Conditions of nontrivial solvability of the homogeneous Dirichlet problem for equations of any even order in the case of multiple characteristics without slope angles
We consider the homogeneous Dirichlet problem in the unit disk $K ⊂ R^2$ for a general typeless differential equation of any even order $2m,\; m ≥ 2$, with constant complex coefficients whose characteristic equation has multiple roots $± i$. For each value of multiplicity of the roots $i$ and $–i$,...
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| Date: | 2010 |
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| Language: | Russian English |
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Institute of Mathematics, NAS of Ukraine
2010
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508882568216576 |
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| author | Buryachenko, E. A. Буряченко, Е. А. Буряченко, Е. А. |
| author_facet | Buryachenko, E. A. Буряченко, Е. А. Буряченко, Е. А. |
| author_sort | Buryachenko, E. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2020-03-18T19:39:51Z |
| description | We consider the homogeneous Dirichlet problem in the unit disk $K ⊂ R^2$ for a general typeless differential equation of any even order $2m,\; m ≥ 2$, with constant complex coefficients whose characteristic equation has multiple roots $± i$. For each value of multiplicity of the roots $i$ and $–i$, we either formulate criteria of the nontrivial solvability of the problem or prove that the analyzed problem possesses solely the trivial solution. A similar result generalizes the well-known Bitsadze examples to the case of typeless equations of any even order. |
| first_indexed | 2026-03-24T02:32:16Z |
| format | Article |
| fulltext |
UDK 517.946
E. A. Burqçenko (Donec. nac. un-t)
USLOVYQ NETRYVYAL|NOJ RAZREÍYMOSTY
ODNORODNOJ ZADAÇY DYRYXLE DLQ URAVNENYJ
PROYZVOL|NOHO ÇETNOHO PORQDKA
V SLUÇAE KRATNÁX XARAKTERYSTYK,
NE YMEGWYX UHLOV NAKLONA
We consider the homogeneous Dirichlet problem in a unit disk K R⊂ 2 for a general typeless
differential equation of arbitrary even order 2m, m ≥ 2, with constant complex coefficients, whose
characteristic equation has multiple roots ± i. For every value of multiplicities of the roots i and – i we
obtain criteria of nontrivial solvability of the problem or prove that the problem has only the trivial
solution. A similar result generalizes the well-known A. V. Bitsadze examples to the case of typeless
arbitrary-even-order equations.
Rozhlqnuto odnoridnu zadaçu Dirixle v odynyçnomu kruzi K R⊂ 2
dlq zahal\noho beztypnoho
dyferencial\noho rivnqnnq dovil\noho parnoho porqdku 2m , m ≥ 2, zi stalymy kompleksnymy
koefici[ntamy, xarakterystyçne rivnqnnq qkoho ma[ kratni koreni ± i. Dlq koΩnoho znaçennq
kratnostej koreniv i ta – i otrymano kryteri] netryvial\no] rozv'qznosti zadaçi abo dovedeno,
wo zadaça ma[ lyße tryvial\nyj rozv’qzok. Podibnyj rezul\tat uzahal\ng[ vidomi pryklady
A.5V.5Bicadze na vypadok beztypnyx rivnqn\ dovil\noho parnoho porqdku.
Vvedenye. V rabote rassmatryvaetsq odnorodnaq zadaça Dyryxle v edynyçnom
kruhe K R⊂ 2
dlq obweho bestypnoho dyfferencyal\noho uravnenyq proyz-
vol\noho çetnoho porqdka 2m, m ≥ 2, s postoqnn¥my kompleksn¥my koπffy-
cyentamy y odnorodn¥m po porqdku dyfferencyrovanyq v¥roΩdenn¥m symvo-
lom. V¥roΩdennost\ symvola oznaçaet, çto korny λ1 , λ2 , … , λ2m xarakte-
rystyçeskoho uravnenyq mohut b¥t\ kratn¥my, a takΩe prynymat\ znaçenyq ± i .
Vopros¥ tryvyal\nosty qdra zadaçy Dyryxle dlq obwyx uravnenyj vtoroho
porqdka s v¥roΩdenn¥m symvolom yzuçalys\ A.5V.5Bycadze [1], V.5P.5Burskym
[2], a dlq uravnenyj çetvertoho porqdka — E.5A.5Burqçenko [3], N.5∏.5Tovmasq-
nom y A.5O.5Babaqnom [4]. Uravnenyq, pryvedenn¥e v rabote [1], ∂ ∂2 2u z/ = 0,
∂ ∂2 2u z/ = 0, ne qvlqgtsq pravyl\no πllyptyçeskymy (v pervom sluçae λ1 =
= λ2 = – i, a vo vtorom λ1 = λ2 = i ), poπtomu dlq nyx vozmoΩna beskoneçno-
mernaq needynstvennost\ reßenyq zadaçy Dyryxle. V to Ωe vremq symvol πtyx
uravnenyj v¥roΩdenn¥j, tak kak suwestvugt tol\ko kratn¥e korny xarakte-
rystyçeskoho uravnenyq. Napomnym, çto v sylu rezul\tatov rabot¥ Q.5B.5Lopa-
tynskoho [5], dlq pravyl\no πllyptyçeskyx uravnenyj çetnoho porqdka s pos-
toqnn¥my kompleksn¥my koπffycyentamy odnorodnaq zadaça Dyryxle ymeet
ne bolee çem koneçnoe çyslo lynejno nezavysym¥x reßenyj. V rabote [3] pol-
nost\g reßen vopros o edynstvennosty reßenyq zadaçy Dyryxle v kruhe dlq
obwyx uravnenyj çetvertoho porqdka. V nej poluçen¥ kryteryy edynstvennos-
ty reßenyq zadaçy v rqde sluçaev: kak v obwem (vse korny xarakterystyçeskoho
uravnenyq prost¥e y ne ravn¥ ± i ), tak y v sluçae kratn¥x kornej, a takΩe kor-
nej, prynymagwyx znaçenyq ± i ; ustanovlena zavysymost\ meΩdu znaçenyem
kratnosty kornej y suwestvovanyem netryvyal\noho reßenyq sootvetstvugwej
odnorodnoj zadaçy Dyryxle.
Vopros suwestvovanyq netryvyal\noho reßenyq zadaçy Dyryxle dlq obwyx
uravnenyj hlavnoho typa, ne ymegwyx kratn¥x xarakterystyk, reßen v stat\e
[6], hde y b¥l dokazan kryteryj suwestvovanyq takoho reßenyq.
