Basic boundary-value problems for one equation with fractional derivatives
We prove some properties of solutions of an equation $\cfrac{\partial^{2\alpha}u}{\partial x_1^{2\alpha}} + \cfrac{\partial^{2\alpha}u}{\partial x_2^{2\alpha}} + \cfrac{\partial^{2\alpha}u}{\partial x_3^{2\alpha}} = 0, \quad \alpha \in \left( \cfrac 12\, ; 1 \right ]$, in a domain $\Omega \subset R...
Збережено в:
| Дата: | 1999 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
1999
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/4582 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510730551296000 |
|---|---|
| author | Lopushanskaya, G. P. Лопушанська, Г. П. |
| author_facet | Lopushanskaya, G. P. Лопушанська, Г. П. |
| author_sort | Lopushanskaya, G. P. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:09:14Z |
| description | We prove some properties of solutions of an equation
$\cfrac{\partial^{2\alpha}u}{\partial x_1^{2\alpha}} + \cfrac{\partial^{2\alpha}u}{\partial x_2^{2\alpha}} + \cfrac{\partial^{2\alpha}u}{\partial x_3^{2\alpha}} = 0, \quad \alpha \in \left( \cfrac 12\, ; 1 \right ]$,
in a domain $\Omega \subset R^3$ which are similar to the properties of harmonic functions. By using the potential
method, we investigate principal boundary-value problems for this equation. |
| first_indexed | 2026-03-24T03:01:38Z |
| format | Article |
| fulltext |
Y/IK 517.956
r . H. d-lonyulaHCbKa (Jlbain. yll-T)
OCHOBHI I'PAHHHHI 3A~IAqI ~JLq O~IHOFO P/BtDIHH$I
B ~POBOBHX IIOXUIHHX
We prove some properties of solutions of an equation ~x ~a + ~ x 2 + ~0x~a = 0, a e ; 1 , in a
domain fl ~ R 3 which are similar to the properties of harmonic functions. By using the potential
method, we investigate principal boundary-value problems for this equation.
--02etuO2aUO2ctu (]1
~oae/Ieni ~aeaKi a.naCrnBOCTi poan'aaKia pinnalnla ~xfft + ~0x2 + ~x~a = 0, a e ; 1 n
o6~aeTi f~ ~ R ~, atta,aoriani B.aaCTUaOCTZ~ rap~ouiiamtx qbyuKuifl. MeTO/IOM noTenttia~y
aoczi/~eno oenonni r'panvlqld 3a/taqi/13~g ttboro pim~mma.
Hexalt ~ ~ o6ZaCTb B R 3, o6Me~ena aa~KHenom noBepxHeio S ~ a c y C ~,
V (X) ~ OpT 3OBHiRIHbOi HopMa.rli RO S y TOqai X. B s poar~taz~aeMo piBrlZHH,q
L u ( x ) - ~ =- *
i=1 ~Xi i=l
1, x ~ ,
~(x) = 0, x ~ ~, ' f~ (x/) =
T y r
{ O(Xi) x~'-l, ai>O,
F ( a i ) i
f:xi+l (xi), Ot i < O,
f(~1,0~2,~3) (X) = f0~l (X 1)" fix2 (X2)" f C t3 (X3),
hl =(1,0,0) , h2= (0,1,0), h3=(0,0 ,1) ,
Kparmom no3HaqeHo onepat~im n p a ~ o r o ~Xo6y~ey, * - - onepa~im 3FOpTKH, ~
oncpauim 3ropTKH y3ara~bneHoi qbynKIfii 30CHOBHOIO, F ( a ) ~ raM~la-qbyHIoli~,
1, x i > O ,
i = 1,2,3.
