On quasiconformal mappings corresponding to the beltrami equation
By using methods of integral equations, we investigate problems of conformal and quasiconformal mappings of close domains.
Збережено в:
| Дата: | 1999 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
1999
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/4738 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510900539097088 |
|---|---|
| author | Samsoniya, Z. V. Samkharadze, I. G. Самсония, З. В. Самхарадзе, И. Г. Самсония, З. В. Самхарадзе, И. Г. |
| author_facet | Samsoniya, Z. V. Samkharadze, I. G. Самсония, З. В. Самхарадзе, И. Г. Самсония, З. В. Самхарадзе, И. Г. |
| author_sort | Samsoniya, Z. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:12:54Z |
| description | By using methods of integral equations, we investigate problems of conformal and quasiconformal mappings of close domains. |
| first_indexed | 2026-03-24T03:04:21Z |
| format | Article |
| fulltext |
Y~IK 517.54
3. B. CaMcomta, H. F. CaMxapa~ae (H.-T B~qXC~.T. MaTemTrfKt~ AH l'py3HH. TSHJmCH)
O K B A 3 H K O H q ) O P M H b I X O T O B P A ) K E H H . q X ,
C O O T B E T C T B Y I O I I I H X Y P A B H E H H I O B E , / I b T P A M H
By using methods of integral equations, we investigate problems of conformal and quasiconformal
mappings of close domains.
,/loc.rfilI~yloTI,C~t aa~aqi KoHqbopMuoro Ta ana3ixouqbopMHoro Bi/~o6pa~Kem, 6~IHSbKHX o6nacTelt a
~tono~oro~o ~eT0~ia im'erpan~,u~x pi~amm.
B ~aHH0i~ pa6oTe ycnnn~a~oTc~ peayn~TaT~, noYs'~eaa~e aBTopaMn paHee (c~. aa-
He~aHae a xoane CTaT~), H npano~aTCa nx HeKOT0pue npa~eHeuna. CnenyeT
TaK~Ke OT~eTHT~, UTO ~ K~aanxoHqbOpM~0M cny~ae OTO6pa~e~a~ 6~maKnX o6nac-
TePI, no-BH/~n~0My, rlayqalOTC.q BnepBue.
1. MeTotI ~uTerpaJmu~x ypaaueuat~. Fno6az~mafl ro~eo~opqbnaM W(z)
~cea KOMnne~CHOfl rtnocxocTn E ypa~HeHna Be~TpaMn
W~ = q(z) W z (1)
n p e a n o n o ~ e ~ n , ~TO q(z) ~ Lp(E), p > 2, I q(z) I < Q0 < 1, MO>KHO CTpOrlT'o nO
cxeMe, npeRno:~eHHOfl B [1 ], C nOMOttII,~O HHTerpa~,max onepaTopoB
F~[C G -- KOHeqHa~q 06JIaCTb IIJIOCKOCTH E. B ~a~bHeflmCM 5y/leM npe/inonaraTb
q(z) /IOCTaTOqHO r~Ia/IKOfl cl~yllKtll4cR B G-, a BHe G q(z) - O.
Cor~acHo [i] (rn. 2), dpyHKU~
WCz) = z + TGf, (2)
r~e f(z) ~ e~HCTBeHHOe petueHHe nnocKoro c~ry~tpHoro ypas~e~ta
f(z) -q(z) Hof = q(z), (3)
ocymecT~nZeT ro~eo~opdpHoe OTo6pa~Kea~e nnOCKOCTa E Ba ce6a r~ 1~/(**) = 0-,
-I i f (z)--> 1 npn Izl ~ TaKaa qbyHKIIrtJl eRrlHCTBeHHa. Z o o
Ec~m q(z)~ C.t(G') a 0 G ~ C~, 0 < a < 1, r~te C~a~K~aCCKOHTypoB
aanyHo~a, TO, cornacno [2], Wz, I~ e Cro, r~e 0 < 70 < min { a , 7},- B cny~iae
f(z) ~ C~ ( G ) l~oag HGf cupa~e~na, mo npeacra~neHr~e [1]
(Of) 1 :a f(~)d~ (4)
~:'(Z) HaBeCTeH 110/:1; Ha3BaHHeM OCHOBHOFO rOMe0MOpdpn3Ma ypan- r0Me0Mopdpn3M
HOHH~ (1).
