MOT SO NHAN XET CHO TJNH Kl Dl CUA CAC HAM SO NHAN GIA TR| TRONG DAI SO CLIFFORD
Doan Thanh Sdn Trudng Dgi hgc Tdi Chinh Qudn Tri Kinh Doanh Dao Viet Cudng Dgi Hoc Giao Thdng Van Tdi Le Hiing Sdn Dgi Hgc Bdch Khoa Hd Ndi Nguyen Dinh Thi Dgi Hgc Su Phgm Ky Thudt Nam Dinh Ngiy nhin bii: 10/7/2019
Ngay nhan bin sfla: 13/8/2019 Ngay duyet ding: 23/8/2019
T6m tdt: Ta biit rdng bdi todn Cousin cgng vd dinh ly thdc trien Hartogs Id nhiing ehd di ca bdn trong gidi tich phiic nhiiu biin. Bdi todn cousin cgng cho phep ta xdy difng cdc hdm phdn hinh tit cdc ki di dia phuang, vd dinh ly thdc trien Hartogs cho ta md rgng hdm chinh hinh cda hdm phiic nhieu bien qua cdc diem ki di khong co lap. Bdi bdo ndy di cap din mgt so nhgn xet vd y kiin cho cdc bdi todn tuang tif trong gidi tich Clifford vd gidi tich Quaternion.
Han nQa, trong bdi bdo ndy chung tdi se ci gdng dua ra mgt soy tudng vd kit qud mdi trong gidi tich Clifford vd Quaternion vdi mdt si chiiu cao han vd dp difng cda nd trong cdc bdi todn vi phuang trinh dgo hdm rieng vd phuang trinh vdt ly todn.
Tiikhoa: Hdm da chinh quy, Dinh ly Hartog, Bdi todn Cousin, Dai so Clifford.
SOME REMARKS ON THE SINGULARITIES OF FUNCTIONS TAKING VALUE IN AN ALGEBRA
Abstract: lit is known that the Additive Cousin Problem and Hartogs Extension Theorem are the fundamental events in Complex Analysis of several variables. The first one allows the construction of the global meromorphic functions by given local singularities and the second one states that holomorphic functions of several complex variables possess non isolated singularities. This paper deals with some remarks and comments concerning the same problems in Clifford Analysis and Quaternion Analysis.
Furthermore in this paper we will try to reflect some new ideas and results on Clifford and Quaternion Analysis of higher dimensions and their applications to Partial Differential Equations, Mathematical Physics and Engineering Problems.
Keywords: Multiregular functions, Hartog theorem. Cousin problem. Clifford algebras.
1. Mfi dan cac him song chinb quy nhin gii tri trong dai sd Him chinh quy nhin gii tri ttong dai sd Clifford Clifford,
da dddc nghSn cflu bdi R.DeIangle, F.Sommen va -prong bai bio nay, chflng toi ngMen cflu ve bam mdt so nhi toin hoc khic (xem [1]). Day la sfl md ^ wu i,- • - , - . A - -^ r'vte A
, -,, , . , , , ;• 1" ou xiiyj ^^ chmh quy nhan gia in ttong dai so Clifford, no rong W nhien cua ham cMnb Mnh Irons ly thuyet ,. , ; ', .„ , ' , , , ' , , , i,x™ J,Af K;SV, .,I,.-*O -0 hfA ^* .•- 1, •- la sd md rong W nMen cua bim cbinh quy va chflng ham mot bien phflc. Y Wdng dau tien cho viec * ,7 ^
md rpng ly thuyet him cMnh Mnh trong him bidn "^"^ m6t so tmh chat cd ban cfla c^ac ham niy, nhfl ohflc nMeu bien tiiudc vl cic nhi toin hpc Brackx ly thuyet ve tinb duy nhat, bieu diSn cdng thflc tich
gfldi da nghien cflu vl phin dang cauchy,...
73 Tap chi Tai chinh - Quan tri kinh doanh
« = / ^a^a : tta £ 1
Ap dung nhflng ket qua nay, chiing toi chiing ]^(in+l)n _ nsCm+DC;^;^) x minh dinh ly Hartogs cho ham da chinh quy. iBt'n+i)^ Cn)1 2. Mot so kien thiic ve dai so CBttord
J.0) = r^rO) ^0) ^my.-^i n.