O(Xi) = 0, X i<0 ,
1. | ]['piHa Ta ~opMy,rIIOBanna oenonriX l"paHnqUHX 3a~aq. YIoaHa-
qI4MO qepe3 C 2 a ( ~ ) KYlac dpyHKRifl U(X), BrlaHaqermx i HenepepBrlO/Ivlqbepeuui-
flOBnnX B ~ , ~t~ta ~ a x aenepepBHi 3ropTKn f_2cth i * (flu), i = 1, 2, 3, qepe3
C2r A C 2 a - l ( ~ ) - - KYlac { tt(x) ~ C2a(~'~) : ~l-2a)hi * (lqU) = (~2-2a)hi *
�9 (rlU))x~ e C ( ~ ) , i = 1 , 2 , 3} , qepea (tp, F ) - - ~ a i I o yaara.rmneHoi qbyru<uii F Ha
OCHOBay 9-
Hexalt u(x ) e C2a(f~) N C 2 a - I ( ~ ) , fi = ~u. ~Lrm ZtoBi~a, noi tp e D ( R 3) MaeMo
3
fl i~:l
�9 F. IT J'IOHYLI.IAHCbKA, 1999
48 I$SN 0041-6053, Yrp. atam. a~ypn..1999 , m. 51, N ~ 1
OCHOBHI FPAHHHHI 3A~AqI s OJXHOFO PIBH$1HHJ;I B s FIOXIJ2HHX 49
3 3
= X ax - -
S i=l i=I [2
3 3
= ~U<X) E( f (2_2~x)h i*~)x iV i (x )dS - f E UxiCX)(f((2-2~x)hi * r +
S i=I S i=l
3
+ f E Uxlxi(f(2-2tx)hi *(p)dx.
[2 i=l
B~e~eMo rpaHaqai onepaTopH
3
(x) -- ( *(n,p) )x ,,,(x).
i=l
3
q,)<x) = E ( *<nq,) ,,,<x).
i=I
I3 rlonepe/IHix piBHOCTeR o~ep~KyeMO rlepmy ~opMy~y FpiHa
fl S [2 ill
ra ~apyry qbopMyJIy Fpina
f (Luq~-uirp)dx = f (BcLuq)-uicL(p)dS, (3)
[2 S
ae u, tpeC2CL([2)NC2a-l(-~), a TaKO:kK (q),L?t) (tp,-(Bau)Ss+B~(u~s)),
.8s--npocTrI~imapHa S, ((p, 8 s) = fsq)ds, ((p, B~(uSs))= fsUBaq)ds.
OrTKe, U(X) e poaa'~taKOM y D' (R 3) p i~nanaz
L~ = - (B a u) 8 s + B~(uSs). (4)
h cni~i~nomea~ (3) xa (4) ~n~ao, mo npapo~ano ~ pi~nannJ~ (1) po3raa~tarn
saOa~y ,aipixne
uls = F~(x) , x e S (5)
Ta saaa*ty muny Heaztana
B(xuls = F2(x ), x e S. (6)
Hexait D(S)= C**(S), D ( ~ ) = C**(~), D'(S), D ' ( -~ )~ npocropn zinia-
nmx nenepepBHrlX ~yHKaiOHanis Ha D(S) Ta D([2) Bi/lJIOBi/~lO.
~op~y, anmanna y~tw.,abnenoi" saaa,d fl[ipix, ae. Hexatt F l e D" (S). 3Hal~TH Ta-
Ky U e D ' ( ~ ) , mo ~ ~taoso:maJte y D'(R 3) pinHmtHJ~
L~ = - F 2 + Ba Fl, (7)
/Ie F 2 - - rlcBHa, rloKrl 11Io HCBi~OMa y~Fa2IbHCHa d~yHKl.ljJl, (qL Ba FI) = (Bvtq ~, FI),
ISSN 0041-6053. Yxp. atam. ~.'vpn., 1999, m. 51, bl e 1
50 F, H. J1OFIYlIIAHCbKA
Oop~tyatoeannst y3azan~,neuoi" 3ttiga,d muny Hetl~ana. Hexat~ F 2 r D' ( S).
3Ha~THTaxy u cD'(-~), mo fi 3a~onozbnar n D'(R 3) piBnanna (7), ~e F 1
Ae~lKa, tiOga mo neBi~OMa yaara.ri~Hena claynKRia.
