Решение краевой задачи Шварца для круга с разрезами

(1)

P ro b le m y M a te m a ty c zn e

11 (1989), 101-107

P e m e i m e i c p a e B o i i I I l B a p r ę a

^ j i j i K p y r a c p a 3 p e 3 a M H

J I . H . n o M e p a H i i e B a

K p a e B a n 3a/raHa IIlB a p u a Ann MHorocBH3HOM oS jia cT H T ecn o cBH3aHa c KJiac- cMMecKow 3aaaM eii /XnpMXJie m c to h k h 3peHnn cymecTBOBaHHH pem enuH M3y - neHa flocTaTOMHO noupoSHO. 4>aKTMHecKoe nocTpoeriM e ee p em en n n ocuoBaHO Ha npMMerieHHH T e o p n n H H TerpajibH bix ypaBHeHHM 4> p earojib M a u n o x o M y n p eu - CTaBJineT 3nann TejibH bie BbiHHCJiHTejibHbie TpyAHOCTM. B CBH3H c 3th m npe/r- cTaBJineT M iiTepec naxo>KAeHMe MHorocBH3Hbix o G jia c T e ii H acTH oro BH/ra, a jih KOTopbix p e m e im e K p a eB oił 3a a a n n m oikho nocTpoM Tb 6 e 3 n c n o jib 30BaHMH T eo- puM H H TerpajibH bix ypaBHeiiMM.

M 3n ecT iib i TaKire KJiaccbi o S jia c T e ii, nanpj-iMep, n n ocK ocT b c p a 3p e 3aMH, jiemamMMM n a oahom ripnMOM MJiM onpy>KHOCTn; aBTopoM pernena 3a a a n a U lB a - pu a aJifl nJiocKocTM c p a 3p e 3aMM n a A B yx B3auMH0 nepnenAH K yjm pH bix npbi- aMbix. H a m ia ri p a S o T a nocBH m ena pem eHHio 3aAa.HH IIlB a p u a a a h oS jiacT H , KOTopan n p eA C T a sjin eT c o 6 o ił n p y r c CHMeTpHHHbiMH p a 3p e 3aMH Ha A B yx B3a- hmho nepn eiiA H K yjin pH bix npnMbix.

KoH CTpyKTM Biioe pem eHHe 3a/łanu IIlB a p u a a jih TaK oil oS n acT H ocHOBano Ha n ocT p oen H H B cnoM oraTejibu oM puManoBOH noBepxHOCTH h nocJieuyioiuM M BbipasceHHH peiueiiHH 3a.AaHH H e p e 3 ocHOBHbie (JiynKUHOHajibi nocTpoeHHOH n o- BepXHOCTH.

I ly c T b D — ( 4

n

+ 1 ) -CBjraaHatf o G jiacT b , n ojiyM en n a a y^ a jien w eM M3

em-HMMHOrO K p y r a Iip.HMOJIMHeMHbIX OTpe3KOB

(2)

102

JI.M.

IloMepaimeBa

Ily c T b L -rp a H H u a o6;iacT n

D. CTaBMTCH

3

a^aMa o KoncTpyKTMBiioM nocTpoeiiMH onepaTopa IIlB a p u a u jih 06-

jracTM

D,

to ecTb onepaTopa, naiom ero pemenue cjienyiomeM KpaeBoii

3

auaMM

R eF ( t ) = c ( t ) , t e L,

ru e

c ( t ) -

3auauHaii neMCTBMTejibHafl MHTerpupyeMaB (tiyHKUH, .F (z)H eH 3 B e

-CTHau anajiMTUHecKau b

D

(JiyHKUHH.

O nepaTop IIlB a p u a

b e jiu c t c h

HiiTerpajibHbiM:

r n e - u p o u 3BOjibiiau ueMCTBUTejibiiau n ocTou iin au .

B o n p o c o nocTpoeiiMH onepaTopa IIlB a p u a paBuocujieH B onpocy o nocTpo-

eiin n u u p a IIlB a p u a ^ ( z ;

t)clt.

Ily c T b

S ( z ] t)dt

uupo IIlB a p u a b tom uacTHOM

c jiy u a e , K orna cym ecTByeT ouuo3HaMiioe peuieuue

3

au au n ; onepaTop IIlB a p u a

b 3Tom c jiy u a e ua3biBaeTcu ouuo3HaMHOM uacTbio onepaTopa IIlB a p u a . P a c-

cmotpmm BcnoM oraTejibHyio o6jiac'rb

D

1, n o jiyn eiiH yio yuajieHueM H

3

njiocKO-

CTH

0

Tpe

3

K

0

B

Ily c T b

S®(z',t)dt

ouuo3H auiiau uacTb u u p a IIlB a p u a

ana

o6jiacTH

D\.