H y c n , KO3dpcl)HIIHeHT q ( z ) ypaBHeHI4a SC:IsTpaMH 3a~aH B HCKOTOpOi~ KOneqHOfl
O/:U-IOCB$13HOI'~ 06.rlaCTH G O H llpHHa,IIdlC~d4T Knaccy Cy(Go), 0 < 7< 1, 0 r G O .
PaCCMOTpnM Ha KOMn.rIOKCHOfl n.UOCKOCTri onHocsa3a3no o6~ac'n, G, G ~: G o,
0 r G, orpaHrlqeaaylo KOHTypor.t F = 0 G, F e Cta, 0 < ~ < 1. HMcx s srt~y
CBOt~CTBa petuemt~ ypaBHeHH~I BC.rlr~TpaMH, qbyHKIJ, rllO W = W (Z), W (0) = 0,
W(zl ) = 1, zl E F, peann3ylomy~o KBa3HKOHOpopMHo00T06paic(0I~rle oSnacT~ G Ha
C,/~HHHHI'-II~II~ Kpyr, 6y/IeM rlCKaTb B BH/~C
�9 3. B. CAMCOHHR, Ho F. CAMXAPA~3E, 1999
ISSN 0041.6053: Yrp. ~tam. :~. pn.. 1999, m. 51,1~ 10 1391
1392 3. B. CAMCOHH~q, H. l". CAMXAPA~3E
W= [W(z)-W(O)]e v(z) = [W(z)-VC(0)]exp ! W(-~- '~z) + iC , (5)
r~e dl,~' = ~ d t + (V~d[. W(z) - - OCHOBHOI~ ro~eoMopqbHaM ypannenHa (1)
(q(z) e 0 SHe G'0. OGo e Ca)" /Ieacvawrem.Haa zoHc'raHTa C a UeacraH'rezu~-
.Ha qbyHzUH~ V(t) onpe~eamoTca HCXOna Ha rpe6oaaHHi~, npea'~aaz~eHHuX Z
W(z).
Tara~ o6paaoM npHxo~n~ K crrpane~mmoc~ cne/X3~omero n p e ~ o ~ e n H a [3]:
~pym~ma W=_B:(z), ~a~a~a~ ~ G qbop~ynofl (5), y~o~ae~opaev ypanHenH~
(1) ~ O~XHOnHCTHO OTO6pa~acT G Ha eRHHHqHUfl zpyr c aunonHeHae~ ycnoaHfl
W(O) = O, W(z~ ) = I, ecJm v(t) Sa~SCTCS pcmemle~ miTerpanbHoro ypaBneH~
H
K F
r
(6)
YpaBHeHHO (6) HMeCT CRHHCTBCHHOe pCIIIOHHe V(t), npnHa~nea<amee HeKOT0pO-
My K.rlaccy F e ~ e p a .
B zonqbopm~o~ czvy~ae Ir = z H, ecam
1 ! v ( t ) d t + - i ! v ( t )
V(z) = ~ t - z ~
IIpHXO~HM K KOHCTpyK~HH, rlpc~nO~KeHHO~ S [4].
(p: I,I), (7)
B npeaonaraeMoR pa6oTe MeTO~ H~IOCKIlX H KOHTypHIaX HHTerpa~HHX ypaBHe-
HHH np~CHRCTCn B KaHOCTBO KOHCTpyKTHBHOrO annapaTa B 3a/~&H~ OToSpa~cHHH
6/IH3KHX 0~dlaCTCPL B IIpHBC~eHHOR HH2K0 IIOCTaHOBKC 9TOT MCTO~ MO2KH0 IIpHMC-
HI4Tb KaK B KOH~OpMHOM, ,TaK H B KBa3HKOH~OpMHOM c31yqagX.