OnepaTop L e OKpeMHM BHIIa]~KOM e~inTriqHoro rlcea/IO/~rldpepeHRiaJ-IbHOrO orte-
paTopa (rl.~LO.). l~oro CriMBO~ a(~) = E~=I ( - i~j)2a, OCKDn, grt
3
-3 ] =
R 3 j=l
3
= (2n) -3 ~ ~ ~ ( - i~ j )2ae i (X-Y '~ )u (y )dyd~ =
R 3 R 3 j=l
3
= (2rc)-a f f E ( - i ~ J ) 2 a e i ( t ' ~ ) u ( x - t ) d t d ~ =
R 3 R 3 j=l
= (2tO -3 f f ~,(-i~j)2aei(X'~)d~*u(x) = f-2ah, *u(x) = Lu(x).
R 3 R 3 j=l j=l
Oneparopn B e Ta /~a r rpa~nqanMH n.Z.o, i3 cHMao~aml
3
bct(x,~) = ~ v j ( x ) (i~j)2~x-I Ta - ba(x,~).
j=L
FpaHHqni 3a~aqi ]IJla n.~t.o, y piannx dpynKRionadlbHHX npocTopax BHBqaJIHCb y
6araT~ox pO6OTaX (/ItlB., narlprlKJla/I, [1 - 10]). j~o8e/xe~i TeopeMrI icrlyBaHttJ~ Ta
e~t~HOCTi, Teope~rI r~a/IKOCTi. Hafl6i~htu ~OB~O aoc~i~ceni rpamtqHi 3a~a~i ~t~la
rinoeJfirrrn~aHX n.~t.o. 3 FJta~KPIMrl a6o O]IHopiI~ntlMH CrlMBOJlaMrl. I I p n BHBqeHHi
aHa.niTriqmix BnaCT~BOCTeia pOaB')~3Ky 3a~aqi Komi ~ a napa6oJaiaaoro rL~I.O, a He-
F~Ia/~KtlM FO~OBHHM CHMBOJIOM [6 - 9] iCTOTnO BHKOpHCTOByeTbC~I dpynjaaMenTaJlb-
HHi'~I pO3B'Z3OK. B~xoI~aqa i3 aaac-rnaocTeia dpyHiaa~eHTanbHOi onepattii onepaTopa
L Ta qbopMya Fpiita, o~ep~ye~o 3o6pa:~eHHa poaa'aaKiB K~acHamtX Ta yaara.nbtte-
HHX OCHOBHHX I'paHnqllHX 3a~aq/~2ta piBHJIHHa (I) i ~tocai~tzxye~o ix B21aCTHBOCTi.
2. IIo6y~aoaa ~ynaa~enTaJ~nO~ c~ynK~iL BnKOpnCTaCMo MeTO~ Pa/~oHa [ I l ].
Po3a'aaor y D'(R 3) piaHaHHa LC0= ~ ~ar a~r~az~
~0(x) = f o~((p ,x))dp, (8)
s3
/xe S3--o/xamtqaacqbepay R 3, p e S 3 , (p,x) = ~ i 3 | pixi =~.
3a B~acTrmOCTJ~Mrl 3FOpTKH
i=l i=l
ToMy
s3 s3 kjffi~
a apaxosymqH ao6pa~enH~t
HJIHH$1 y aropTzax
8(x) = - ( 8 9 2 ) -t IS3 ~"(~)dp [I1, c. 961, MaeMo piB-
ISSN 0041-6053. Yrp. ~gtm. ~.'vlm., 1999, m. 51, IV ~ 1
OCHOBHI FPAHHHHI 3A22Aql ]]21~I O~HOFO PIBHflHHfl B 22POBOBHX I-IOXI]2HHX
( L ~ . * % ) ( g ) = - 8 ~ ~ p?~ a"(g).
i=1
3Bi]~CH
51
3BincH BHA.O, tUO CO(X) = o (i xl2<"-3).
3. ~eaKi n.~aca~m~c'li pO3B'~I3KiB.
TeopeMa I. YIKu~o u (x) ~ C (G) i e y3aza.abnenum poze'.~,Kom pi6n.~mo~ (1),
mo u(x) ~ C**(D) i e u.aacu,cnum poze'.azKo~t ~bozo pisn.~nn.~.