© E. A. BURQÇENKO, 2010
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5 591
592 E. A. BURQÇENKO
V rabotax [7, 8] b¥ly obobwen¥ rezul\tat¥ statej [3, 6] na sluçaj obwyx
uravnenyj proyzvol\noho çetnoho porqdka 2m, m ≥ 2, ymegwyx kratn¥e xarak-
terystyky, a takΩe prost¥e korny ± i xarakterystyçeskoho uravnenyq; usta-
novlena zavysymost\ meΩdu znaçenyem kratnosty kornej, ne ravn¥x ± i, y su-
westvovanyem netryvyal\noho reßenyq sootvetstvugwej zadaçy Dyryxle v
kruhe.
Vopros netryvyal\noj razreßymosty zadaçy Dyryxle v sluçae suwestvova-
nyq kratn¥x kornej ± i ostavalsq otkr¥t¥m. ∏tomu voprosu posvqwena na-
stoqwaq rabota.
Otmetym takΩe, çto odnoj yz osobennostej sluçaq suwestvovanyq kornej
± i xarakterystyçeskoho uravnenyq (ne hovorq uΩe ob yx kratnosty) qvlqetsq
tot fakt, çto dlq πtyx kornej ne suwestvuet uhlov ϕ naklona sootvetstvug-
wyx xarakterystyk (reßenyj uravnenyq −tg ϕ = ± i ), poskol\ku ranee prakty-
çesky vo vsex rabotax (sm., naprymer, [2 – 4, 6]) podobn¥e kryteryy edynstven-
nosty reßenyq hranyçn¥x zadaç b¥ly zapysan¥ ymenno v termynax ϕ j uhlov
naklona xarakterystyk.
1. Postanovka zadaçy. Rassmotrym odnorodnug zadaçu Dyryxle v edynyç-
nom kruhe K R⊂ 2
dlq obweho bestypnoho dyfferencyal\noho uravnenyq
proyzvol\noho çetnoho porqdka 2m, m ≥ 2, s postoqnn¥my kompleksn¥my
koπffycyentamy y odnorodn¥m po porqdku dyfferencyrovanyq v¥roΩden-
n¥m symvolom:
L ux( )∂ = a
u
x
m
m0
2
1
2
∂
∂
+ a
u
x x
m
m1
2
1
2 1
2
∂
∂ ∂− + …
… + a
u
x x
m
m
m2 1
2
1 2
2 1− −
∂
∂ ∂
+5 a
u
x
m
m
m2
2
2
2
∂
∂
= 0, (1)
u K∂ = 0 , ′ =∂u Kν 0 , … , u m
K
ν
( )−
∂
=1 0 , (2)
hde
�
ν — edynyçn¥j vektor vneßnej normaly, ∂x =
∂
∂
∂
∂
x x1 2
, , a Ci ∈ , i =
= 0, 1, … , 2m.
V¥roΩdennost\ symvola oznaçaet, çto korny λ1 , λ2 , … , λ2m xarakterys-
tyçeskoho uravnenyq L( , )1 λ = 0 mohut b¥t\ kratn¥my, a takΩe prynymat\
znaçenyq ± i .
Dlq kaΩdoho znaçenyq kratnostej p = 2, … , 2m kornq i y q = 2, … , 2m
kornq – i xarakterystyçeskoho uravnenyq poluçen¥ kryteryy netryvyal\noj
razreßymosty zadaçy (1), (2) yly dokazano, çto zadaça ymeet tol\ko tryvyal\-
noe reßenye. Podobn¥e rezul\tat¥ obobwagt yzvestn¥e prymer¥ A.5V.5Bycad-
ze (sm. v¥ße) na sluçaj bestypn¥x uravnenyj proyzvol\noho çetnoho porqdka
2m, m ≥ 2.
Napomnym (sm., naprymer, [2]), çto uhlom naklona xarakterystyky, soot-
vetstvugwej nekotoromu korng λ j i≠ ± xarakterystyçeskoho uravnenyq,
budem naz¥vat\ kakoe-nybud\ reßenye uravnenyq − =tg ϕ λj j , j = 1, 2, … , 2m.
Ohranyçenye λ j i≠ ± , j = 1, 2, … , 2m, kak raz svqzano s tem, çto uravnenye
tg ( )x i= ± ne ymeet reßenyj.
2. Sluçaj kratnoho kornq i (yly – i ) xarakterystyçeskoho uravne-
nyq. PredpoloΩym vnaçale, çto sredy kornej xarakterystyçeskoho uravnenyq
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
USLOVYQ NETRYVYAL|NOJ RAZREÍYMOSTY ODNORODNOJ ZADAÇY DYRYXLE …593
L( , )1 λ = 0 ymeetsq koren\ i (yly – i ) kratnosty k > 1. Ostal\n¥e korny
prost¥e y ne ravn¥ ± i . DokaΩem kryteryj netryvyal\noj razreßymosty zada-
çy (1), (2) v zavysymosty ot znaçenyq kratnosty k kornq i . Sluçaj – i rassmat-
ryvaetsq analohyçno.
Osnovn¥m rezul\tatom πtoho punkta qvlqetsq sledugwaq teorema.
Teorema 1. 1. PredpoloΩym, çto çyslo i qvlqetsq kornem xarakterys-
tyçeskoho uravnenyq y ymeet kratnost\ k < m. Tohda dlq netryvyal\noj raz-
reßymosty zadaçy Dyryxle (1), (2) v prostranstve C Km2 ( ) neobxodymo y
dostatoçno v¥polnenyq sledugweho uslovyq dlq nekotoroho n ∈N , n > 2m –
– 1:
∆1 = det A = det ( , , , , )�A A A Ak k m+ + …1 2 = 0. (3)
Zdes\ bloky
�A , Aj , j = k + 1, k + 2, … , m, matryc¥ A ymegt vyd
�
�
�A
e e
e e
in i n k
in i n k
k k
k
=
+ +
+
− −( )
− −
ϕ ϕ
ϕ
1 1
2
2 1
2
( )
( 11
2 1
2
2 2
)
( )
( )
− −( )
+
ϕ
ϕ ϕ
k
m me ein i n k
� � �
�
,
A
n j n j
n
j
k k
=
− −( ) − −( )
−
+ +cos ( ) sin ( )
cos
2 1 2 1
2
1 1ϕ ϕ
(( ) sin ( )
cos ( )
j n j
n j
k k−( ) − −( )
− −(
+ +1 2 1
2 1
2 2ϕ ϕ
� �
)) − −( )
ϕ ϕ2 22 1m mn jsin ( )
,
j = k + 1, k + 2, … , m.
Pry v¥polnenyy uslovyq (3) suwestvuet netryvyal\noe polynomyal\noe
reßenye zadaçy (1), (2).
2. Esly k = m, to zadaça Dyryxle (1), (2) ymeet tol\ko tryvyal\noe re-
ßenye.