Cho E'"'*"^ la Ichong gian Euclid vdi cac vec td ^ ° ' i ' ' " ' ' " / ' • '
dan vi {e„,e,,ej,...,e„J va phep nhan (ti'ch trong) Ta xet ham f xac dinh tiong n nhan gia tri trong codang =/Zttlclainhxa/':a ^ c/?ldiiddfcddang
<x.y> = yx,yp / - ^ ' ' " W e . ;x = (x<» ...,x<"))
= (x„»> V " ' ) Dai sd Clifford <A xic dinh dai sd 2"° ehilu (cd
tmh kdt hdp, khdng giao hoin) dUdc xay dflng bdi ^^^^S ^^ ^ ^ W l i bam thflc cua (m^l)n bidn e^, Cj, ej,...,e^ v i phep nhan theo luit sau: -*-0 '—••'^m
eiCj + e;e,- = - 2 6 y eo; rj" = 1 m Tavidt/ ^ c c a ^ ) ; C^Cac/i); i.pCa./l),.., t h e o / G C ( a ) ; C'^(n); Lp(fl),... tfldng flng.
trong do 5:; la ly MSu Cronecker va en = 1.
-' Toan W Cauchy - Rieman dfldc xiy dflng dfldi Mpt phan tfl aeJl c6 ci'u true dang ^ Q
d a n g D , o , = 2 . ^ ' Z X « ' - ' = l n . i=0 "•^i
""' Binh nghia 2.1. Ham so / £ C'(n,t/!) duac trong do a={a,,.. ..a,^ ),(a,<a<a^) la t|p con cija goi !a da chinh quy neu no th6a man h$ Cauchy - tap iV={l,...,m} va e^ = ^ai •••^ah- Khi t-=^ ta Iceman sau:
"''="=''»"'• B,^"V = 0; ; = 1 n . ( D Mpt vec to trong B"-*' dUOc ddng nha't vdi mot ^hu y rSng, mi8ng hap n = 1 thi ham da chinh vec to trong M bflr phep nhung q„y ^ j ,^-„^ ^^^ ^ ^ , ^ q^^, ^j^^^ ^ j ^ ^^^^ j ^ ^ ^ ^ A: G E"*"*"^, a: = (Xfl, a:i,..., Xn) [1]. Vdi « = 2 thi ham da chinh quy khac vdi ham
^ song chinh quy dildc dinh nghla bdi Brackx vS -> y XjBi := xe</l. Pincket (xem [2,3]).
f~*^ Vdi mot nghTa khac, ham da chi'nh quy lit ham Phep lien hap dUdc dinh nghla bSi anh xa chi'nh quy theo nhiSubiS'nArO'JE B'"*!).
g !-» a vdi ej = -ej,j = 1,2,..., m; e^ = GQ va Dudi day, ta ky hieu tap ham da chinh quy trong ab = ba. fl bdi MRia, A).
Phep "tich trong" trong Jl dildc xic dinh bdi 3. Ham da chinh quy
^ o a Ai, thi A/^O, trong do:
Do do chu^n trong ^ dtiac xac dinh bdi ^~ — - - + — [ - . L ,
r y/^ < ' a;.™' ^a:f«'
Hallo = ({a, a>o)''^ = 2™/^ ( 2 j " ° j CAiiKj minA.- Dinh nghia
Dieu nay dan dSh Jl la mdt dai so Banach vdi D 0) = ^o —77; - / e, — r r . 2"chiSu. a^?' fef 3^,^'
Mpt so dinh nghia khic cfla dai sd Clifford ta cd ' Ta cd:
the xem trong [1]. m Cho n li mdt tap con (mdt miln) ttong khdng ^x^'^^x^^^ = ^x^^^xO) = ^ xd^ ~ 2^—n-
gian Euclid v i _ t=i "-^^i Sd 15 thang 9 nam 2019 74 ---"-^^^i^^^rti^-z^^iz^ . ioanh
do A / e M f i ( a t ^ ) nen:
A ^Uif = B^o) {D^tj)f) = 0 vdi moi j = i n.
Do dd A/^0 (chiing minh xong).
Bo ei 3.1 ndi ISn ring nSii / 6 C'^ia.jVj va da chinh quy thi m5i thanh ph^n f^ la ham dieu hoa, vi vSy chiing la cdc hkn gi&i tich thilc cua Xo''^ ..., JCm*"' e n. Do dd, dinh IJJ ve tinh duy nhat 1^ dling cho cac ham ttky.
Cho n la mot mien cd dang H = Hi x... x ii^
; trong dd Ilj la mien trong E'-"'"*"^-'(x^^) vdi cac biSn trdn tuang flng 3flj.
