Phragmén-Lindelöf theorem for solutions of elliptic differential equations in a banach space
For a second-order elliptic differential equation considered on a semiaxis in a Banach space, we show that if the order of growth of its solution at infinity is not higher than the exponential order, then this solution tends exponentially to zero at infinity.
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| Datum: | 2007 |
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| Sprache: | Ukrainisch Englisch |
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2007
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509408128139264 |
|---|---|
| author | Gorbachuk, M. L. Горбачук, М. Л. |
| author_facet | Gorbachuk, M. L. Горбачук, М. Л. |
| author_sort | Gorbachuk, M. L. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:51:39Z |
| description | For a second-order elliptic differential equation considered on a semiaxis in a Banach space, we show that if the order of growth of its solution at infinity is not higher than the exponential order, then this solution tends exponentially to zero at infinity. |
| first_indexed | 2026-03-24T02:40:37Z |
| format | Article |
| fulltext |
UDK 517.9
M. L. Horbaçuk (In-t matematyky NAN Ukra]ny, Ky]v)
TEOREMA FRAHMENA – LINDEL|OFA DLQ
ROZV’QZKIV ELIPTYÇNYX DYFERENCIAL|NYX RIVNQN|
U BANAXOVOMU PROSTORI
∗∗∗∗
For a second-order elliptic differential equation considered on the semiaxis in a Banach space, we show
that if the order of growth of its solution at infinity is not higher than the exponential order, then this
solution exponentially tends to zero at infinity.
Dlq dyfferencyal\noho uravnenyq vtoroho porqdka πllyptyçeskoho typa na poluosy v bana-
xovom prostranstve pokazano, çto esly porqdok rosta na beskoneçnosty eho reßenyq ne v¥ße
πksponencyal\noho, to πto reßenye πksponencyal\no stremytsq k nulg na beskoneçnosty.
Prototypom prostoru rozv’qzkiv abstraktnoho dyferencial\noho rivnqnnq
eliptyçnoho typu u banaxovomu prostori, rozhlqduvanoho na pivosi, [ prostir
harmoniçnyx funkcij u ( t, x ) v oblasti G = ( 0, ∞ ) × [ 0, 1 ] , wo zadovol\nqgt\
krajovi umovy u ( t, 0 ) = u ( t, 1 ) = 0. Klasyçnyj pryncyp Frahmena – Lindel\o-
fa u c\omu vypadku stverdΩu[: qkwo taka funkciq pry t → ∞ prqmu[ do ne-
skinçennosti povil\niße za e tπ , to vona [ obmeΩenog u pivsmuzi G i navit\
prqmu[ do nulq na neskinçennosti qk e t− π
. U cij statti zaznaçenyj pryncyp po-
ßyrg[t\sq na rozv’qzky dyferencial\noho rivnqnnq vyhlqdu
′′y t( ) = By t( ), t ∈ ( 0, ∞ ) , (1)
de B — pozytyvnyj operator u banaxovomu prostori X . Ce da[ zmohu dovesty,
wo pryncyp Frahmena – Lindel\ofa spravdΩu[t\sq i dlq rozv’qzkiv dostatn\o
ßyrokoho klasu dyferencial\nyx rivnqn\ iz çastynnymy poxidnymy ne obov’qz-
kovo eliptyçnoho typu. Vidmitymo, wo abstraktnyj pidxid do poßyrennq c\oho
pryncypu dlq harmoniçnyx funkcij na rozv’qzky deqkoho klasu eliptyçnyx riv-
nqn\ u pivcylindri, buv zaproponovanyj v [1]. Cej pidxid bazu[t\sq na moΩlyvo-
sti kompaktnoho vkladennq prostoru takyx rozv’qzkiv u prostir rozv’qzkiv toho
samoho rivnqnnq, ale u pivcylindri, stroho vuΩçomu za vyxidnyj. U vypadku, ko-
ly dim X = ∞ (a ce vaΩlyvo z ohlqdu na zastosuvannq), take vkladennq moΩly-
ve lyße todi, koly operator B ma[ kompaktnu rezol\ventu. Pidxid, wo propo-
nu[t\sq, vykorystovu[ opys usix rozv’qzkiv rivnqnnq (1) na ( 0, ∞ ) i klasyçnyj
pryncyp Frahmena – Lindel\ofa dlq skalqrnyx analityçnyx useredyni kuta
funkcij.