3. Esly xarakterystyçeskoe uravnenye ymeet koren\ i kratnosty k > m,
to zadaça Dyryxle (1), (2) vsehda ymeet netryvyal\noe reßenye (pry lgb¥x
suwestvugwyx ϕ j ).
Dokazatel\stvo. S uçetom razloΩenyq symvola L( )ξ = a m
0 1
2ξ +
+ a m
1 1
2 1
2ξ ξ− + … + a m
m
2 2
2ξ = ξ, a1 ξ, a2
… ξ, a m2
predstavym uravnenye
(1) v vyde
∇ … ∇ =, , ,a a ua m1 2 2 0∆ , (4)
hde a j ∈C2 , j = 1, 2, … , 2m, — kompleksn¥e vektor¥, opredelqem¥e koπffy-
cyentamy uravnenyq (1), a b, = a b1 1 + a b2 2 — skalqrnoe proyzvedenye vekto-
rov. V dal\nejßem budem rassmatryvat\ takΩe vektor¥ �a j = −( )a aj j
2 1, =
= ( cos− ϕ j , sin )ϕ j , j = 1, 2, … , 2m.
PredpoloΩym, çto dlq nekotoroho s = 1, … , 2m λ s i= — koren\ xarakte-
rystyçeskoho uravnenyq, tohda a ias s
1 2+ = 0. Sledovatel\no, a ias s
1 2= y s —
mnoΩytel\ v razloΩenyy symvola L( )ξ uravnenyq (1) — prynymaet vyd ξ1 1a s
+
+ ξ2 2a s = − +ia is
2 1 2( )ξ ξ t.5e. s toçnost\g do mnoΩytelq raven ξ ξ1 2+ i , çto y
budem yspol\zovat\ v dal\nejßem. Analohyçno dlq kornq – i : ξ ξ1 2− i .
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
594 E. A. BURQÇENKO
1. Neobxodymost\. Sohlasno utverΩdenyg [2, c. 199, 200], suwestvovanye
netryvyal\noho reßenyq zadaçy Dyryxle (1), (2) v prostranstve C Km2 ( ) vle-
çet suwestvovanye netryvyal\noho analytyçeskoho v C2
reßenyq w Z( )ξ ∈
uravnenyq
( ) ( ) ( )∆ξ ξ ξ+ { } =1 0m L w .
(Klass Z zdes\ opredelen kak prostranstvo obrazov Fur\e funkcyj vyda θK v ,
v ∈C m2 2( )R , θK — xarakterystyçeskaq funkcyq kruha.)
Razlahaq funkcyg v( )ξ = L w( ) ( )ξ ξ v stepennoj rqd, dlq mladßej netry-
vyal\noj odnorodnoj polynomyal\noj çasty vN ( )ξ rqda poluçaem uravnenye
∆ξ ξm
Nv ( ) = 0 . (5)
S uçetom sootnoßenyj (32.15) – (32.22) v [9] yly formul¥ Al\mansy [10,
s. 208] obwee polynomyal\noe reßenye �v( )ξ uravnenyq (5) moΩno predstavyt\
v vyde
�v( )ξ = Re ( ) ( )f z z f zm
m1
1+ … +{ }− + i g z z g zm
mRe ( ) ( )1
1+ … +{ }− ,
hde z = ξ1 + iξ2 , f zi ( ) = f zin
n∑ , g zi ( ) = g zin
n∑ , i = 1, 2, … , m , — nekoto-
r¥e polynom¥. Otsgda
�ν ρ ϕ( , ) = ρ α ϕ β ϕ ϕn
n n n
n
n n a n1 1 2
0
2cos sin cos ( )− + −(
=
∞
∑ –
–5 β ϕ2 2n nsin ( )− + … + a n mmn cos ( – )−( )2 1 ϕ –
–5 β ϕmn n msin ( )− −( ) )2 1 . (6)
Zdes\ postoqnn¥e αin , βin stroqtsq po koπffycyentam razloΩenyq polyno-
mov f zi ( ) = f zin
n∑ , g zi ( ) = g zin
n∑ , i = 1, 2, … , m , sledugwym obrazom:
Reαin = Re , ( )fi n i− − 1 , Im αin = Re , ( )gi n i− − 1 , Reβin = Im , ( )fi n i− − 1 , Im βin =
= Im , ( )gi n i− − 1 , n N∈ , i = 1, … , m. Yspol\zuq formul¥ ∏jlera, yz (6) poluça-
em uravnenye
�ν ρ ϕ( , ) =
=
1
2 0
2 1ρ α
β
α
βϕn
n
jn
jn i n j
jn
j
i
e
=
∞
− −( )∑ −
+ +( ) nn i n j
j
m
j
m
i
e
− − −( )
==
∑∑ 2 1
11
( ) ϕ
yly
�ν ρ ϕ( , ) =
=
1
2 0
2 1ρ α β α βϕn
n
jn jn
i n j
jn jni e i
=
∞
− −( )∑ + + −( ) ( )( ) ee i n j
j
m
j
m
− − −( )
==
∑∑
2 1
11
( ) ϕ ,
kotoroe moΩno perepysat\ v vyde
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
USLOVYQ NETRYVYAL|NOJ RAZREÍYMOSTY ODNORODNOJ ZADAÇY DYRYXLE …595
�ν ρ ϕ( , ) =
1
2 0 1
2 1ρ α β ϕn
n
jn jn
j
m
i n ji e
=
∞
=
− −( )∑ ∑ +( ) ( ) +
+
1
2 0 1
2 1ρ α β ϕn
n
jn jn
j
m
i n ji e
=
∞
=
− − −( )∑ ∑ −( ) ( ) =
=
1
2
1 1
0
2
2ρ α β ρ α βϕ ϕn in
n n
n
n i n
ne i e i( ) (( )+ + +
=
∞
−∑ 22n ) + …
… +5 ρ α βϕn i n m
mn mne i− −( ) +
2 1( ) ( ) +
+
1
2
1 1
0
2
2ρ α β ρ αϕ ϕn in
n n
n
n i n
ne i e−
=
∞
− −− + −∑ ( ) (( ) ii nβ2 ) + …
… +5 ρ α βϕn i n m
mn mne i− − −( ) −
2 1( ) ( ) =
=
1
2
1 1
0
2 2
2 2
4z i z i zn
n n
n
n
n n
n( ) ( )α β ρ α β ρ+ + + +
=
∞
−∑ −− +4
3 3( )α βn ni + …
… +5 ρ α β2 1 2 1( ) ( )( )m n m
mn mnz i− − − + +
+
1
2
2
1 1
0
2 2 2
2 2ρ α β ρ α βn n
n n
n
n n
n nz i z i( ) (− + −
=
∞
− −∑ )) +
+ ρ α β2 4 4
3 3
n n
n nz i− − −( ) + … + ρ α β2 2 1 2 1n m n m
mn mnz i− − − − −
( ) ( )( ) =
=
1
2
1 1
0
2 2
2 2z i z in
n n
n
n
n n( ) ( )α β ρ α β+ + +
=
∞
−∑ +
+ ρ α βϕ3 3
3 3z e in i
n n
− − +( ) + … + ρ α βϕm n m i m
mn mnz e i− − − +
( ) ( )2 +
+
1
2
2
1 1
0
2 2 2
2 2ρ α β ρ α βn n
n n
n
n n
n nz i z i( ) (− + −
=
∞
− −∑ )) +
+ ρ α βϕ2 3 3
3 3
n n i
n nz e i− − −( ) + … + ρ α βϕ2 2n m n m i m
mn mnz e i− − − −
( ) ( ) .