~oseOennst. Hexa~ u e C (D) i e pO3B'J~aKOM piBH~IHH~I (1) B y3araylbHeHoMy
cenci, acaMe: I o u L ~ 0 d x = 0 A~aAOBi~bnoi ~0 e D(D) . 13y~o noKaaaHo, mo fi
3a/IoBo~bnae y D'(R 3) piBHaHHa (1). flOrO pOaB'a30K fi = (Lfi)*0). OCKi~bKH
Lfi = L.0 = 0 B R 3 \ D , TO supp(Lfi) c S . OT>Ke,
Ae q0, ~ ~ D(R3), ~= 1 BOZO~i S. 3ozpeMa,/I~AoBiamuoi ~p ~ D(D)
(% fi) = (q), u) = I (~(y) o3(x- y), Lfi(y)) q0(x)dx. (9)
fl
OTis<e,
u(x) = ( ~ ( y ) t O ( x - y ) , (Lfi)(y)) , x ~ ~'1, (10)
a TOMy n ~ AoBi~Isnoro ~yasT~iaAe~ca k
Dku(x) = (~(Y)Dx~tO(x-y) , Lfi(y)) ~ C ( ~ ) .
3oKpe~a,
(Lu)(x) = (~ (y )Lxr .O(x -y ) , (Lfi)(y)) = (0, (Lfi)(y)) = O, x e ~ .
Bi/xzHaaH~O me AezKi BJIaCTHBOCTi pO3B'$I3KiB piBH~IHH~I (1), aaa~oriqHi Bi/xIIO-
Bi/XHI4M B~aCTHBOCTZM y3ara~bHCHHX pO3B'J~3KiB /~HC1)epeHlXia21bHHX piBHJ~Hb 3i CTa-
JII4HH KOCdpiLIieHTaMH.
1. Hexalt u(x)--po3B'Jt3OK pissJtHHIl (1) B R 3, g ~/IOBi/IbSa qbiHiTHa yza-
ra~bHeHa qbyHKaia. ToAi u * g TaKOar 3a~oBo~H~e piBHZHH~ (1).
/liltCHO, L(u * g) = Lu* g = 0* g = 0 (3a acouiaTHBHiCTm aropTKa). 3Bi~CH,
B3~Bttm g = D t S , oAepacycMo, mo D t u ( x ) 3a~OBO~mHZr piBHZHHZ (1) y R 3 Ta y
D, anpH g = 5 ( x - y ) , u ( x - y ) 3a~OBO~H~eaepiBHaHHZB R 3.
ISSN 0041-6053. Yrp. mare..~.'vpn., 1999, m. 51, N e I
%(~)=-8,: ~ p?'~ W~,*B")(~) = -8,~ ~ p?~ .6;~(~).
i=1 i=1
BpaxoBymqa, mo f " ((p, x)) = A x f ((p, x)), oAepz~ye"MO
)_l , , , ,
CO(x) = - Ax Z p?(X f2a((p,x))dP = - - . ^ . - x j ~ a '
S 3 i=1 8921t z~) s3Pl +P2 +P3
~+~ = {~x, ~>o,
0, ~<o.
52 F.H. JIOHYLIIAHCBKA
2. ~ a poas'aazy u �9 C2et(~'~) N C2et-l(~) piBHJIHH$1 (1) Ta jlosi~mnoi r~aazoi
noBepxai t~ c (f~)
I Bet ud~J = 0. (11)
t~
Re 8~nmmae ia nepmoi qbopMy~m Fpina rrpa
{ 9(x) = 1, x e f ~ ,
o, x~Uga),
Ae US(~) ~8-ozi .n o6stacri ~2.
~.rI~ poaB',qaKy U e cZet(~'~) ~ C2~-1(~) piBH.qHa$l (1) ia (4) ra (10) o~ep~yeMo
u(x), x e ~ , (12)
I u ( y ) ( B a c o ) ( x - y ) d S - I (Be tu) (y ) to (x -y )dS = f i ( x ) = t O, x ~ .
s s
3. HexaR Se(x) = {y: ly-xl = e} . Toai
l(x) = I (Bet(y)oa)(x- y)dS = 1. (13)
S~(x)
~i~cHo,
) = I O(x -y ) vi(Y) dS = I(x)
Sdx) i=i
$~,(x) $3 i = l
Oczi,rn>z~
(D) ' (f l-2et *tOp) -- f l f-2et * ~ p - f i * 8~2 p?et 8" 8~ 2 p?et
#
\ i=1
TO
3 p?et dp s,.
l(x) = 8-~-iy_xl=e~Sa i=1..