1. Nexaj X — banaxiv prostir z normog ⋅ , E ( X ) — mnoΩyna vsix wil\no
vyznaçenyx v X zamknenyx linijnyx operatoriv, L ( X ) — alhebra obmeΩenyx
linijnyx operatoriv u X, I — odynyçnyj operator, D ( ⋅ ) — oblast\ vyznaçennq
operatora, ρ ( ⋅ ) , σ ( ⋅ ) , σc ( ⋅ ) , σp ( ⋅ ) — joho rezol\ventna mnoΩyna, spektr, ne-
perervnyj ta toçkovyj spektry vidpovidno.
Navedemo deqki neobxidni u podal\ßomu vidomosti z teori] odnoparametryç-
nyx pivhrup operatoriv.
Sim’q ( ( ))T t t ≥0 linijnyx neperervnyx operatoriv u X utvorg[ C0 -pivhrupu,
qkwo: a) T I( )0 = ; b) ∀ ≥t t1 2 0, : T t t T t T t( ) ( ) ( )1 2 1 2+ = ; c) dlq dovil\noho
x=∈ X T t x x( ) − → 0 pry t → 0. Henerator A C 0 -pivhrupy ( ( ))T t t ≥0
vyznaça[t\sq qk
Ax = lim
( )
t
T t x x
t→
−
0
, x A∈D ( ),
∗
Vykonano za pidtrymky DerΩavnoho fondu fundamental\nyx doslidΩen\ Ukra]ny (proekt
14.1-003).
© M. L. HORBAÇUK, 2007
650 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 5
TEOREMA FRAHMENA – LINDEL|OFA DLQ ROZV’QZKIV … 651
de D ( )A — sukupnist\ vektoriv x ∈ X , dlq qkyx cq hranycq isnu[. Henerator
A zavΩdy naleΩyt\ do E ( X ) . C 0 -pivhrupu z heneratorom A poznaçatymemo
çerez ( )etA
t ≥0 .
Pivhrupa ( )etA
t ≥0 nazyva[t\sq rivnomirno obmeΩenog, qkwo isnu[ stala
M ≥ 1 taka, wo
∀ ≥t 0 : etA ≤ M .
Zhidno z [2], pivhrupa ( )etA
t ≥0 nazyva[t\sq obmeΩenog analityçnog z kutom
ϕ π∈( , / ]0 2 , qkwo etA
dopuska[ prodovΩennq do operator-funkci] ezA , anali-
tyçno] v sektori Σϕ = { }: argz z∈ <C ϕ , syl\no neperervno] v nuli na bud\-
qkomu promeni c\oho sektora, i dlq bud\-qkoho ′ <ϕ ϕ isnu[ stala c ′ >ϕ 0 taka,
wo
ezA ≤ c ′ϕ pry z ∈ ′Σϕ = { }: argz z∈ ≤ ′C ϕ .
Operator A E X∈ ( ) [ heneratorom obmeΩeno] analityçno] C0 -pivhrupy z kutom
ϕ todi i til\ky todi, koly Σϕ π ρ+ ⊂/ ( )2 A i dlq dovil\noho ′ ∈ϕ ϕ( , )0 isnu[
stala M ′ϕ taka, wo
∀ ∈ ′ +λ ϕ πΣ /2 : RA( )λ ≤
M ′ϕ
λ
,
de RA( )λ = ( )A I− −λ 1
— rezol\venta operatora A .
TverdΩennq,1 [2, c. 61]. Pivhrupa ( )etA
t ≥0 [ obmeΩenog analityçnog to-
di i til\ky todi, koly vona dyferencijovna na ( 0, ∞ ) i isnu[ stala c > 0
taka, wo
∀ >t 0 ∀ ∈n N : A en tA ≤ c n tn n n− .
Skriz\ u podal\ßomu vvaΩatymemo, wo dlq pivhrupy ( )etA
t ≥0 vykonu[t\sq
umova ker etA = { }0 dlq bud\-qkoho t > 0 .
