8] pokazano, çto dlq uravnenyq Laplasa v kvadra-
te suwestvugt kraev¥e zadaçy s operatorom v kraev¥x uslovyqx, spektr koto-
r¥x ne8qvlqetsq dyskretn¥m. Asymptotyçeskoe povedenye sobstvenn¥x znaçe-
nyj odnoj kraevoj zadaçy dlq dyfferencyal\no-operatorn¥x uravnenyj vto-
roho porqdka s kusoçno-postoqnn¥m koπffycyentom pry vtoroj proyzvodnoj y
s uslovyqmy soprqΩenyq yzuçeno v [9].
Avtor blahodaren professoru S.8Q.8Qkubovu za obsuΩdenye poluçenn¥x re-
zul\tatov.
1. Lyons+Û.-L., MadΩenes+∏. Neodnorodn¥e hranyçn¥e zadaçy y yx pryloΩenyq. – M.: Myr,
1971. – 3718s.
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obratn¥e zadaçy teoryy rasseqnyq. – Kyev, 1981. – S.83 – 13.
3 R¥bak+M.+A. Ob asymptotyçeskom raspredelenyy sobstvenn¥x znaçenyj nekotor¥x hranyç-
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5. Horbaçuk+V.+Y., Horbaçuk+M.+L. O samosoprqΩenn¥x hranyçn¥x zadaçax s dyskretn¥m
spektrom, poroΩdenn¥x uravnenyem Íturma – Lyuvyllq s neohranyçenn¥m operatorn¥m
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Poluçeno 21.01.2005,
posle dorabotky — 28.11.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8
|
| id | umjimathkievua-article-3519 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:44:03Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/7d/cb05cd25fc9633435b4ec10384151e7d.pdf |
| spelling | umjimathkievua-article-35192020-03-18T19:56:35Z Asymptotic behavior of the eigenvalues of a boundary-value problem for a second-order elliptic operator-differential equation Асимптотическое поведение собственных значений одной краевой задачи для эллиптического дифференциально-операторного уравнения второго порядка Aliev, B. A. Алиев, Б. А. Алиев, Б. А. We study the asymptotic behavior of the eigenvalues of a boundary-value problem with spectral parameter in the boundary conditions for a second-order elliptic operator-differential equation. The asymptotic formulas for the eigenvalues are obtained. Вивчається асимптотична поведінка власних значень однієї крайової задачі зі спектральним параметром у граничних умовах для еліптичного диференціально-операторного рівняння другого порядку. Одержано асимптотичні формули для власних значень. Institute of Mathematics, NAS of Ukraine 2006-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3519 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 8 (2006); 1146–1152 Український математичний журнал; Том 58 № 8 (2006); 1146–1152 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3519/3775 https://umj.imath.kiev.ua/index.php/umj/article/view/3519/3776 Copyright (c) 2006 Aliev B. A. |
| spellingShingle | Aliev, B. A. Алиев, Б. А. Алиев, Б. А. Asymptotic behavior of the eigenvalues of a boundary-value problem for a second-order elliptic operator-differential equation |
| title | Asymptotic behavior of the eigenvalues of a boundary-value problem for a second-order elliptic operator-differential equation |
| title_alt | Асимптотическое поведение собственных значений одной краевой задачи для эллиптического дифференциально-операторного уравнения второго порядка |
| title_full | Asymptotic behavior of the eigenvalues of a boundary-value problem for a second-order elliptic operator-differential equation |
| title_fullStr | Asymptotic behavior of the eigenvalues of a boundary-value problem for a second-order elliptic operator-differential equation |
| title_full_unstemmed | Asymptotic behavior of the eigenvalues of a boundary-value problem for a second-order elliptic operator-differential equation |
| title_short | Asymptotic behavior of the eigenvalues of a boundary-value problem for a second-order elliptic operator-differential equation |
| title_sort | asymptotic behavior of the eigenvalues of a boundary-value problem for a second-order elliptic operator-differential equation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3519 |
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