E
i=l
Biao~o [ 1 1 , c . 9 5 ] , m o 8 ( ~ ) = F [ ~ . + I ) i . Toai
L~75 --) t~-~
ISSN 0041-6053. Yr.p. ~ra. a~pn., 1999, m. 51, N 21
OCHOBHI I"PAHHqHI 3AB, AHI J2.rI~l OB, I-IOFO PIBH$1HI-Lq B s POBOBHX FIOXI/],HHX 53
-T- - - - - I ~ p/2Ct-I . .~,[(x-y, p)l~'-ls.g.n(x-,, p) , , . ) . I
[(X)= ~ 2 ~ s 1,-'1 =e`=. vitY, F ( ~ . f ) a~JapL=_ 1
i=l
BBiBmrt Ha St(x ) cqbepHqny CrlCTeMy xoop/IrmaT, npoBiBmrI ni/I~iK BLa sexTopa p,
3HaIf/IeMo
I ~ P2ct-lvI(Y)I(x-Y' P)lX-lsgn(x-Y' p)dS =
SE(x ) i=l
2x
= - f dqo ] ~ p?a-I picosq/~X-llcoslltlX-I sgn cosqt E2sinlg d~I/ =
0 0 i=1
3
= - X P/2ct ex+t2n I ]c~ s inv dV =
i=1 0
= _ ~.p?CteX+141tlyXdy = 4X e:X+l
X+I i=1 0 i=1
OTX~e,
I(x) = - 8 7 ~ F X=-I 2rc2
_ ~. e x+l ]
r(X+3 l = 1.
\ -"2" , t ~.=-t
PisnicTt, (13)/XoBe~e~Io.
4. Hexall S m noaepxzta :larryrloaa, "ro~
I ( Bc~ (Y) Co) (x - y) dS =
S
~L E X+l 41t ] =
I 2 X=-I
1, X E ~ ,
1 x~S ,
2'
O, x ct f2.
(14)
~OBO]IHTbCJI Bi]IOMHM CrlOCO60M [12], a BHKOpHCTaHHJIM (13).
5. Ia qbopMy.rta (14) ai~oMrIM CnOCO6OM [12] armo~rIMO qbopMy:ta ~a~a rpanaqmix
anaqenb noaepxneBriX noTemtia.MB, a came: ~.qa/IOBiSmnOi g E C(S) piBI-IOMipHo
CTOCOanO X 0 r S iCH3qOTb
1
y)dS = + -~ It(x o) + I It(y)( tt~ (y)co)(x o - y )dS , (15)
s
lira Ig (Y)( ~ ( y ) c O ) ( x -
x ---> xo E S $
xE~
xER 3 \~
1
lim I It(y)(Bc~ (Xo)co)(x- y)dS = ~- ~ It(x o) + I It(y)(/}a (Xo)co)(x- y)dS. (16)
x'-~ x~ S $
xE~
xER3 ~'~
Is ,aesm, ~aoBe~erIoi s [ 13], Ta cnissizmomerm (15), (16) nnrlmmae, mo piBrIOMiprIO
si/mocHo y e S ~ ~osi.m, rIOi qu E D (S)
ISSN 0041-6053. Yxp. ~am. acypn., 1999, m. 51., 1r ~ I
54 F.H. J'IOHYIIIAHCbKA
lim [ {p(x~)(Bc~(y)co)(xe-y)dS = + I ~- ,o~ - 2 (P(Y)+ fs <p(x)( ba(y)o~)Cx- y)dS, (17)
1
lim f q~(xe)(Ba(x)o3)(x~-yldS = r, ~ q)(y)+ fq~(x)(Ba(x)o3)(x-y)dS, (18)
r 0S~z S
de S+~ = {x~= xr-~v(x), x~S}.