Poznaçymo çerez X At− ( ) popovnennq X za normog
x t− = e xtA , t > 0.
Normy ⋅ −t [ uzhodΩenymy i porivnql\nymy. Pry t t< ′ ma[mo wil\ne i nepe-
rervne vkladennq X A X At t− − ′⊂( ) ( ). Poklademo
X A−( ) = proj lim
t
tX A
→
−
0
( )
(wodo proektyvno] hranyci banaxovyx prostoriv dyv. [3]). X A−( ) — povnyj zli-
çenno-normovanyj prostir. NevaΩko perekonatys\, wo operator etA
dopuska[
neperervne rozßyrennq S t( ) z X na X At− ( ), pryçomu pry t t< ′ S t X At
( ) ( )′
−
� =
= S t( ) .
Vyznaçymo na X A−( ) operator êtA
( )t ≥ 0 qk ê xtA = S t x( ) . V [4] pokaza-
no, wo sim’q ( )êtA
t ≥0 utvorg[ C0 -pivhrupu v X A−( ) (]] henerator poznaçymo
çerez  ). Pivhrupa ( )
ˆ
etA
t ≥0 = ( )êtA
t ≥0 ma[ taki vlastyvosti:
1) ∀ >t 0 : e X A Xt ( )− ⊆ ;
2) e x e xtA tAˆ
= dlq x ∈ X , t ≥ 0;
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 5
652 M. L. HORBAÇUK
3) ∀ ∈ −x X A( ) ∀ >t s, 0 : e xt s A( ) ˆ+ = e e xtA s = e e xsA t .
Rozhlqnemo rivnqnnq
′y t( ) = Ay t F t( ) ( )+ , t ∈ ∞( , )0 , (2)
de F C X∈ ∞1 0([ , ), ).
Vektor-funkcig y ( t ) : ( 0, ∞ ) � D ( A ) nazyvatymemo rozv’qzkom rivnqnnq
(2) na ( 0, ∞ ) , qkwo vona syl\no neperervno dyferencijovna na ( 0, ∞ ) i tam za-
dovol\nq[ ce rivnqnnq. ZauvaΩymo, wo Ωodni umovy na povedinku y ( t ) v okoli
toçky 0 ne nakladagt\sq. Vzahali kaΩuçy, v toçci nul\ y ( t ) moΩe buty nevy-
znaçenym. V [4] dovedeno take tverdΩennq.
TverdΩennq,2. Qkwo pivhrupa ( )etA
t ≥0 zadovol\nq[ umovu ker { }etA = 0
dlq bud\-qkoho t > 0, to vektor-funkciq y ( t ) [ rozv’qzkom rivnqnnq (2) na
( 0, ∞ ) todi i til\ky todi, koly vona zobraΩu[t\sq u vyhlqdi
y ( t ) = e y e F s dstA
t
t s Aˆ ( ) ( )0
0
+ ∫ − , y X A0 ∈ −( ).
2. Nexaj A — dovil\nyj operator z E ( X ) . Çerez X A+( ) poznaçymo mnoΩy-
nu vsix cilyx vektoriv operatora A :
X A+( ) =
= x A c c x k A x c k
n
n k k k∈ ∀ > ∃ = > ∀ ∈ = ≤
∈N
N N∩ ∪D ( ) ( , ) { } :α α α0 0 00 .
U prostori X A+( ) vvedemo topolohig proektyvno] hranyci banaxovyx pro-
storiv
X Aα( ) = x A c c x k A x c k
n
n k k k∈ ∃ = > ∀ ∈ ≤
∈N
N∩ D ( ) ( ) :0 0 α , α > 0,
z normog
x X Aα ( ) = sup
k
k
k k
A x
k∈N0
α
.
OtΩe,
X A+( ) = proj lim
t
X A
→0
α( ).
Qkwo operator A obmeΩenyj, to X A+( ) = X .
U vypadku, koly
X = L a b2(( , )) , – ∞ < a < b < ∞ , A x ( t ) =
dx t
dt
( )
, D ( A ) = W a b2
1([ , ]) ,
prostir X A+( ) zbiha[t\sq z mnoΩynog vsix cilyx funkcij, qka [ wil\nog v X .