H c rio jib

3

yu npMuuun cwMMeTpuu, nocTpouM

3

aMKuy'ryio puMaHOBy noBepx-

HocTb /?, na KOTopyio mo*ho aHaJiMTUMecKU npouoji>KMTb uupo IIlB a p u a

S °

( z ;

t)dt.

H jih

3Toro

ueTbipe 3K3eMUJiflpa

oGjiacTn

D\

ckjiohm no

cjien yiom eM y

3a-Kouy: B/iojib KoiiTypoB

[(

—

l ) (a[. ^

;

(

— 1

)*&][. ^ ]

;

l

= 1 , 2 cKJieuBaHue

npo-M3B0UMTCH 110 3aKOIIbI nOUCTailOBKU

(

1 2 3 4

\

1 2 1 4 3

J

B/ioJib KOUTypoB [ ( —

1

) ^ .

2

; ( —

l ) ldki], l

=

1 , 2

no

3

aKouy

n O U C T a llO B K M

Cna6>Kan noBepxnocTb ecTecTBemiou KOHOopMHon CTpyKTypon, nojiyuaeM

opueiiTMpyeMyio puM auoBy noBepxnocTb

R. Ila n u e n o aJire6panuecK o e y p a -

B iiueiine nonepxHOCTU

R.

(3)

Pemenne KpaeBoii 3a^aMM IIlBapua

103

rae

tti = 7r1( z ) - I I

( z 2 - al ) ( z 2 ~ b2

k) ( z 2 - ak2) ( z 2 - bk2)

k

7

r

2

=

7

r2( z ) = n

( - 2

+ CfcK

^ 2

+ d l ) ( z 2 + c k 2) { z 2 + dk 2)

k

3H an ypaBHeHHe noB epxnocTM , motkho n o c jp o u T b 6a3nc a5 ejieB b ix 7in(J)(&epeH- u n ajioB n e p B o ro p o u a , ,ueHCTBHTe.nbHbix n a rp a n n u e o6jiacTM

D

(3aMeTHM, mto a n a jin T n u e c K n e b oS -n a cin

D

n

aePic

TBMTejibHbie na rp a n n u e o6 jiacT M

D

a6 e;ieB b i uncJjiiepeHUMajibi riep B o ro p o u a coBna,naioT c ^encTBHTejibiibiMM n a rpaH H ue o 6 jia c T H

D

aSeJieBbiiun ancJ)(l)epenn.HajiaMn n e p B o ro p o a a noBepxHO- c t m R ) :

(

2

)

dxm( t )

—

t 2k~l dt

( x 2 - P

2 + P

1 )

=

1,

.

..,4 7 1 - 1 )

x t 2kdt

( X2 - P

2 + P

1 )

4

n,

. . . , 6 n — 1)

t 2k

( P i + P

2 )

( x 2 — P 2

+

P\)x

k

= 0 , . . . ,

2n —

1;

m

= 6 n , . . . ,

8n

— 1 ) r a e 7T i , 7T2 3 a .a a H b i < J )o p M y jia M n ( 1 ) P i = TTl ( t ) ' , P2 = 7T2( ł ) .

.fljia t o t o , MToSbi n o c T p o n T b Saanc a 6 e jie o b ix unciHlJepeHunajioB, aeHCTBHTejib- h h x Ha rp a H n u e o6jia.CTM

D ,

n au o no/ioSpaTb Taune KONiGmiaunn 9 JieMeHTOB 6a3Mca ( 2 ) , KOTopbie uenuTBHTejibHbi ne to jiu k o n a r p a n n u e o 6 jia c T n

D\,

h o h n a eunnnMHon OKpys+cnocTH. H jih s t o t o bm.hchhm, Kanne ajieMeHTbi 6 a 3 n ca (2 ) b CHMMeTpnuHbix OTnocnTeJibHo onpy>KHOCTn TOMKax npnnnM aiOT conpn>KeH- Hbie 3nauennH n c tp o h m 6a3nc a6eJieBbix un(J)(t)epenunajioB, uencTBnTeJibHbix Ha

L:

( 3 )

dVk = dxm

+

d xi

;

l

= 4

n — m;

m

= 1 , . . . ,

2n

;

k

= 1 , . . . , 2 n

(4)

104 JI.I4. floNiepanueBa

dVk = dxm + dxi ;

1 = 12n — m — 1;

m = 4 n , . . . , 6n — 1;

fc = 2n + 1 , . . . , \n

r\ge

dxm

3a ;ia n M (JiopM yjiaM M ( 2 ) .