2. O 3a/laqe KOH~OpMHoro OTo6pameHHa 6JIH31~X o6~acTefl. Hpe~nono-
~ M , wro yp~HCHHe r p a n . i ~ F aa/Iano B napaMeTpn'4ecKo~ qbopMe
t = g ( ~ ) , 0 < x < 2 n ; g ( 0 ) = g ( 2 ~ ) .
Pacc~ovpHM/~pyrym O~HOCBaaHym o6~acrb (~ ~ n a G Ha KOMnJTeKCHO~ mTOC-
ZOCa~ C ypasHeHHeM rpaumu~ F,
t = ~'('0, o_<z_<2~; ~,(o) = ~,C2~).
Onpe/lememle. 06~acmb C, ~asoae~t e.-8nuz~o~ (~ > O) ~ o6nacmu G .
eC~lU MBO/UtglorllcJl yCAOOU.~
I g ( x ) - ~ ( x ) l -< ~, I lg ' (x) -F(x)IIc= -< e. (8)
~ [ ~ nm6oro ~ > 0, paayMeeTcx, o6pa3yeTca 6eczoHeqHoe HHomec~o o6na~rel~
s-6~m~z~x K G. Tahoe ~moacecrno o6oaaa~HM uepea Gs.
Ht.rrerpa.r~uoe ypaaHeHHe (6) npe/Ic'raaH~ B KoMn~cKCHOM nu41e
1 [ 1 K(t;to)V(t)d t = _lnltol, to~ r , (9) V(to) + ~ t-;
r~c
ISSN 004 i-6053. YKp. ~lam. ~.'vpn,, 1999, m. 51,1~- 10
0 KBA3HKOH~I)OPMHHX OTOEPA)KEHM~IX, COOTBETCTBYIOIHHX... 1393
t~ t -to _ ~
t - t O K(t; to) = t ' - '~ + t_' -'[0 +
t - t O t - t o ~ - i 0 "
t" = dg( 'O �9 ( d g ( x ) ' ~ ., ToE [0,2n].
dx " to = \ dx "/x=xo
AaanorH'e,~o, arrrerpaxn, rIOe ypaanerme ~rta o6zaem (~ EIMOCT
~ / ( t o ) + - ~ i f l K ( t ; t o ) ~ / ( t ) d t = - l n l , o I , , o e I ' , (9')
F
r~e K(t;t0) noJ1yqaeTca aa K( t ; to ) B npearto~o)KeHrIH, q'ro t = ~(X), to= ~,(X0).
CnpaBenmma cJ~e~ylomaa TeopeMa.
TeopeMa 1. Ecnu zpanut~bt odnacmet~ G u G e G 5, 0 < e < ~ , npunaOne-
:ram r.naccy C:z , 0 < cc < 1, mo cnpaeeO:zuao nepaeencmoo
IIv(x)- ~'(x)llc~_~ < Ao(l~)llvilc~_~e, (10)
zOe v ( t ) u ~('c) ~ eauncmeennbte p e m e n u a unmezpanbnt~x y p a s n e n u a (9) u (9')
coomeemcmeenno . K o n c m a n m a Ao( f~) u ~tanoe ~uc,~o e o onpetgenatomca 3aOa-
nue~t ucxoOnoa odnacmu G; ~ ~ ntodoe nono:rumenbnoe ~ucao, ~tenbutee Ct.
Z Imcasameabcmoo . H c n o ~ y a napaMeTp "c, ypaBnenna (9) H (9') npeacTasHM
B BH]~(~
v('Co)+ ~ / ~ [-~;-e-~% ~ ~,ctg 2 x~ + II~v(l:)dx = -lnlg(x~ (11)
1__ ~=S(xo ;x)M(x;:o)(etgX- xo + OleiX _eiX. I' rc(x)dx = - ln]~( '~o)[ , (11')
V(X0) + 2/ti 0 leiZ_e,X, ,ll~ \ 2
rz;e v ( x ) = v [ g ( x ) ] , ~'(x)= ~'[,~(x)].