Ostann[, vzahali kaΩuçy, moΩe ne vykonuvatys\ dlq dovil\noho A ; napryklad,
dlq zvuΩennq A0 zaznaçenoho vywe operatora A na D ( A0 ) = { ( )x A∈D
x b( ) }= 0 X A+( )0 = { }0 . ZauvaΩymo, wo A0 [ heneratorom pivhrupy styskiv.
Prote dlq heneratora analityçno] C0 -pivhrupy ma[ misce take tverdΩennq
(dyv. [5]).
TverdΩennq,3. Qkwo pivhrupa ( )etA
t ≥0 [ analityçnog, to mnoΩyna
X A+( ) vsix cilyx vektoriv operatora A [ wil\nog v X . Operator-funkciq
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 5
TEOREMA FRAHMENA – LINDEL|OFA DLQ ROZV’QZKIV … 653
exp ( )zA : = z
k
A
k
k
k !=
∞
∑
0
[ cilog v prostori X A+( ) . Sim’q (exp ( ))zA z∈C
utvorg[ hrupu linijnyx nepe-
rervnyx operatoriv v X A+( ), i qkwo t ∈R1, a x X∈ + , to exp ( )t A x = e xtA
pry t ≥ 0 i exp ( )t A x = ( )e xtA− −1
pry t < 0.
U prostori X A+( ) vvedemo cilu operator-funkcig
sinh ( )zA
A
= z
k
A
k
k
k
2 1
2
0 2 1
+
=
∞
+∑ ( )!
.
NevaΩko pereviryty, wo
sinh ( )zA
A
=
0
2
z
z s A ds∫ −exp(( ) ) . (3)
V [5] dovedeno nastupne tverdΩennq.
TverdΩennq,4. Nexaj – A — henerator analityçno] C 0 -pivhrupy v X .
Dlq toho wob vektor-funkciq y ( t ) bula rozv’qzkom rivnqnnq (2) z F t( ) ≡ 0,
neobxidno i dostatn\o, wob
y ( t ) = exp ( )At y0 , y X A0 ∈ +( ).
3. Nexaj teper B — pozytyvnyj operator v X . Ce oznaça[, wo B E X∈ ( ),
( , ] ( )− ∞ ⊂0 ρ B i isnu[ stala M > 0 taka, wo
∀ ≥λ 0 : RB( )− λ ≤ M
1 + λ
. (4)
Qk pokazano v [6],
∃ >γ 0 ∃ ∈θ π[ , )0 : Σγ θ, = z z∈ − ≤{ }C : arg( )γ θ ⊇ σ ( B ) ,
a zovni sektora Σ ′ ′γ θ, ( , )0 < ′ < < ′ <γ γ θ θ π dlq rezol\venty operatora B vy-
konu[t\sq ocinka (4) z konstantog M = M ′ ′γ θ, . Velyçyny γ = γ ( B ) = inf :{�λ
λ σ∈ ( )}B i θ θ= ( )B nazyvatymemo verßynog i pivkutom operatora B .
Dlq operatora B vyznaçeno stepeni B
α, 0 ≤ α ≤ 1, i γ α( )B = ( ( ))γ αB ,
θ α( )B = αθ( )B . Zvidsy vyplyva[, wo operator A B= − 1 2/
heneru[ obmeΩenu
analityçnu C0 --pivhrupu z kutom
π θ−
2
, typ qko]
ω = ω ( A ) : = lim
ln
t
tAe
t→∞
= – γ ( )B ,
tobto
∀ >ε 0 ∃ >cε 0 : etA ≤ c e t
ε
ω ε( )+ . (5)
Teorema,1. Nexaj B — pozytyvnyj operator v X z verßynog γ i
pivkutom θ , a y ( t ) — rozv’qzok rivnqnnq (1) na ( 0, ∞ ) , qkyj pry t ≥ t0 > 0
( t0 fiksovane) zadovol\nq[ umovu
∃ >a 0 ∃ >ca 0 : y t( ) ≤ c ea
atβ
, t t∈ ∞( , )0 , (6)
de β < π
π θ+
. Todi dlq dovil\noho ε > 0 isnu[ stala cε > 0 taka, wo
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 5
654 M. L. HORBAÇUK
y t( ) ≤ c e t
ε
ω ε( )+ pry t ≥ t0 > 0, (7)
de ω ω= ( )A — typ pivhrupy ( )etA
t ≥0 .