T e n e p b m o jk h o n p u c T y n u T b k n o c T p o e H H io s i r p a I I l B a p u a u jih o 6 u a c T U D . iI ,n p o I I l B a p u a S ® ( z ' , t ) d t a j i h B c n o M o r a T e jib H o ii o 6 jia c T n D H M e eT b u u [8 ]:

fjo,

w _ K

- ^T 2)^ +

z f a w i

- £ i g r )

^

b 1 [ z , t ) d t — a i , ( x ; - P 2 + Ą ) ( * 2 - .z2) ru e (JjyHKuuu i ° + y/ ^ 2°> x l = * 3/ ie cb + \ / 7r i 0 ; 4 " \ / ^ 2° ; + v / Ą ° ; \ / ^ ° T e (J)MKcupoBaHHbie BeTBH K o p n e u , K O T o p b ie n p u Z\ t --- > OO M M dO T p a 3JIOH<eHMU 4 n i . -,4n , . / n j 4n , . / n j 4 n

¥ T ~ ^ 47l+ ...;

v p i ~

V^2 ~ * 4n+ ...

H c x o u u M3 n a p a IIlB a p u a

a n n

o 6jiacT n

D \

nocTpouM nap o

S ° ( z ] t ) d t , koto-

poe o 6aaaaeT cneayioiuHMM cBoiłcTBaMu:

1) S °( z' ,t )d t ~ 441^ npH t --- > ^

2)

n p u t

^

z ,

i / 0,

OyuKuuu 5 ° ( z ; t ) d t aH anuTuuecKH 3aBucuT OT Z U OT t B 0 6jiaCTM D .

3 ) S ° ( z ; t ) d t — 0 , e c jiu z

E

L , t

E

L ;

4 ) n p u 2 = 0 $yHKuuu HMeeT n o jiro c , npuueM Bce KoattKjmueHTbi rjiaB H ou uacTU p a 3Jio»teHHfi b OKpecTiiocTU u y jiu u b jih io t c h JiHHeuHbiMU ko m6h

-

HauHHMH ajieMeiiTOB 6 a3u c a

(3).

BBeueM (J)yHKuuu:

S2(z\t)dt = S ° ( z ~ 1; w i ( z ~ 1) ; t ~ 1; x ( t ~ 1) ) d t ~ 1

= S ° { z ~ 1] w 2{ z ~ 1); r 1; x 2( r 1) ) f / r 1;

S °( z ; t)dt = S ° ( z ~ 1; w 2( z ~ 1) - t - , x 1( t ) ) d ł ]

S ° ( z ; t ) d t = S° (z; w2( z ) ] ź-1 ; x ( i ~ 1) ) d t ~ 1 —

= S ° ( z ; w i { z _1); t ~ 1; x 2{ t ~ 1) ) d ł ~ 1;

(5)

PemeiiMe KpaeuoM 3a/iaHn LLlDapua

105

r\ae w k (I)MKcnpoBannbie b c tb w (J>yiiKnnii w ( z )

Wi(z) --

jy t F T 0 +

;

u;3 =

i y / x

- y/tt^0

w

2

{z

) =

- i^/ Wf

+

;

w

4

= - i

y / F ^ - 0 ^ °

x k -

wk( t ) ] t e L

To r\ aa cyM M a i 4 n ( 4 ) S ° P ; 0 * = ( r ) £ S°k( r , t ) d t

k = l

o 6 jia .aae T cnoHCTsaMM 1 ) - 4 ) , yKa3anubiMH Bbime u, c jie ^ o B a T e j ib n o , BBJi.aeTC.fi oano3iiaMiioM MacTbio fu ip a I l l a a p u a j i ji h o C j u i c t w D . T o ecTb (JjynKUHfl F

\z) = ^ - [ s ° ( z - t ) c ( t ) d t + i p

Ztt i Jl p e m a e T 3 a i i a M y IIlB a p u a /ijtji o S j i a c T M D b n p e ^ n o j i o > K e i i M H , m t o c y m e c T B y e T o i i u o 3 H a M i i o e p e m e n n e 3 a j i a H H , t o e c rr b B b i r i o j i n n i o T C . f i y cJ T O B M fi ( 5 ) j c ( t ) d v k( t ) = 0; k = 1 , . . . , 4rc,

/

L

r/ie

dvk

/in(f)(|)epeimMajibi 6a3nca (

3

).