KpoMe TOrO, -ue~ema c.ne,ay~ottme o6oaHa,~emi.a:
d t _ g'('c)d'c = M(,~0;x)(ctg'~2, o + i ) d ' r
t - t o g(x) - g(Xo)
r~ae diDyHZUH~l
M(ZO ; .r = g'('O eiZ - eiX~
2ie ix g('C) - g('CO) (12)
y/zoa.rteTaop.qeT yCJ'IOBHIO Fem,/Iepa c noKaaaTeJleM a OTHOCHTeJI/aHO CBOHX apry-
MenTos, T. e. npHHazl, neacwr z~accy Ca(F• F) ;
S(zo; ~) = g(x) - g(Xo) K [ , ~ ) ; g(Xo) ],
g'('O
03)
S(xo;X)~ c~(rxr ) , S(xo;Xo) = o,
[3 -- .rm)6oe q.c~to .a (0. ct). IIo~zo6no onpe~extmorca aua,~enHJ~ ~-tx aupa~xenn~
M(~o, x), g(~0, ~).
Hpe~zcTaaaM rmverpa~,mae ypemnemls (I 1) ri (I 1") a oneparopnos~ a ~ e
ISSN 0041-6053. Yrp. ,~eam. ,~.'ypn., 199O. m. 51, bl e 10
1394 3. B. CAMCOHH~I, H. F. CAMXAPA,~3E
Av = v + H v = fo , (14)
�9 4~' = v + /'t~' = fo, (14')
r a e f o = - l n l g ( ' C o ) I, ~o = - I n l ~ , ( x o ) l , A = I + H , A = I + ~I, a H V H f t g - -
mcrcrpan~a~e onepaTopu, sxo~amae B a e s u e q a e m ypaBHCHHfl (11) a (11') COOT-
B~TCTBCHHO.
Pacc~oTpHM aupa~enne
2n
(H- / - t )V(Xo) = 2-~ii![(*( '~o;'C)(ctg'~2"c~ (15)
r/~e
K*(~:O ;X) = S(1:0 ;x)M(Zo ;X) ' S(ZO ;~:)/Q(~:O ;q;)
leiX_ei%l [~
BocnoJIb3yeMcJt C~lel~ylomrlMrl pe3yYlbTaTaMrI H3 [5 ];
a) qbynKRrm ~ayx nepeMeHauX
q~(t) - q~(t0) (t; t o e F) ,
W(t;t~ = I t - ' i0 I~" '
y/~osneT~opJ~eT Ha F ycnoBmo Fenb~epa c noKa3aTeneM B0 - ~,
e. C~o (F) a 0 --< 9~ < I~ o < 1 ; 6once roro , cnpaaeztnnaa oRenKa
Ilw(t; to)llc, o_ x < A*(1 +X) ,
rze A * _> 19(0 - q~(to)l (cM. w 5 H 6 yKaaarInOg pa6oTU);
i t _ t01~.
6) cI~yHKRH~I Bnaa (15)
0~[ t('CO) ] =
(16)
ecsm cp(t) e
2~
1 r .-., . . . . v(x)dx
J / ~ (Xo;.~) r('co ;X) leiX _ el% 11-13 '
0
rze r(x 0 ;q:) = e x p ( - i arg(e ix - eiX~ npHHa~neacHT Knaccy C8 an.q nio6oil
orparaHqenuog qbyaKamI V (X) H 8 = a - 13 (cM. w 51 yKaaarmog pa6OTU).