Bitlh If 3.1. (CSng thflc tich phan Cauchy) C h o / € C^(ilA) n A r R ( n , c 4 ) , k h i d d :
«>m+l
FTCD _ r & ) Tda,,!
| „ ( 1 ) _ v C D l m + l u™ •••
| l , ( n ) _ j . ( n ) | m + l u^ ' trong dd
;cO) = 2r=oe^^F^, uO-' = E r = o " f ^ e , ; d a , ( o = S r = o ( - l ) ' e i d i i f ^
( i u f ^ = d u J ^ A ... A d u , 9 l \ A . . . A d u [ i . \ A . . . A d l i ^ ^ va a>ni+i la dien tich mit cau ddn vi trong E^'""'"^'.
Chdng minh: Sfl dung cdng thdc tich phin Cauchy cho ham f theo tflng bidn x^' vi ap dung dinh \f Fubini ta dfldc (2).
4. He Cauchy - Rieman suy rong khong thuan nhait
Trong muc niy ta xet hS
D . « ) « = fflO) <p">; ; = l , . . . , ) i (3)
Y^e,Ei7])
-J^eMlwJ-jjj
ttong E(D) =
n =ZS
do 1
.ofJiei 1 111"
•,n.
i=0
= 1, ..
i+T '
E R ; a y t " dg
« ' " '««,"'
lay tai ( j : 0 ' + I ) , x < « , .,.,>:(")).
Ap dung bd de 1.2 (xem [6]) ta cd th^ chflng minh cac dinh l^ sau:
Binh ly 4.1. (xem [6])
Gi5 sfl » ) « . p W e C o ' C l R t " ' * " - . . / ! ) , k> 2. Khi do he (3) edit nhat mdt nghiem a E Co**
khi va chi khi dieu kign sau dfldc thda man:
I ^ei£(,,)
dx (1) dr,
dtpm dx]
nsCm+i) i=0
IR(m+l) i=0 vdij=l....,n va vdi mpi
ttong d o ^^(T}-^ -^^^ol" l^y g*^ tri tai [x^^^ + 7], x^'^^,.... x^^^) va Co la bam tiiudc Idp C CO gii compact.
Chiing minh:
a) Biiu kien cdn
tin
.(ft+l)^™(m+l)n trong dd tp'j'' e CHii.c^), vdi n=l thi he (3)
dfldc xet trong [1].
B ^ a l 4 . J . (xem[6])
Cho g E C'^Cn.t/J) la nghiem cua (3), khi dd
Cho fl G £•[
(3). T h e o b 6 d e l . 2 t a c d :
c/V) la nghiem cua
g,,(,)_l_=g,/,(,,,_J_l,,
0) "^ a (1) "•dx'f
K ) ( 6 )SA:^ •(7) 75 Tap chi Tai chinh - Qu^n tri kinh doanh
Doa e C
( m + l ) „ . ^ ^
Tir bieu dign Pompeiu cua g (xem [6], bo d l 3.2) keo theo
|,(m+l) 1=0 " - ' i
= I ^ e i J ? ( i ) ) D , ( ^ ) d . , . (8)
K(m+1) i=0 ° ^ i Tfl (6), (7) vii (8) ta cd (5).
b) Dieu kien dil
Gia sit (5) dfldc thda man. Dat
a W = - j E(y ^ xm-),pm{y,xO> x'''>)dy
Igm + l
= - J E(7,)^(«(xm+7),xra x<"))dy. (9) K h i d d D , w 3 = - J D^O)(E(J))<P'"(A:<"
^ m + l
+ 7j,xO) x'-"''>)')dn
= - J [o,o)ff(';)'P<"+2,«''^W^]'*''-(io)
K'" + l ' = 0 '
(i) Neu j = 1, D^jE(n) = OdoEla chinh quy theo x!"
(il) Neu j > 1; D^fi(n) = 0 do E khong phii thuoc vao y^'
V I vay, (10) din den 0,0)9 = - J eiF(j))
Va do dd, til (5) ta cd dq,»
dr,.
,n,g{x^'\x^^^ / " ^ ) :
cd ||;x:|| dfl Idn. Mat khac g la him da cMnb quy bSn di cua him/.
Sd 15 thang 9 nam 2019
ngoai tip compact cfla R '•"'"^''". Ap dung dinh ly ve tinb duy nhat din den g xac dinh bdi (9) cd gii compact.