Dovedemo spoçatku dvi lemy, qki magt\ i samostijnyj interes.
Lema,1. Vektor-funkciq y ( t ) [ rozv’qzkom rivnqnnq (1) na ( 0, ∞ ) todi i
til\ky todi, koly ]] moΩna podaty u vyhlqdi
y ( t ) = e f e ftA tAˆ
1 2+ − , f X A1 ∈ −( ), f X A2 ∈ +( ) , (8)
de A = – B1 2/
.
Dovedennq. Nexaj y ( t ) — rozv’qzok rivnqnnq (1) na ( 0, ∞ ) . Oskil\ky
A B2 = , to rivnqnnq (1) moΩna zapysaty qk
d
dt
A d
dt
A y t+
−
( ) = 0.
Poklademo z ( t ) = d
dt
A y t−
( ). Zrozumilo, wo vektor-funkciq z ( t ) naleΩyt\
do C X1 0(( , ), )∞ i [ rozv’qzkom rivnqnnq
dz t
dt
( )
= – Az t( ), t ∈ ∞( , )0 .
Vraxovugçy, wo operator A = – ( – A ) heneru[ obmeΩenu analityçnu C0 -piv-
hrupu, i tverdΩennq=4, pryxodymo do vysnovku, wo
z ( t ) = exp ( )− t A f , f X A∈ +( ),
a tomu vektor-funkciq y ( t ) na ( 0, ∞ ) zadovol\nq[ rivnqnnq (2), v qkomu F ( t ) =
= exp ( )− t A f — cila vektor-funkciq. Todi tverdΩennq=2 i formula (3) zumov-
lggt\ rivnist\
y ( t ) = e g e e f dstA
t
t s A s Aˆ ( )+ ∫ − −
0
= e g
t A
A
ft sinh ( )+ , (9)
de g X A∈ −( ), f X A∈ +( ).
Poklademo f g f1 2= + , f A f2
1= − . Oskil\ky 0 ∈ρ( )A , to A X A−
+
1 ( ) =
= X A+( ), a otΩe, z (9) oderΩu[mo zobraΩennq (8). Bezposeredn\og perevirkog
nevaΩko perekonatysq, wo vektor-funkciq vyhlqdu (8) [ rozv’qzkom rivnqnnq
(1) na ( 0, ∞ ) , wo j zaverßu[ dovedennq lemy.
Lema,2. Nexaj B — pozytyvnyj operator v X z pivkutom θ , A = – B1 2/
,
x X A∈ +( ). Qkwo isnu[ stala a > 0 taka, wo
e xtA− ≤ c ea
atβ
, t ∈ ( 0, ∞ ) ,
de β < π
π θ+
, 0 < ca = const, to x = 0.
Dovedennq. Zafiksu[mo dovil\ne t0 > 0. Iz hrupovo] vlastyvosti e t A−
u
prostori X A+( ) i tverdΩennq=3 vyplyva[
y ( t ) = e xt A− = e y tt t A( ) ( )0
0
− , t ∈ [ 0, t0 ] ,
zvidky
y tn( )( ) = A e y tn t t A( ) ( )0
0
− .
Todi, zhidno z tverdΩennqm=1,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 5
TEOREMA FRAHMENA – LINDEL|OFA DLQ ROZV’QZKIV … 655
y tn( )( ) = A e y tn t t A( ) ( )0
0
− ≤ c n t t y tn n n( ) ( )0 0− − .
Pokladagçy t = 0, t0 = n1/β, pryxodymo do spivvidnoßennq
A xn
0 = y n( )( )0 ≤ c n c e nn n
a
an n− /β = c c e na
n an n( / )1 1− β .