4. E~HHiCTb po3u'aaKiB rpaHHXlHHX 3 a ~ a q . Hexa l l u ~, u 2 ~ / ~ B a pO3B'$I3KH
C 2ct(~2) A C 2a-l('~) 3a~aui (1), (5), u = u I - u 2. Tozd L u = 0 B ~ , u IS = K~iacy
= 0. h neptuoi qbopMy~,4 Fpiaa (2) npH r = 11 u, spaxoBymaa, mo ~1 u I S = 0, o]xep-
myeMo
~ Uxi(f(2_2a)h ' ~'('QU)x i ) d x = O,
i=I
3Bi~C~
3
i=l
BI4KOp~4CTOByIOqrl nepeTBopeHH~I (Dyp'r 3 , MaCMO
3
i=1 i=l
TO6TO
o2o_2) _ I = o
i=l R 3 i=l
(19)
I$SN 0041-6053. Yrp. ~am. ~. pn.,1999 , m. 51, N ~ 1
3Bi~Crt ~/3=t r [3 [rl u][ 2 = 0. OTa<r supp ~ [1"Iu] = { 0 }. ToNy r lu - - no.~iHOM
[12]. OCKi.rmKH (~ U)(X) r E'(R3), TO (r I U)(X) ~ C = const, x e ~ . Aae TOni
(3a HeuepepBrdC'nO rlu ) rlu [ S -- C. OT~r C = 0.
3ayBa~m4Mo, mo CtmHiCTb pO3B'~3Ky BHyTpimHbOi 3a~aqi T~rIy He,Maria ~OBO-
/II4TSC~q TaK caMo. I3 dpopMyz~H (2) npH r = TI u, BpaxoBy~OqH, mO Bc~ u IS = 0, oaep-
mycMo ~ u ~ C = const.
OTme, npH aenepepBHil~ F Ha S pOZB'Z30K aaaa,~i j~ipix~e (1), (5) e~rmH~, pOa-
B'a30K 3a~aqi T~ny HeiaMaaa h~a piBI-I~IHHJI (I) S/~I4HI41~ 3 TOqHiCTIO /I0 a/IHTHBHOI
CTaaOi.
TaK caMo ~aoso~rrrbca e~HiCTb pO3B'~13Ky 3OBHilIIHbO1 aazxa~i ~ipix~e
Lu(x)=O, xe f l , , Uls=Fl(x), x eS , u(x)-->O npr~ Ixl-->** (20)
Ta 30BHimHbOi 3a~aqi T~ny He,Maria
Lu(x)=O, xef~ e, BaUls=F2(x), xeS , u(x).-->O nprx Ix[--> oo. (21)
1, x r e
Hexalt 111 (x) = 0, x ~ ~e" Iz nepmoi ~bopMy~a rpina ~tza o6.rtaeri ~ , npr~
= rl ~ u, u = u ~ - u ~ , u ~, u z ~ ~ a ~ao~i~mHi p o z s ' a z z a aa~a,~i (20) (a6o (21)) ,
o~epmyeMo
OCHOBHI FPAHHqHI 3A~AqI s O~HOFO PIBHYlHHH B LIPOISOBHX 1-1OXI]2HHX 55
3
I Z (:,.-..,., = o.
~e i=1
a OT)KC, (19) np~t aaMiHi ~(X) Ha TI l(X).
BI4KopaCTOByIOq;t nepeTBopeHHg q)yp'r MaTHMeMo, mo 11 ! u - - no~iHoM. A OC-
ziabKH u(x)--->O npH [ x l ~ o o , TO Tllu---- 0, TO6TO U(X) -- O, X ~ ' 2 e. OT)Ke,
pO3B'II30K 3a~aqi (20) r i pO3B'}I3OK 3a]laqi (21) CZlaHIafl.