5. Dinh ly thac trien
Cho n Ii mien ttong E *'^^'*° vdi H e n . Ham / G MR{h,Jl) dUdc gpi la Mac ttien cua
/ e >fR(n, =/i) ndu / = / ttong n.
Tfl dinh ly vl tinh duy nhit keo theo ring, nlu / cd the thic trien thinh f thi thic trien nay li duy nhit. Cho n > 2 ta cd
Dinh ly 5.1. (Dinh ly Hartogs)
Cho K l i mpt tip compact cfla t|p md g c i[5;(m+l)n sao cho g\X l i liSn thdng. Hdn nfla, cho / G MR(,g\X, A). Kbi dd ton tai him / e MR{g, A) sao cho / = / trong QXJC. Theo mot nghia khic, mpi ham da chi'nb quy ttong G\K deu cd the thic trien ra toin miln 5.
Chiing minh:
Lay (p G C^ ( 5 ) sao cho q) = 1 ttong mdt lan can md cua JC, dat / t " ) = (1 - (p)f kM do / W l i da chi'nh quy ttong Q\supp(p-
Dinh nghla
{ 0 TOI Z E K \ 5
»W) = Co*(E('»+«",c/Z)vdikduldn.
X6thtD^ij)g = <p^K j=l,...,n. (11) D i thS'y rang (p^ ^ thda man (5) (xem [7]).
Dinh ly 4.1 chi ra rang ton tai nghiem 5 E Co* (»'"•+"",=/!) cfla (11).
Bay gid ta dat / = / " " - 5 khi dd ' ' x O ) / = B;,0)/™ - D;,0)5 = 0 trong 5.
Hon nfla / = / trong g\X. do dd / la thac trien cfla / .
(van di chi tiet han co the tham khdo trong [7]) 6. Bai toan Cousin cong
Trfldc tien chiing ta dinh nghia khai niem vl diem "chi'nh quy" va diem "ki di" cua ham da chinh quy.
- Diem x E »("•+!)" dfloc gpi la diem chi'nh quy cfla/(j:) neu ton tai mdt lan can md (/ cua .x sao cho / e >ffi (V, JV), ngfldc lai ta ndi x la diem Id
0) =
<f khiddcp"
i^l,...n,
f- i"«l^sJiitfl'i^Sl^WTO9OUIHM)Ili0anll
-Tttdinhnghiankytadiay.nSu/ E MR(U,A) t h i / i a chinh quy tai moi x E fl neu/cd diem ki di trong a ta viet / S MR(n, A)..
Cho x^aia^l Ifj mpt phfl md cua fl, trong moi Ua , oho htai / E MRin, A) sao cho:
/ « - / ^ eMR{UcriUi,,A)
trongU^nUf. (12) Bay gid ta tim h a m / E >ffi(fl,c/Z) sao cho:
f -f^e MR(Ua, JV) vdi mdi « S /. (13) Neu c/l = C thi bai toan n^y trd thanh bai toan Cousin cdng (xem [10]), d dd ta cd the x^t dfldc
\^ thuyet ham gidi lich nliieu bien phiic (xem [5], [12]).
Phfldng phap Shema cho bai loan Cousin (xem [8]).
Dat Kp:=fa-fp. (14) IChidd:
(i) haf&MR[U^nUf,M) (ii) ha/i + h^a = 0 trong (/„ n V/, (iU) hall + hfy + h^ = <i t r o n g U a r \ U p t - \ U y \ ( n o c , p , - y € 1.
Cho {^j} la mpt phan hoach con cfla phu Wa]ael, khi dd t ; e Co°°(W„^) vdi oCy E I.
Tai mdi /? E / cd dinh, ta dinh nghla ham:
-1
= > h- trong dd
h- _\'^K,i!trongU^^nUf 0 trong lJp\U„
j
j
= Y^1>iiha,a + K^f,
j
Dieu nay keo theo
D;t</)(5o - 3 / ) ) = D,ij-:haf = <i
trong t / „ n ! / ^ . (15) Ta dinh nghia:
<P, •= {D«0) gavoixe Uai (16) theo (15), ham ^j (x) hoiin toan xac dinh tren fl.
Binh ly 6.1. XethS:
O,0)'l=-«'u) (17) trong do 0. dfldc xac dinh bdi (16) vdiy = 1, ..., n.
Neu he nay cd nghiem h(x) eC^ (fl, JV) thi ton tai ham
f eJvTRiUa.A) sao cho f-fx£ MR(JJa,A) vdi moi ael Chitng tnlnh:
Gpi h(x) la nghiSm cfla hS (17). Ta dinh nghia:
ha = ga-t-hvaix^Ua (18) khiddD^m'ia = D ^ 0 ) 5 a + D^O)'!