(7)
Pry delenyy celoj funkcyy �ν ξ( ) = L w( ) ( )ξ ξ� na symvol L( )ξ , kotor¥j v
sluçae kratnosty k < m kornq i xarakterystyçeskoho uravnenyq ymeet vyd
L( )ξ = ( )ξ ξ1 2+ i k
ξ, ak + 1
ξ, ak + 2
… ξ, a m2
, dolΩen poluçyt\sq poly-
nom. V sluçae, kohda koren\ xarakterystyçeskoho uravnenyq ymeet kratnost\
k < m, poluçaem sledugwye uslovyq:
α β1 1 0n ni+ = , α β2 2 0n ni+ = , … ,α βkn kni+ = 0 ,
(8)
�ν ϕ ϕ= − +
=
k 1
0 , �ν ϕ ϕ= − +
=
k 2
0 , … , �ν ϕ ϕ= − =
2
0
m
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
596 E. A. BURQÇENKO
Perv¥e k uslovyj voznykagt posle delenyq funkcyy (7) na mnoΩytel\ (ξ1 +
+ i kξ2 ) symvola L( )ξ . Poskol\ku ( )ξ ξ1 2+ i k = zk = ρ ϕk ike , k > 1, posle de-
lenyq budem ymet\
�ν ρ ϕ
ξ ξ
( , )
( )1 2+ i k
=
=
1
2
1 1
0
2 2
2z i z e in k
n n
n
k n ik
n
−
=
∞
− − −+ + +∑ ( ) (α β ρ αϕ ββ ρ2
3 3
n
k nz) + − − ×
× e i z e ii k
n n
k n i k
n
− + − − − ++ + +( ) ( )( ) (1
3 3
4 4 2
4
ϕ ϕα β ρ α ββ4n ) + …
… +5 ρ α βϕm k n m i k m
mn mnz e i− − − + − +
( ) ( )2 +
+
1
2
2
1 1
2 2 2ρ α β ρ αϕ ϕn k n ik
n n
n k n ikz e i z e− − − − − −− +( ) ( 22 2
0
n n
n
i−
=
∞
∑ β ) +
+ ρ α βϕ2 3 3 1
3 3
n k n i k
n nz e i− − − − −( ) −( ) + …
… +5 ρ α βϕ2 2n k m n m i k m
mn mnz e i− − − − − −( ) −
( ) ( ) . (9)
V sylu toho, çto v¥raΩenye (9) dolΩno b¥t\ celoj funkcyej, voznykagt per-
v¥e k uslovyj. Poslednye 2m – k uslovyj voznykagt vsledstvye toho, çto
ravenstvo ξ, a j = 0 πkvyvalentno ravenstvu ϕ = −ϕ j , j = k + 1, … , 2m, t.5e.
�ν ρ ϕ ξ( , ) ( )L = 0 = 0. Podstavlqq (7) v (8), pryxodym k systeme lynejn¥x urav-
nenyj otnosytel\no postoqnn¥x αin , βin , matryca kotoroj ymeet vyd
( , , , , , , )C C C C Ck k m1 2 1… …+ , (10)
C
i
n n
n n
k k
k k
1
1 1
2 2
1
0 0
0 0
=
+ +
+ +
� �
cos sin
cos cos
ϕ ϕ
ϕ ϕ
�� �
cos sinn nm mϕ ϕ2 2
,
C
i
n n
n
k k
2
1 1
0 0
1
0 0
2 2
2
= − −
−
+ +
� �
cos ( ) sin ( )
cos ( )
ϕ ϕ
ϕϕ ϕ
ϕ ϕ
k k
m m
n
n n
+ +−
− −
2 2
2 2
2
2 2
sin ( )
cos ( ) sin ( )
� �
,
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
USLOVYQ NETRYVYAL|NOJ RAZREÍYMOSTY ODNORODNOJ ZADAÇY DYRYXLE …597
C
i
n k n kk
k k
= − −( ) − −( )+
0 0
0 0
1
2 1 2 11
� �
cos ( ) sin ( )ϕ ϕ ++
+ +− −( ) − −( )
1
2 22 1 2 1cos ( ) sin ( )
cos
n k n k
n
k kϕ ϕ
� �
−− −( ) − −( )
2 1 2 12 2( ) sin ( )k n km mϕ ϕ
,
C n j n jj
k k
= − −( ) − −( )+
0 0
0 0
0 0
2 1 2 11
� �
cos ( ) sin ( )ϕ ϕ ++
+ +− −( ) − −( )
1
2 22 1 2 1cos ( ) sin ( )
cos
n j n j
n
k kϕ ϕ
� �
−− −( ) − −( )
2 1 2 12 2( ) sin ( )j n jm mϕ ϕ
,
j = k + 1, k + 2, … , m.
UmnoΩym kaΩd¥j çetn¥j stolbec matryc¥ (10) na i, prybavym eho k pre-
d¥duwemu stolbcu y, razloΩyv opredelytel\ poluçennoj matryc¥ po stolb-
cam, poluçym opredelytel\, stoqwyj v levoj çasty (3). Poskol\ku dvojstven-
naq zadaça (5) ymeet netryvyal\noe reßenye, suwestvuet nenulevoj nabor pos-
toqnn¥x αin , βin , n = 1, 2, … , m . Znaçyt, lynejnaq systema ymeet nenulevoe
reßenye, çto vleçet obrawenye v nul\ ee opredelytelq, t.5e. v¥polnenye us-
lovyq (3).
Dostatoçnost\. Pry v¥polnenyy uslovyq (3) dlq nekotoroho n N∈ , n ≥
≥ 2m, postroym netryvyal\noe reßenye zadaçy (1), (2) v qvnom vyde. Tem sam¥m
budet dokazana dostatoçnost\ uslovyq (3) dlq netryvyal\noj razreßymosty za-
daçy (1), (2).