ZIjih nocTpoeiiMH onepaTopa IIlB a p u a

b

cjiyM ae oTcyrcTBMfi o.ano

3

iiaHHoro

peiueiiMH paccMoTpuM BM^0M3Meiiemiyio

3

aaaMy IIlB a p u a :

(6 )

Re<j>{t)

=

c{t) - Y ^c tkFi e ( idv

k ,

k—l J c d

me dvk

.aHdHjiepeimnajibi

6

a

3

n ca (

3

).

a t - n o c T O J i i u i b i e ; ie M i j . T B M T e j i b i i b i e H H C J ia , n y T b m i T e r p n p o B a n n . f i r i o j n i o c T b i o n a - XO.ZIHTC.fi B o S j i a C T H D .

FIoaSepeM cv^. TaKMM o

6

p a

3 0

M, h to 6 w Bbinojiiifuiocb ycjiOBHe (

5

):

/ ( c( 0 -

Q^ e [ idvk)dvi = 0,

/ =

4n.

k=l

cl i

(6)

106 JI.J4. IloMepaHneBa

O n p e a e j i H T e j i b c u c T e M b i O T J iM u e H o t n y j i n ; p e i n a n c u c T e M y c n o c o 6 o M K p a - M e p a . , n a x o / i.M M a k . T o r u a p e i u e H u e 3 a u a M n ( 6 ) o r ip e . a e ji. f ie T C . f i ( t > o p M y j i o u

l r

471 r t

<t>(z) = i (3 + — / ( c ( t ) - Q k R e / i d v k ) S ° ( z ; t ) d t ,

Liri JL

Jc\i

r u e S ° ( z ' , t ) d t Haxo.aMTC.fi no (J)opMyjie ( 4 ) , /9- n p o n 3B ojibH afl ueMCTBMTejibHa.fi nocTOBunafl. T o r u a (JiyHKUHH

4n

F ( z ) = <f)(z) + Y , a k [ i d v k

(—1

Jci'

k=i

pern aeT 3a u a u y IIlB a p u a

/um

o6jiacTM D .

J l n T e p a T y p a

[ 1 ] C n p u H r e p J \ m . B B e u e i m e u T e o p u i o p u M a n o B b i x n o B e p x H o c T e i t , M .M .J I. 1960 [ 2 ] H e 6 o T a p e B I I .r . T e o p u n a j i r e 6 p a u > - ie c K H X (JfynKUMM, r o c T e x u 3 u a T , 1948 [ 3 ] M ycxejiuuiB M Jiu B .I 4 . C M i i r y j i f l p u b i e H H T e r p a J i b H b i e y p a B H e i i M H , M . ” H a -y n a ”

[4]

r a x o B <£./!. K p a e B b i e 3auaHM,

M.

4>M3MaTru3, 1963

[ 5 ] 3 B e p o B u u 3 . M . K p a e B b i e 3auaMU TeopMM anajiMTMuecKMX diyHKUHH, Y M I I 26: 1 ( 1 5 7 ) , 1971

[ 6 ] 3 B e p o B M U 3.14. n o c T p o e n u e b h b h o m B M u e a n a jio r a n a p a K o m u n a pu-M a H O B b ix n o B e p x n o c T f i x H e K O T o p b i x a j i r e 6 p a u u e c K H X (JiyH K U M H , M a T e M .

3a,MeTKM 8: 6, 1970

[ 7 ] A n cen T b eB J I.A . IIo c T p o e n u e o n e p a T o p a IIlB a p u a M eTouoM cuM M eTpuu, T p y u b i c e M H i i a p a n o KpaeBbiM 3 a u a u a M , Bbin. 4, H 3 U . K a3 a H C K o ro y H -T a, 1967

(7)

Pemeuiie KpaeBott 3auaHH IIlBapua

107

[ 8 ] IIo M ep a H u eB a J I.M . O KOHCTpyKTUBHOM nocTpoeiiM H o n e p a T o p a IIlB a p u a

U J iu o u h o h M H o r o c B U 3 a H o i ł oSjiacTM , H 3B. y u e 6 . 3aBeueHMH, M a T e M a - TMKa, 7^ 5 6 9 0 - 7 3 /Ien ., K a 3 a ii b , 1973 H e p K a c c K u u neuuHCTMTyT H e p u a c c b i 2 57000 yji. K . M a p u c a 24 Re c e i ved before 23. 12. 1988