Cq ocinka pokazu[, wo vektor-funkciq
h ( z ) = ( )I zA x− −1 = z A xk k
k =
∞
∑
0
[ cilog. Porqdok ]] rostu
ρ : = lim
n nn
n n
A x→∞ −( )
ln
ln
1 ≤ lim
n n
n n
n→∞ −
ln
ln ( / )1 1β =
β
β1 −
≤
π
π θ
π
π θ
+
−
+
1
= π
θ
.
Oskil\ky A heneru[ obmeΩenu analityçnu C0 -pivhrupu z kutom
π θ−
2
, joho
rezol\venta R zA( ) [ analityçnog v sektori Σπ θ− /2 , i qkwo 0 < ε < 2π θ− ,
to
h z( ) = z A z I x− − −−1 1 1( ) ≤ Mε pry z ∈ − +Σπ θ ε( )/2 . (10)
Beruçy do uvahy, wo porqdok rostu ρ vektor-funkci] h ( z ) menßyj za π θ/ ,
vybyra[mo ε π θ∈ −( , )0 2 tak, wob vykonuvalas\ nerivnist\ ρ π
θ ε
<
+
. Todi z
(10) vyplyva[, wo cila vektor-funkciq h ( z ) , porqdok qko] ρ π
θ ε
<
+
, [ obme-
Ωenog na storonax kuta z z∈ − ≤ +{ }C : arg( ) ( ) /θ ε 2 . Za teoremog Frahmena
– Lindel\ofa (dyv. [7, c. 69]) h z( ) < Mε vseredyni c\oho kuta, a nerivnist\
(10) obumovlg[ obmeΩenist\ h z( ) u vsij kompleksnij plowyni. Za teoremog
Liuvillq
h ( z ) = ( )I zA x− −1 ≡ x1 ∈ D ( A ) ,
tobto x x zAx= −1 1, a ce moΩlyvo lyße pry Ax1 0= . Vraxovugçy, wo
0 ∈ρ( )A , pryxodymo do vysnovku, wo x1 0= , a otΩe, x0 0= .
Lemu dovedeno.
Dovedennq teoremy,1. Nexaj y ( t ) — rozv’qzok rivnqnnq (1) na ( 0, ∞ ) . Za
lemog=1 y ( t ) moΩna zobrazyty u vyhlqdi (8). Iz vlastyvostej=1 i 3 pivhrupy
( )
ˆ
etA
t ≥0 i nerivnosti (5) vyplyva[, wo dlq dovil\nyx t ≥ t0 > 0 i ε ∈ ( 0, ω )
e ftÂ
1 = e e ft t A t A( ) ˆ− 0 0
1 ≤ c e e ft t t A
ε
ω ε( )( ) ˆ+ − 0 0
1 ≤ �c e t
ε
ω ε( )+ , (11)
de �cε = e e ft t A− +( ) ˆω ε 0 0
1 . Todi nerivnosti (6) ta (11) zumovlggt\ pry t ≥ t0 >
> 0 ocinku
e ft A−
2 ≤ �c ea
atβ
.
Zhidno z lemog=2, e ft A− ≡2 0, a otΩe, y t e ft A( )
ˆ
= 1 i zavdqky (11) spravdΩu[t\-
sq ocinka (7), wo j potribno bulo dovesty.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 5
656 M. L. HORBAÇUK
U vypadku, koly X = � — hil\bertiv prostir, a B — normal\nyj pozytyv-
nyj operator v � , teorema=1 dopuska[ utoçnennq.