V rabote [6] b¥lo dokazano, çto funkcyq
u x C F a xj j
j
j
m
( ) = −( )
=
∑ �
1
2
, (11)
hde F yj ( ) — nekotor¥e hladkye funkcyy odnoho arhumenta, C j , j = 1, … , 2m,
— nekotor¥e postoqnn¥e, qvlqetsq reßenyem odnorodnoho uravnenyq proyz-
vol\noho çetnoho porqdka 2m v sluçae, kohda suwestvugt uhl¥ naklona vsex
prost¥x xarakterystyk. V kaçestve F yj ( ) v ukazannoj rabote yspol\zovana
funkcyq, qvlqgwaqsq s toçnost\g do postoqnnoj (m – 1)-j pervoobraznoj po-
lynoma Çeb¥ßeva pervoho roda porqdka N = n – m + 1:
F yj ( ) = F y dy dy dy T t dtm
y y
m
y
N
ym
− −= …∫ ∫ ∫
−
1 1
0
2
0
2
0 0
1 3
( ) : ( )
mm −
∫
2
, j = 1, … , 2m,
yly
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
598 E. A. BURQÇENKO
F y
C N T y C N T y
m
r N r r N r
−
+
+ +
−
− −
=
+{ }
1
2 1 2 1
( )
( ) ( ) ( ) ( ) ,, ,
( ) ( ) ( ) (
r
l
r N r r N r
m l
D N T y D N T
=
+
+
−
−
∑ − = +
+
0
2 2
1 2 1
yy m l
r
l
) , ,{ } − =
=
∑
0
1 2
(12)
hde C Nr
+ ( ) , C Nr
− ( ) , D Nr
+ ( ) , D Nr
− ( ) — postoqnn¥e, zavysqwye tol\ko ot N y
voznykagwye pry v¥çyslenyy (m – 1)-j pervoobraznoj polynoma Çeb¥ßeva
pervoho roda porqdka N = n – m + 1. V sluçae, kohda koren\ i xarakterysty-
çeskoho uravnenyq ymeet kratnost\ n < m, razloΩenye symvola L( )ξ pryny-
maet vyd
L i a a ak k k m( ) , , ,ξ ξ ξ ξ ξ ξ= +( ) …+ +
1 2
1 2 2 , k = 1, 2, …, m – 1.
Uravnenyg L ux( )∂ = 0 udovletvorqet funkcyq vyda
u x C x ix C x ix C x ixn n n( ) ( ) ( ) ( )= + + + + +− −
1 1 2 2 1 2
2
3 1 2
44 + …
… + C x ix C F a xk
n k
j j
j
j k
m
( ) ( )( )
1 2
2 1
1
2
+ + −− −
= +
∑ � , (13)
vsledstvye ortohonal\nosty vektorov a j
y �a j
pry kaΩdom j = k + 1, … , 2m.
Kolyçestvo slahaem¥x, stoqwyx do summ¥ C F a xj j
j( )− � v (13), ravno krat-
nosty kornq i xarakterystyçeskoho uravnenyq. Rassmotrym, dlq opredelen-
nosty, sluçaj çetnoho m: m – 1 = 2l + 1 (dlq neçetnoho m rassuΩdenyq ana-
lohyçn¥). Funkcyy F a xj
j( )− � , j = k + 1, … , 2m, podobran¥ po formulam (12).
Uçyt¥vaq, çto na ∂K ( )− �a xj = − +cos ( )τ ϕ j , N + m – 1 = n, ymeem
u C N N r C N rK r
r
l
j j
j
∂
+
=
= + + + +∑ ( ) cos ( ) cos ( )2 1 2 1
0
τ ϕ
== +
∑
k
m
1
2
–
–5 C N r C N rr
r
l
j j
j k
m
+
= = +
+ + + +∑ ∑sin ( ) sin ( )2 1 2 1
0 1
2
τ ϕ +
+5 C N r C N rr
r
l
j j
j k
m
−
= = +
− − − −∑ ∑cos ( ) cos ( )2 1 2 1
0 1
2
τ ϕ –
–5 C N r C N rr
r
l
j j
j k
m
−
= = +
− − − −∑ ∑sin ( ) sin ( )2 1 2 1
0 1
2
τ ϕ +
+5 C n i n C n i n1 2 2 2(cos sin ) cos ( ) sin ( )τ τ τ τ+ + − + −( ) + …
… + C n k i n kk cos ( ) sin ( )− −( ) + − −( )( )2 1 2 1τ τ = 0. (14)
Posle pryvedenyq podobn¥x slahaem¥x oçevydno, çto ravenstvo (14) spravedly-
vo tohda y tol\ko tohda, kohda ymegt mesto sledugwye sootnoßenyq:
C N l q Cj j q
j k
m
cos ( )+ + − −( ) + =
= +
∑ 2 1 2 1 0
1
2
ϕ ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
USLOVYQ NETRYVYAL|NOJ RAZREÍYMOSTY ODNORODNOJ ZADAÇY DYRYXLE …599
C N l q iCj j q
j k
m
sin ( )+ + − −( ) + =
= +
∑ 2 1 2 1 0
1
2
ϕ
dlq kaΩdoho q = 1, 2, … , k, a takΩe
C N rj j
j k
m
cos ( )− − =
= +
∑ 2 1 0
1
2
ϕ , C N rj j
j k
m
sin ( )− − =
= +
∑ 2 1 0
1
2
ϕ ,
(15)
C N rj j
j k
m
cos ( )+ + =
= +
∑ 2 1 0
1
2
ϕ , C N rj j
j k
m
sin ( )+ + =
= +
∑ 2 1 0
1
2
ϕ
dlq kaΩdoho r = 0, 1, … , l – k, tak kak dlq r = l – (k – 1), … , l analohyçn¥e us-
lovyq zapysan¥ v¥ße (r = l – (q – 1), q = 1, 2, … , k ).
Dann¥e sootnoßenyq predstavlqgt soboj systemu 2m uravnenyj, oprede-
lytel\ kotoroj raven nulg (pry N + m – 1 = n on sovpadaet s opredelytelem,
stoqwym v levoj çasty (3)). Sledovatel\no, funkcyq u x( ) , zadannaq po for-
mule (13), udovletvorqet pervomu hranyçnomu uslovyg v (2). Ysxodq yz doka-
zannoho v [6] rezul\tata, moΩno sdelat\ v¥vod, çto funkcyq u x( ) zadannaq po
formule (13), budet udovletvorqt\ y ostal\n¥m hranyçn¥m uslovyqm zadaçy,
tak kak
u p
K
ν
( )
∂
= C P N p N rj
j k
m
r j
r
l
= +
+
=
∑ ∑ + + +{
1
2
0
2 1( , ) cos ( ) ( )τ ϕ +
+ P N p N rr j
− − − + }( , ) cos ( ) ( )2 1 τ ϕ
s nekotor¥my postoqnn¥my. Sledovatel\no, uslovyq u p
K
ν
( )
∂
= 0, p = 0, 1, …
… , m – 1 snova pryvodqt k systeme (15) otnosytel\no neyzvestn¥x C1 , C2 , …
… , C m2 .