= D^mga -<P(j) = 0 bdi vi (17) va hj,- ha = {hp + h) - (h,^ + h)
= gp- gii= hap trong [/„ n [/^.
KSt hpp (18) vdi (14) tacd:
hp-ha = fa-fp hoac
fa+K=fp+ K irongUar^Up. (19) Ta vidt
/ = /„ vai :t e t/«, (20) kbi do / dddc xac dinh ttSn toan 11 va nd la
nghiem cfla bai toin.
7. Tong ket
Bai bio dua ra dddc mot sdkdt qua ve bai toin Cousin cdng va dinh ly thac ttien Hartogs ttong dai sd Quaternion va dai sd Clifford co dien, nhflng bii toan nay cdn cd the phit trien cho ly thuyet dai sd Clifford phu thupc tham so vdi cau true dddc tiinh bay trong [4]. Chflng ldi hy vpng cic bai toin nay cdn dddc giii quyet trong dai sd Clifford cd ciu trflc tdng quit hdn va mdt sd bii toan vl gia tri ban dau cua phfldng trinh dao bam rieng (xem [11,13,14,15]).
Tap chi Tai chinh - Qu^n tri Idnh doanh
Tai lieu tham Ictiao [I] Brackx F., Delanghe R., Somen F., Clifford Analysis. Pitman 76, 1982.
[2] Brackx, F.; Pincket, W., Two Hartogs theorems for nullsolutions of overdetermined systems in Euclidean space. Complex Variables Theory Appl. 4 (1985), no. 3, 205-222.
[3] Brackx, Freddy; Pincket, Willy, A Bochner MartinelU formula for the hire gular functions of Clifford analysis. Complex Variables Theory Appl. 4, (1984), no. 1, 39-48.
[4] Dao Viet Cuong and Trinh Xuan Sang, Function Theory in Clifford algebras Depending on Parameters and its Applications, Joumal of Mathematical Applications, Vol XVI, Noi, 2018, pp.
19-36.
[5] Hoermander L.,An Introduction to Complex Analysis in Several variables, North-Holland PublisMng Co., Amsterdam, 1990.
[6] Le Hung Son, Some applications of integrated operators and Cauchy Pompeiu representations in Clifford Analysis. Complex Variables and Elliptic Equations 53 (2008), No.5.
[7] Le Hung Son and Nguyen Thanh Van, The Hartogs extension theorem for multi-regular functions taking values in a Clifford algebra. Adv.Complex Anal.Appl.,2, Kluwer Acad. Publ. Dordrecht, 2004, pp 309-322.
[8] Le Hung Son, Nguyen Thanh Van, The additive Cousin problem and related problems for regular function with parameter taking values in Clifford algebra. Complex Var. Theory Appl. 48 (2003),
no. 4, 301-313.
[9] Le Hung Son, Nguyen Thanh Van, Some extension theorems for regular functions of several quatemionic variables. Complex Var, Theory Appl. 47 (2002), no. 3, 259-269.
[10] Le Hung Son, The Cousin Problems in the view point of partial diferential equations. Geometry Seminars, 1988-1991 (Italian) Bologna, 1988-1991, 63-77, Univ.Stud.BoIogna, 1991.
[II] Le Hung Son; Nguyen Thanh van. Differential associated operators in Clifford analysis and their applications. Complex analysis and its apptications, 325-332, OCAMI Stud., 2, Osaka MuMc. Univ.
Press, Osaka, 2007.
[12] B.V.Shabat(1992), Introduction to complex analysis. Part II: Functions of several variables.
Mathematical Monographs, Vol.110, AMS, Providence Rhode Island.
[13] Le Hung Son, Le Cuong, Differential operators in Clifford analysis and applications. Report in the 16th. ICFIDCAA, DonggMe University, gyeongju Korea, 2008.
[14] Le Hung Son, Nguyen Thanh Van, Necessary and sufficient Conditions for Associated Pairs in Quatemionic Analysis. Quantemionic and Clifford Analysis, Trends in Mathematics. Birkbauser Verlag Basel-Swtzeland, pp.207-222, 2008.
[15] Tutschke, Wolfgang, Solution of initial value problems in classes of generalized analytic functions, Springer-Verlag, Berlin, 1989. 188 pp
So IS thang 9 nam 2019 78 1 > ||'i".^-.',uj:iunAQniSMiiin,ij,tanh