Teorema,2. Nexaj B — pozytyvnyj normal\nyj operator v � z verßynog
γ i pivkutom θ . Qkwo rozv’qzok y ( t ) rivnqnnq (1) na ( 0, ∞ ) pry t ≥ 1 za -
dovol\nq[ umovu
y t( ) ≤ c e t
ε
γ ε( )−
(12)
z deqkymy cε > 0 i ε γ∈( , )0 , to
∀ ≥t 1 : y t( ) ≤ ce t− γ . (13)
Dovedennq. Qk i v teoremi=1, moΩna pokazaty, wo ocinky (5) i (12) dlq roz-
v’qzku y ( t ) zumovlggt\ nerivnist\
e ft A−
2 ≤ �c e t
ε
γ ε( )− . (14)
Poznaçymo çerez E ( λ ) spektral\nu miru operatora –=A . Todi nerivnist\ (14)
ekvivalentna spivvidnoßenng
e e ft t A− − −2
2
2( )γ ε =
σ
γ λ ε λ
( )
( Re ) ( ( ) , )
−
− − −∫
A
te d E f f2
2 2
( ( , )⋅ ⋅ — skalqrnyj dobutok v � ). Oskil\ky dlq λ ∈ σ( )− A e t− − −2( Re )γ λ ε →
→ ∞ pry t → ∞ , to za lemog Fatu pro hranyçnyj perexid pid znakom in-
tehrala v nerivnostqx ( ( ) , )E f f∆ 2 2 = 0 dlq dovil\no] borel\ovo] mnoΩyny
∆ ∈ −σ ( )A , a tomu j f2 0= . Vraxovugçy, wo dlq f ∈�
e ftA =
σ
λ λ
( )
Re ( ˜ ( ) , )
A
te d E f f∫ ≤ e ft− γ 2
(
˜ ( )E λ — spektral\na mira operatora A ), dlq y t e ftA( ) = 1, f X A1 ∈ −( ), na
osnovi rivnosti e ftÂ
1 = e e ft t A t A( ) ˆ− 0 0
1, de e f Xt A0
1
ˆ
∈ , oderΩu[mo (13).
Teoremu dovedeno.
4. Proilgstru[mo navedeni vywe rezul\taty na prykladi krajovo] zadaçi
∂
∂
2
2
u x t
t
( , )
= ( )
( , )− ∂
∂
1
2
2
m
m
m
u x t
x
, m ∈N ,
(15)
u tk( )( , )2 0 = u t lk( )( , )2 = 0, k = 0, 1, … , m – 1,
u smuzi t ∈ ( 0, ∞ ) , x ∈ [ 0, l ] , 0 < l < ∞ .
Poklademo
X = L l2 0(( , )), ( B f ) ( x ) = ( ) ( )( ) ( )−1 2 2m mf x ,
D ( B ) = f W l f f l k mm k k∈ = = = … −{ }2
2 2 20 0 0 0 1 1([ , ]) : ( ) ( ) , , , ,( ) ( ) .
Operator B [ samosprqΩenym i dodatno vyznaçenym, a tomu pozytyvnym.
Joho spektr σ ( B ) = σp ( B ) = { / }( )λ πk
m
kl= ∈
2
N dyskretnyj i prostyj, γ ( B ) =
= ( )/π l m2 . Z teoremy=2 todi ma[mo takyj naslidok: qkwo rozv’qzok u ( t , x )
zadaçi (15) pry t ≥ 1 zadovol\nq[ umovu
∃ >ε 0 :
0
2
l
u t x dx∫ ( , ) ≤ c e l tm
ε
π ε2(( / ) )− ,
to
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 5
TEOREMA FRAHMENA – LINDEL|OFA DLQ ROZV’QZKIV … 657
0
2
l
u t x dx∫ ( , ) ≤ ce l tm−2( / )π , t ≥ 1, 0 < c = const.
Analohiçne tverdΩennq moΩna otrymaty, qkwo prostir L 2 ( [ 0, l ] ) zaminyty
na Lp ( [ 0, l ] ) ( p > 1 ) , abo C ( [ 0, l ] ) .
Pry m = 1 rozv’qzky zadaçi (15) [ harmoniçnymy funkciqmy v pivsmuzi G =
= ( 0, ∞ ) × [ 0, l ] , wo zadovol\nqgt\ umovu u ( t, 0 ) = u ( t, l ) = 0. Takym çynom,
pry m = 1 oderΩu[mo zhadanyj u vstupi rezul\tat.
ZauvaΩymo takoΩ, wo v zadaçi (15) zamist\ operatora ( )− ∂
∂
1
2
2
m
m
mx
moΩna
vzqty eliptyçnyj dyferencial\nyj operator v obmeΩenij oblasti.