Herpyzmo y6e~rrnbca a cnpaBe~gmr~oc'rrt cne~yIotrarm nepasenca~ [6]:
ll[g'fx) - g'(Xo)] - < 21z;
ei.~X.-e ixO g'('~) e i x - e l ' q , ~'(Z)[ <
AlE;
IIg(x) - g(xo) 2ie ix - ~(x) - ~(Xo) 2ieiZ [c~
[gCx) - g(Xo) ~'Cx) - ~'('I:0) I < A2E;
(Zo)
rain Ig'(z)l
min Ig'(x)l.
npH 8 <
2
r~e ace coliCp~KattlHec~ rt OROHKaX KOHCTaI-ITbl He aanHca'r OT G.
< .A38,
I~'(x)l > [o.2n]
2
_ g( o)
ISSN 004 ! -6053. Yrp. .warn. ~.'ypu., 1999. m. 51, N e ] 0
0 KBA3HKOH~3PMHMX OTOBPA~EHHflX, COOTBETCTBYIOI~HX... 1395
Y,:rn-uBa.a (12), (13) a pcayHbTaT a) B (16), a TaK~KC npHBC~CHH~V HcpaBeHer~a
Ha [6] npH Ma.rlhlX 3HaHCHHHX napaMexpa 8, y6ea:aaeMc~ s cnpaBemmBOCTH Hopa-
BCHCTBa
I k'(=o:)llc._,
KOTOpOr B CBOIO oqCpc~b, cornacHo pe~yJ-mTaTy 6) Ha (15) npHso~rr K OI~CHKr
[[(H- t)Vlc._ ~ <-Co(P)"v"c._, It. (17)
r~c C0([3 ) , Cl ( [ 3 ) - a6COHmTHUr KOHCTaHTm, a V ( X ) - mo6a~ qbyHxmla Knacca
ca_~.
I'Ipe/~HOHO~KHM tencpb, wro V(~) H V(~) -- petueHH.q ypaBHeHHI~I (11) H (11')
COOTBeTCTBeHHO. Toraa n caay (14), (14") H~ee~
v - 9 = / ~ - ' ( H - / ~ ) v - / t - I (j~ 0 - f0). (18)
Ho CorHacHo (I 6)
IIA-AIIc~_~ = sup I I (A-A)vlIc~ ~ -< sup Co(13)IIvlIc~:~ = Co(13)it
Ilvll= 1 - Ilvll= 1
tt
l t ( .~ -A)A- I I Ic=_~ <_ IIA-AIIc~_~ ItA-~llc=_~ < Co(fJ)llA-111c=_~ It.
EcnH Teneps npeano~or~rrb,,wro It < 1 / (Co(13) IIA -] llc~_p ), TO o6paTnull one-
paTop A-] 6y~eT paBHo~epHo (OT~OCrrren~HO 1 ~ = 3 G ) orpanH~enHu~ B rrpo-
CTpm~cT~e Ca-p ,
IIA-t Ilco.p
II~-tllc=_~ <
1 -I1(/~ - H)A-~IIc=_~ "
HMeJt B sa~y r n oqeBHl~rlylo OI~eHKy ZUI~I Iif o - f0llc~_~, n3 (18) rlCHocpe/~-
CTBeHHO noHyaaeM Hepa~eHCT~O (10), cnpa~r A ~ Bcex it <~o, r/~r
min; min Ig(~:)l. min lg'(x)l 1 l
e0 ] J 2 ' 2 ' Co(~)IIA-~IIc~_~
A0 (13) - - KOHCTaaTa, 3aBHcmuaa TOZ~KO OT G H 13 ( 13 - - mo6or no~oacrrrea~nor
~acno, MeH~mee ~) . TeopeMa ~toxa~a~a.
Ouem~a (10) no3Bon.qeT c nOMOtt0~IO CI:)yHKI/JIH V(t), onp~02lOltHOfl 3a~aI~eM
HCX0~-mfl o6~ac'm G, CTp0HT~ npH6Hm~eHH~ K KOH~Op~o O T o 6 p a ~ x a m ~
~yHKaa~ w = ~ (z ) (~>~(0) = 0, ~ (0 ) > 0) (CH~ay~ [41; C~. 4~op~ar (7))
nporI3Bo.rlbHOfl o6.rIac'l~ G E Ge Ha I w I < 1.