2. Rassmotrym sluçaj, kohda kratnost\ k kornq i xarakterystyçeskoho
uravnenyq ravna m. Tohda razloΩenye symvola uravnenyq (1) prymet vyd
L( )ξ = ( )ξ ξ1 2+ i m
ξ, am + 1
ξ, am + 2
… ξ, am m+
. Perv¥j mnoΩytel\ dan-
noho v¥raΩenyq moΩno zapysat\ v vyde ( )ξ ξ1 2+ i m = zm = ρ ϕm i me . Çtob¥
pry delenyy funkcyy (7) na πtot symvol poluçylsq polynom, s neobxodymo-
st\g budem ymet\
α β1 1 0n ni+ = , α β2 2 0n ni+ = , … , α βmn mni+ = 0 ,
(16)
�ν ϕ ϕ= − +
=
m 1
0 , �ν ϕ ϕ= − +
=
m 2
0 , … , �ν ϕ ϕ= − =
2
0
m
,
Podstavlqq funkcyg (7) v uslovyq (16), snova pryxodym k systeme lynejn¥x
uravnenyj otnosytel\no postoqnn¥x αin , βin , opredelytel\ kotoroj ymeet
vyd
∆2 = det
( ) ( )
e e e e
in i n i n m i nm m mϕ ϕ ϕ+ + +− − −( ) −1 1 12 2 2 2� (( )
( ) ( )
m
in i n i n m
m
m me e e
−( )
− − −( )
+
+ +
1
2 2 2
1
2 2
ϕ
ϕ ϕ � ϕϕ ϕ
ϕ ϕ
m m
m m
e
e e
i n m
in i n
+ +− −( )
−
2 2
2 2
2 1
2
( )
( )
� � � � �
� ee ei n m i n mm m− −( ) − −( )
2 2 2 12 2( ) ( )ϕ ϕ
=
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
600 E. A. BURQÇENKO
= e e e
i n m i n m i n mm m− −( ) − −( ) − −+ + …2 1 2 1 2 11 2( ) ( ) ( )ϕ ϕ (( ) ϕ2m ×
×
e e
e e
i m i
i m i
m m
m m
2 1 2
2 2 2
1 1
2 2
1
1
( )
( )
−
−
+ +ϕ ϕ
ϕ ϕ
�
� � �
� 11
.
V¥polnym zamenu z1 = e
i m2 1ϕ + , z2 = e
i m2 2ϕ + , … , zm = e i m2 2ϕ
, tohda
∆2 = e e e
i n m i n m i n mm m− −( ) − −( ) − −+ + …2 1 2 1 2 11 2( ) ( ) ( )ϕ ϕ (( ) ϕ2m ×
×
z z z
z z z
z z z
m
m
m
m
m m
1
1
1
2
1
2
1
2
2
2
1 2
1
1
1
1
−
−
−
�
�
� � � �
�
=
= e e e
i n m i n m i n mm m− −( ) − −( ) − −+ + …2 1 2 1 2 11 2( ) ( ) ( )ϕ ϕ (( )
≤ < ≤
−∏ϕ2
1
m z zi j
j i n
( ) .
Ravenstvo ∆2 = 0 vozmoΩno tohda y tol\ko tohda, kohda z zi j= , no πto ne tak
(ϕ ϕi j≠ ). Sledovatel\no, ysxodn¥j opredelytel\ otlyçen ot nulq. Takym
obrazom, dvojstvennaq zadaça ymeet tol\ko tryvyal\noe reßenye, y, sohlasno
utverΩdenyg [2, c. 199, 200], odnorodnaq zadaça Dyryxle (1), (2) ymeet tol\ko
tryvyal\noe reßenye.
3. V sluçae, kohda kratnost\ k kornq i xarakterystyçeskoho uravnenyq k >
> m, ymeem sledugwee razloΩenye symvola:
L i a am l m l m( ) ( ) , ,ξ ξ ξ ξ ξ= + …+ + +
1 2
1 2 ,
hde l = 1, 2, … , m.
Dal\nejßye rassuΩdenyq analohyçn¥ takov¥m pry dokazatel\stve utverΩ-
denyj pervoho y vtoroho punktov: çtob¥ pry delenyy funkcyy (7) na symvol
L( )ξ = ( )ξ ξ1 2+ +i m l
ξ, am l+ +1
… ξ, a m2
poluçylsq polynom, dolΩn¥ v¥-
polnqt\sq sledugwye m uslovyj:
α β1 1 0n ni+ = , α β2 2 0n ni+ = , … , α βmn mni+ = 0 .
TakΩe ymeem 2m – (m + l) = m – l sootnoßenyj dlq uhlov naklona xarakte-
rystyk:
�ν ϕ ϕ= − + +
=
m l 1
0 , �ν ϕ ϕ= − + +
=
m l 2
0 , … , �ν ϕ ϕ= − =
2
0
m
.
Takym obrazom, dlq opredelenyq koπffycyentov αin , βin poluçaem systemu
m + m – l = 2m – l uravnenyj s 2m neyzvestn¥my. Takaq systema vsehda ymeet
nenulevoe reßenye. Sledovatel\no, zadaça Dyryxle (1), (2) pry lgb¥x suwest-
vugwyx ϕ j ymeet netryvyal\noe reßenye, j = m + l + 1, … , 2m, l = 1, 2, … , m.
Esly znaçenye kratnosty k kornq i bol\ße polovyn¥ porqdka uravnenyq,
t.5e. k > m, to sootvetstvugwaq zadaça Dyryxle (1), (2) ymeet sçetnoe çyslo
lynejno nezavysym¥x reßenyj. Dejstvytel\no, razlahaq operator L porqdka
2m : L m2 =
∂
∂
+
∂
∂
+
x
i
y
m l
Lm l− , hde Lm l− — dyfferencyal\n¥j operator
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
USLOVYQ NETRYVYAL|NOJ RAZREÍYMOSTY ODNORODNOJ ZADAÇY DYRYXLE …601
porqdka m – l, l = 1, 2, … , m, xarakterystyçeskoe uravnenye kotoroho ne ymeet
kornej ± i, netrudno zametyt\, çto nabor funkcyj
u z zz P zk
m l
k( ) ( ) ( )= − + −1 1 , k = 0, 1, 2, … ,
hde P zk ( ) — proyzvol\n¥e polynom¥ stepeny k, udovletvorqet uravnenyg
L u
x
i
y
L u
z
L um
m l
m l
m l
m l m l2 =
∂
∂
+
∂
∂
=
∂
∂
=
+
−
+
+ − 00
y uslovyqm Dyryxle na hranyce ∂K = z C∈{ : z 2 = zz = 1} edynyçnoho
kruha:
u z = =1 0 , ′ =
=
u
zρ 1
0 , … , u m
z
ρ
( )−
=
=1
1
0 , ρ = z .