5. Teoremy=1 i 2 vtraçagt\ pravyl\nist\, qkwo zamist\ umovy pozytyvnosti
dlq operatora B vymahaty lyße joho slabku pozytyvnist\ (dyv. [6]), tobto za-
minyty (4) na umovu
∀ >λ 0 =: RB(– )λ ≤ M
λ
z deqkog stalog M > 0. Qkwo 0 ∈σc B( ), to rozv’qzky, wo zadovol\nqgt\
ocinku (6) ((12)) v teoremi=1 (teoremi=2), prqmugt\ do nulq na neskinçennosti,
ale porqdok c\oho prqmuvannq moΩe buty qkym zavhodno.
1. Lax P. D. A Fragmen – Lindelöf theorem in harmonic analysis and its applications to some
questions in the theory of elliptic equations // Communs Pure and Appl. Math. – 1957. – 10 . –
P. 361 – 389.
2. Pazy A. Semigroups of linear operators and applications to partial differential equations. – New
York etc.: Springer, 1983. – 300 p.
3. Kantorovyç L. V., Akylov H. P. Funkcyonal\n¥j analyz v normyrovann¥x prostranstvax. –
M.: Fyzmathyz, 1959. – 684 s.
4. Horbaçuk M. L., Horbaçuk V. I. Pro odne uzahal\nennq evolgcijnoho kryterig Berezans\ko-
ho samosprqΩenosti operatora // Ukr. mat. Ωurn. – 2000. – 52, # 5. – S.=608 – 615.
5. Gorbachuk V. I., Gorbachuk M. L. Boundary value problems for operator differential equations. –
Dordrecht etc.: Kluwer Acad. Publ., 1991. – 347 p.
6. Krejn S. H. Lynejn¥e dyfferencyal\n¥e uravnenyq v banaxovom prostranstve. – M.: Nau-
ka, 1967. – 464 s.
7. Levyn B. Q. Raspredelenye kornej cel¥x funkcyj. – M.: Hostexteoryzdat, 1956. – 632 s.
OderΩano 27.02.2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 5
|
| id | umjimathkievua-article-3336 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:40:37Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/30/b14f351e7d3bc01b985a0e22671e1730.pdf |
| spelling | umjimathkievua-article-33362020-03-18T19:51:39Z Phragmén-Lindelöf theorem for solutions of elliptic differential equations in a banach space Теорема Фрагмена – Ліндельофа для розв'язків еліптичних диференціальних рівнянь у банаховому просторі Gorbachuk, M. L. Горбачук, М. Л. For a second-order elliptic differential equation considered on a semiaxis in a Banach space, we show that if the order of growth of its solution at infinity is not higher than the exponential order, then this solution tends exponentially to zero at infinity. Для дифференциального уравнения второго порядка эллиптического типа на полуоси в банаховом пространстве показано, что если порядок роста на бесконечности его решения не выше экспоненциального, то это решение экспоненциально стремится к нулю на бесконечности. Institute of Mathematics, NAS of Ukraine 2007-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3336 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 5 (2007); 650–657 Український математичний журнал; Том 59 № 5 (2007); 650–657 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3336/3416 https://umj.imath.kiev.ua/index.php/umj/article/view/3336/3417 Copyright (c) 2007 Gorbachuk M. L. |
| spellingShingle | Gorbachuk, M. L. Горбачук, М. Л. Phragmén-Lindelöf theorem for solutions of elliptic differential equations in a banach space |
| title | Phragmén-Lindelöf theorem for solutions of elliptic differential equations in a banach space |
| title_alt | Теорема Фрагмена – Ліндельофа для розв'язків еліптичних диференціальних рівнянь у банаховому просторі |
| title_full | Phragmén-Lindelöf theorem for solutions of elliptic differential equations in a banach space |
| title_fullStr | Phragmén-Lindelöf theorem for solutions of elliptic differential equations in a banach space |
| title_full_unstemmed | Phragmén-Lindelöf theorem for solutions of elliptic differential equations in a banach space |
| title_short | Phragmén-Lindelöf theorem for solutions of elliptic differential equations in a banach space |
| title_sort | phragmén-lindelöf theorem for solutions of elliptic differential equations in a banach space |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3336 |
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