I'IOHO~Km,t
~'(~) = ~'L ~o ~(~)-~ + ;~ ~,(~)i )' (19)
rae v (x) - - peme~me y p a ~ . e m ~ (9):.
(w 18) H n~pa~sc'mo (10), TO: ycra~o,,m~ c n ~ m m m o c ~ c a m ~ m e ~ " ~ .
ISSN 0041-6053, Yr.p. ~mn. .~cypH.. 1999,m. 51, N ~ lO
1396 3. B. CAMCOHH$t, H. F. CAMXAPALI3E
r~e
~(;o) + Re • t " ~g)d; = - l n l ~ o - W o [ ~0~ ~:(r), (20")
~(r = ~[~-'(g(,o))], ~(r ~[~-'(a(,o))], ,o~ ~o,2=],
W o = I ~ ( 0 ) , ~ = ~i:[g(z)] s(20'), ~ = ~/[~(x)] B(20").
Tax.H o6paaoH, M~ pacnoaaracM CacaymmHM. ~a.maHH [2]:
KpoHe TOrO,
C o c r a n ~ pa3H0CTb
_ , . 1), ( ~ ) ( . - _ , _, w[g( ,~) ~ - w[~( ,~) ~ = ~ [ g ( ' ~ ) ] g , - w~[g(~) ] g~ + ~ [ ~ ( , ~ ) ] ~ , -
- ~[~<,~)]z~ + ~[~e~)]g~- ~[z('~)]~ +-~[g('~)] ~~- ~[~('(~] ~;.
Y~a'r~ma.a ~ Hr ~cpa~c~c'r~a �9
, I I z ' ( '~ ) - ~'( '~)l lc~,o_, ~ < ( c o n s 0 1 I Iz ' ( '~) - ~ ' ( ' ~ ) l l c ~ �9
I l c m - Pmll~o_, -- (co.so~ I I ~ ' m - ~'mll~ .
I 1 ~ - " II [g('t)] - Wt[g (I:)] < (const)3 Ig(x) - ~(x)[ n
CTO -q "
r~r 0 <11 < 7o, n caay (8) noay,~acM Hepa~cac'mo
ISSN 0041-6053. Y~cp. ,uam. ~'vptL. 1909, m. 51, N e 10
~r ~(~)~ c~,,[0,2~l, ~[~(~)]~ q,,[o,2~j, ~[~(~)]~ q,,[0,2~].
B cHay (8) ~Me~M
I~g(~)l-~:[~(~)] < Q(W;Go)Ig(x)-~,(x)l <- CI(-I~Z;Go)E. (21)
= ~,[g(z)] g'(,) + ~[g(~)] g'(,),
= #,[~(~)]g(~) + ~ [~c , ) ]~ ' (~ ) ,
TeopeMa 2. @ynKt4u~ w = ~pv(z.) (q)v(0) = 0, q)v(0) > 0), sacgam-ta..t s
d#op~o,.aoa (19), ztge (~ e G~, 0 ,~ ~ < I, 8 G cgonycr.aem o~en~,
I~r, Cz)- ~vCz)l < QI ~,
zae Koncmanma QI zaaucum mo/tbKo o m o&tacmu G .
3. O KBa3HKOH~OpMHOM OTO6paXeHHH 6JIH3KHX o6aacTefl. BocnonbayeHca
IIDHCMOM BI~XO/Ia B rUIOCKOCTb OCHOBHOrO FOMCOH0pCl~H3Ma ~/(Z). B IIJIOCKOCTH
# ( Z ) HHTcrpaJIbHblC ypaBHCHH~I (9) H (9') HpHHHMalOT BH]~
~(~o) + Rel J" l.t(~) d~ = _ In[~o- Wol ~o ~ I~(F), (20')
~z ~,(r) ~ -~o
O KBA3HKOH~OPMHHX OTOBPA~ZKEHH,qX. COOTBETCTBYlOUlMX... 1397
II- . _ w ll - c~,
(22)
r~e ~ ~ = 70 - ~ ; rI ~ ~no6oe ancno ~e~y 0 n 70- OnTHMaJ-a, HI,L~ ~aprmrrro~
]~aHHOM cJ~yqae Hart npe~tcTa~aaeTca rl = T0/2.