Pryvedenn¥j prymer obobwaet yzvestn¥j rezul\tat A.5V.5Bycadze [1] dlq
uravnenyj vtoroho porqdka:
∂
∂
+
∂
∂
=
x
i
y
u
2
0
∂
∂
=
2
2 0
z
u
y
∂
∂
−
∂
∂
=
x
i
y
u
2
0
∂
∂
=
2
2 0
z
u ,
xarakterystyçeskye uravnenyq kotor¥x ymegt korny i (y – i sootvetstvenno)
kratnosty k = 2 > m = 1 — polovynu porqdka uravnenyq.
Teorema dokazana.
Zameçanyq. 1. Analohyçn¥j rezul\tat s zamenoj i na – i ymeet mesto y
dlq sootvetstvugwyx kratnostej kornq – i xarakterystyçeskoho uravnenyq.
2. V sluçae, kohda odnovremenno suwestvugt korny i y – i, operator L po-
rqdka 2m moΩno razloΩyt\ v vyde proyzvedenyq operatora Laplasa y opera-
tora porqdka 2(m – 1) : L m2 = ∆ ⋅ L m2 2− .
Vtoraq çast\ teorem¥51 v stat\e [8] svydetel\stvuet o tom, çto s toçky zre-
nyq edynstvennosty reßenyq zadaçy Dyryxle dlq operatora L, xarakterysty-
çeskoe uravnenye kotoroho odnovremenno ymeet prost¥e korny i y – i, eho pra-
vyl\no πllyptyçeskaq çast\ (operator Laplasa) znaçenyq ne ymeet, y takye
uravnenyq vedut sebq kak y uravnenyq porqdka 2(m – 1), poskol\ku uslovye
det
cos sin cos ( ) sin ( )n n n m n mϕ ϕ ϕ3 3 32 2 2 2� − −( ) − −( ))
− −( ) − −
ϕ
ϕ ϕ ϕ
3
4 4 42 2 2 2cos sin cos ( ) sin ( )n n n m n m� (( )
− −( )
ϕ
ϕ ϕ ϕ
4
2 2 22 2
� � � � �
�cos sin cos ( ) sin n n mm m m nn ( )n m m− −( )
2 2 2ϕ
= 0,
sohlasno rezul\tatam rabot¥ [6], qvlqetsq kryteryem narußenyq edynstven-
nosty reßenyq zadaçy Dyryxle v kruhe dlq uravnenyj hlavnoho typa porqdka
2(m – 1).
3. Sluçaj kratn¥x kornej i y – i xarakterystyçeskoho uravnenyq.
DokaΩem teoremu suwestvovanyq netryvyal\noho reßenyq zadaçy Dyryxle (1),
(2) v sluçae, kohda xarakterystyçeskoe uravnenye ymeet kratn¥e korny i y – i .
Kak y v utverΩdenyqx pred¥duweho punkta, uslovyq narußenyq edynstvennos-
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
602 E. A. BURQÇENKO
ty reßenyq zadaçy Dyryxle budut zavyset\ ot znaçenyj kratnostej k1 y k2
kornej i y – i sootvetstvenno.
Teorema 2. Pust\ çysla i y – i qvlqgtsq kornqmy xarakterystyçeskoho
uravnenyq kratnostej k1 y k2 sootvetstvenno ( k k1 2+ = 2, 3, … , 2m ,
k k1 2 ≠ 0) y dlq opredelennosty k2 > k1 . PoloΩym l = k2 – k1 > 0.
Tohda:
1. Esly l < m, to dlq netryvyal\noj razreßymosty zadaçy Dyryxle (1),
(2) neobxodymo y dostatoçno v¥polnenyq sledugweho uslovyq dlq nekotoroho
n N∈ , n ≥ 2m:
∆3 1 2 1
0= = … =+ + −det det ( , , , , )B B B B Bl l m k
� .
Zdes\ bloky matryc¥ B ymegt vyd
�
�
�
B
e e
e e
in i n l
in i n l
l l
l
=
+ +
+
− −( )
− −
ϕ ϕ
ϕ
1 1
2
2 1
2
( )
( 11
2 1
2
2 1 2 1
)
( )
( ) (
( )
− −( )
+
− −
ϕ
ϕ ϕ
l
m k m ke ein i n l
� � �
� ))
,
B
n j n j
n
j
l l
=
− −( ) − −( )
−
+ +cos ( ) sin ( )
cos
2 1 2 1
2
1 1ϕ ϕ
(( ) sin ( )
cos ( )
j n j
n j
l l−( ) − −( )
− −(
+ +1 2 1
2 1
2 2ϕ ϕ
� �
)) − −( )
− −ϕ ϕ2 21 1
2 1( ) ( )sin ( )m k m kn j
,
j = l + 1, l + 2, … , m – k1 .
2. Esly l = m, to zadaça Dyryxle (1), (2) ymeet tol\ko tryvyal\noe re-
ßenye.
3. Esly m < l ≤ 2m, to zadaça Dyryxle (1), (2) pry lgb¥x suwestvugwyx
ϕ j vsehda ymeet netryvyal\noe reßenye.
Dokazatel\stvo. PredpoloΩym, çto xarakterystyçeskoe uravnenye
L( , )1 λ = 0 ymeet odnovremenno korny i y – i kratnostej k1 y k2 sootvet-
stvenno ( k k1 2+ = 2, 3, … , 2m , k k1 2 ≠ 0). Rassmotrym raznost\ kratnostej
kornej l = k2 – k1 > 0 (dlq opredelennosty). V sluçae, kohda k1 > k2 , doka-
zatel\stvo analohyçno (s zamenoj i na – i ).
Operator L porqdka 2m moΩno razloΩyt\ v vyde proyzvedenyq operatora
Laplasa stepeny k1 y operatora porqdka 2(m – k1 ) : L m2 = ∆k1 ⋅ L m k2 1( )− =
= ∆k k ki1 2 1
1 2( )ξ ξ+ − ξ, ak k2 1 1− + ξ, ak k2 1 2− +
… ξ, a m2
. No xarakterysty-
çeskoe uravnenye, sootvetstvugwee operatoru L m k2 1( )− , uΩe ne ymeet kornq
– i, no ymeet koren\ i kratnosty k2 – k1 = l, poπtomu k nemu moΩno prymenyt\
teoremu51 yz pred¥duweho punkta s zamenoj k na l y m na m – k1 , çto y za-
verßaet dokazatel\stvo dannoho utverΩdenyq.
1. Bycadze A. V. O edynstvennosty reßenyq zadaçy Dyryxle dlq πllyptyçeskyx uravnenyj s
çastn¥my proyzvodn¥my // Uspexy mat. nauk. – 1948. – 3, v¥p.56. – S. 211 – 212.
2. Burskyj V. P. Metod¥ yssledovanyq hranyçn¥x zadaç dlq obwyx dyfferencyal\n¥x urav-
nenyj. – Kyev: Nauk. dumka, 2002. – 316 s.
3. Burqçenko E. A. O edynstvennosty reßenyj zadaçy Dyryxle v kruhe dlq dyfferencyal\-
n¥x uravnenyj çetvertoho porqdka v v¥roΩdenn¥x sluçaqx // Nelynejn¥e hranyçn¥e zada-
çy. – 2000. – 10. – S. 44 – 49.
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
USLOVYQ NETRYVYAL|NOJ RAZREÍYMOSTY ODNORODNOJ ZADAÇY DYRYXLE …603
4. Babayan A. O. On unique solvability of Dirichlet problem for fourth order property elliptic equa-
tion // Izv. Nats. Akad. Nauk Armenii. – 1999. – 34, # 5. – P. 5 – 18.
5. Lopatynskyj Q. B. Ob odnom sposobe pryvedenyq hranyçn¥x zadaç dlq system dyfferen-
cyal\n¥x uravnenyj πllyptyçeskoho typa k rehulqrn¥m yntehral\n¥m uravnenyqm // Ukr.
mat. Ωurn. – 1953. – 5, # 2. – S. 123 – 151.
6. Burskyj V. P., Burqçenko E. A. Nekotor¥e vopros¥ netryvyal\noj razreßymosty odnorod-
noj zadaçy Dyryxle dlq lynejn¥x uravnenyj proyzvol\noho çetnoho porqdka v kruhe //
Mat. zametky. – 2005. – 74, # 4. – S. 1032 – 1043.
7. Burqçenko E. A. Razreßymost\ odnorodnoj zadaçy Dyryxle v kruhe dlq uravnenyj porqdka
2m v sluçae kratn¥x xarakterystyk, ymegwyx uhl¥ naklona // Mat. metody ta fiz.-mex. po-
lq. – 2008. – 51, # 1. – S. 33 – 41.
8. Burqçenko E. A. Netryvyal\naq razreßymost\ odnorodnoj zadaçy Dyryxle v kruhe dlq
uravnenyj porqdka 2m v sluçae xarakterystyk, ne ymegwyx uhlov naklona // Visn. Donec.
nac. un-tu. Ser. fiz.-mat. nauk. – 2007. – 50, # 2. – S. 10 – 21.
9. Vekua Y. N. Nov¥e metod¥ reßenyj πllyptyçeskyx uravnenyj. – M.: OHYZ, 1948. – 296 s.
10. Bycadze A. N. Nekotor¥e klass¥ dyfferencyal\n¥x uravnenyj v çastn¥x proyzvodn¥x. –
M.: Myr, 1981. – 448 s.
Poluçeno 26.08.09,
posle dorabotky — 01.03.10
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
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| id | umjimathkievua-article-2890 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:32:16Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/40/78d42cb7e009e42adaa7014df4398a40.pdf |
| spelling | umjimathkievua-article-28902020-03-18T19:39:51Z Conditions of nontrivial solvability of the homogeneous Dirichlet problem for equations of any even order in the case of multiple characteristics without slope angles Условия нетривиальной разрешимости однородной задачи Дирихле для уравнений произвольного четного порядка в случае кратных характеристик, не имеющих углов наклона Buryachenko, E. A. Буряченко, Е. А. Буряченко, Е. А. We consider the homogeneous Dirichlet problem in the unit disk $K ⊂ R^2$ for a general typeless differential equation of any even order $2m,\; m ≥ 2$, with constant complex coefficients whose characteristic equation has multiple roots $± i$. For each value of multiplicity of the roots $i$ and $–i$, we either formulate criteria of the nontrivial solvability of the problem or prove that the analyzed problem possesses solely the trivial solution. A similar result generalizes the well-known Bitsadze examples to the case of typeless equations of any even order. Розглянуто однорідну задачу Діршге в одиничному крузі $K ⊂ R^2$ для загального безтипного диференціального рівняння доцільного парного порядку $2m,\; m ≥ 2$, зі сталими комплексними коефіцієнтами, характеристичне рівняння якого мак країні корені $± i$. Для кожного значення кратностей коренів $i$ та $–i$ отримано критерії нетривіальної розв'язності задачі або доведено, що задача має лише тривіальний розв'язок. Подібний результат узагальнює відомі приклади А. В. Біцадзе па випадок безтиппих рівнянь довільного парного порядку. Institute of Mathematics, NAS of Ukraine 2010-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2890 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 5 (2010); 591–603 Український математичний журнал; Том 62 № 5 (2010); 591–603 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/2890/2531 https://umj.imath.kiev.ua/index.php/umj/article/view/2890/2532 Copyright (c) 2010 Buryachenko E. A. |
| spellingShingle | Buryachenko, E. A. Буряченко, Е. А. Буряченко, Е. А. Conditions of nontrivial solvability of the homogeneous Dirichlet problem for equations of any even order in the case of multiple characteristics without slope angles |
| title | Conditions of nontrivial solvability of the homogeneous Dirichlet problem for equations of any even order in the case of multiple characteristics without slope angles |
| title_alt | Условия нетривиальной разрешимости однородной задачи Дирихле для уравнений произвольного четного порядка в случае кратных характеристик, не имеющих углов наклона |
| title_full | Conditions of nontrivial solvability of the homogeneous Dirichlet problem for equations of any even order in the case of multiple characteristics without slope angles |
| title_fullStr | Conditions of nontrivial solvability of the homogeneous Dirichlet problem for equations of any even order in the case of multiple characteristics without slope angles |
| title_full_unstemmed | Conditions of nontrivial solvability of the homogeneous Dirichlet problem for equations of any even order in the case of multiple characteristics without slope angles |
| title_short | Conditions of nontrivial solvability of the homogeneous Dirichlet problem for equations of any even order in the case of multiple characteristics without slope angles |
| title_sort | conditions of nontrivial solvability of the homogeneous dirichlet problem for equations of any even order in the case of multiple characteristics without slope angles |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2890 |
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