Ta~HM 06pazou, ycnO~Ha (8) ~ nnOCKOCTH roMeoMop~Hz~ta l~a0 (Z) aaMena-
mrca y c n o ~ a a ~ (21) vt (22). IIpa~eHaa Tenep~ Teope~y 1, y 6 c ~ a e ~ c a ~ cnpane~t-
nrmOCTn cnelBnomero npeRno)KeHHa.
Teope~a 3. Ecnu od~acmu G ~ Go u G e G ~, G ~ Go, zpanu~bt ~omopt~x
npunacg.ne.,wam ~.aaccy C~, 0 < cc < 1,
ocex s ~ (0; s cnpaoeOauoa o~enxa
s Opye Opyey ~ c~btcne (8), mo 8n~
C(~(W; G; fS)II~ll%_q_~ e n ,
unmezpanbnbtx ypaonenut~ (20') u (20") coomoemcmoenno, 0 < 70 < min { a , y} , 1~
nto6oe nono.~ume.abnoe qucno, ~enbutee 3[0 - ~ , a rl e ( 0 , Yo)- K o n c m a n m a
C~ u ~ta.noe ,~uc.ao E l ono.ane onpe~e.aa~omca aaOanue~t ucxocgno[l oS.nacmu G u
zo~teo~wpqbu~ta l~' ( z ) .
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no~y~eHmae s [7].
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y~eTo~ s nell qbop~yn~a (4) n s~apa~enaa T~o(z n ~m) a~ [1, c. 44], r~e n, m
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1SSN 0041-6053. Yxp, ,~tam. m'T. pn., 1999, m. 51, bl ~ 10
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| id | umjimathkievua-article-4738 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:04:21Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/0d/8f65cc9ac5ef8db7970b0afc7f98360d.pdf |
| spelling | umjimathkievua-article-47382020-03-18T21:12:54Z On quasiconformal mappings corresponding to the beltrami equation О квазиконформных отображениях, соответствующнх уравнению Бельтрами Samsoniya, Z. V. Samkharadze, I. G. Самсония, З. В. Самхарадзе, И. Г. Самсония, З. В. Самхарадзе, И. Г. By using methods of integral equations, we investigate problems of conformal and quasiconformal mappings of close domains. Досліджуються задачі конформного та квазіконформного відображень близьких областей з допомогою методів інтегральних рівнянь. Institute of Mathematics, NAS of Ukraine 1999-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4738 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 10 (1999); 1391–1397 Український математичний журнал; Том 51 № 10 (1999); 1391–1397 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/4738/6177 https://umj.imath.kiev.ua/index.php/umj/article/view/4738/6178 Copyright (c) 1999 Samsoniya Z. V.; Samkharadze I. G. |
| spellingShingle | Samsoniya, Z. V. Samkharadze, I. G. Самсония, З. В. Самхарадзе, И. Г. Самсония, З. В. Самхарадзе, И. Г. On quasiconformal mappings corresponding to the beltrami equation |
| title | On quasiconformal mappings corresponding to the beltrami equation |
| title_alt | О квазиконформных отображениях, соответствующнх уравнению Бельтрами |
| title_full | On quasiconformal mappings corresponding to the beltrami equation |
| title_fullStr | On quasiconformal mappings corresponding to the beltrami equation |
| title_full_unstemmed | On quasiconformal mappings corresponding to the beltrami equation |
| title_short | On quasiconformal mappings corresponding to the beltrami equation |
| title_sort | on quasiconformal mappings corresponding to the beltrami equation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4738 |
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