5. P03B'~g3KH OCHOBHHX I"paHHqHHX 3a~aq npH HenepepsnHX FpaHHqHHX
~aHHX. I3 qbopMy~n (12) BHII.rlHBa~, mO pO3B'JI30K 3a~aqi ~(ipix~e MO)KHa 3a_rlrICaTH
y a~rna/li
u(x) = ~ F l ( y ) ( B a ( y ) o ) ) ( x - y ) d S - ~ V (y ) co (x - y )dS , x ~ , (22)
s s
ae V(Y) 3HaXO]IHTbCJ~ i3 yMOBH
V(y) oJ(x-y)dS = f Fl(x)(Ba(Y)CO)(x-y)dS, xE~'~ e. (23)
s s
Hexafi CnO~aTKy Ft(y) - 0. ~)yaKttia U(X) = Uo(X) = -- IS rl(y) CO(x-- y)dS, ari-
]IHO 3 (23),/IopiBHIOr HyJIeBi B R3\ ~ . To~ti
lim Uo(X) = 0, (24)
x --~ xo E S
x c R3 \ -~
a TaKO)K lira Ba (x 0) u o (x) = 0, TO6TO (3a qbopMy~o~ (16))
x--# x o r
x ~ R 3 \ ' ~
] V ( x 0 ) + J V(y)(Ba(xo)CO)(xO-Y)dS = O, Xoe S. (25)
S
OTz~e, c13yHKRi~I Uo(X ) 3a~oBo~IBHJ/e piBHZrlIdZ (1) B O6J~acTi ~2 i 3ri/IHO 3 (24)
Ta 3a HenepepBHiCTtO noBepxHeBoro noTenttia.ny Tnny npocToro tuapy B R3;
Uo] s = lim Uo(X) = 0. (26)
x-..> x o ~ S
x ~
TaKrlM qHHOM, U 0 (X) e pOaB'HaKOM 3a~aqi ~ipix.rle/I~J~ piBH~Hn~t (1) 3 Hy.rlbOBOIO
rpaI-IHqHOtO yMOBOIO. Ia e~arIHOCTi pOaB'$13Ky aa~aqi ~ipix.ne u 0 (x) - 0, x e ~ , a
TO~y lira Ba(xo)uo(x)=O. IaqbopMya-a(16)
x --d, Xo ~ S
x ~
V(x0)= lim Ba( x o ) uo (x ) - lim Ba(xo)u(x ) = 0 .
x...d, x o r x- - ) x o r
xe R3 ~ -~ xefl
OT)Ke, ~iHittHe OaHOpi~ae iHTerpaa~ue piaHJraHa (25) Mae Jvatue TpnBiaJmnnll poa-
a'SaOK. Toni O/IuoanaqltO poaB'~layBaHe piBaaHHg
I v ( x o ) + ] V(y)(Bct(Xo)O~)(xo-y)dS = ,(:Co), Xor S, (27)
s
g(xo) = lim Ba(xo)uo(x) f Fl(y)( Ba (y ) co) ( x - y) dS,
x"> x ~ S S
x e R 3 \'~
ISSN 0041-6053. YKp. ~tam. ~'vpn., 1999, m. 51. IW 1
56 F.H. JIOIIYlIIAHCbKA
/ao azoro Tax caMo rtprlxo/IrlMO i3 (23) npn F t(Y) ~ 0. BazopncTosymaa e/IrlHiCTb
pOZB'aazy aazlaai Tnny HeltMaHa B R 3 \ ~ , /IOSOZlnMO, UlO piBaanHa (27) eZBiBa-
~enTHe piBaarmm (23). Taxns ~nao~,/IoBe~eno aacTynHe raepa~erraa.
TeopeMa 2. Hexa~ F l ~ C ( S) . lcnye eOunu~ pozs ' ~ o ~ aaba~i ]lipix~e (1), (5),
arug auana~aem~c~ zzi3no z (22), (27).
PoaraaHe~o Tenep 3a~a~y (1), (6). Ia qbopsysta (3) Bnnn~sae neo6xizlaa yMo~a ii
pO3B' It3HOCTi
I F2dS = O. (28)
S
13 (12)/1ha poaB'aaKy 3allaai HaXBHe ao6pa~erma
u(x ) = I l l t ( Y ) ( B a ( Y ) C ~ I F 2 ( Y ) ~ x e (29)
s s
~e ~r aa~o~oa~nae piBHanna
V(y)(~(y)co)(x-y)dS = ] F~(y)co(~-y)dS, ~ ~ e3 \ f t . (30)
S S
E~iBa~euTnn~ ~o (30) e i nTe rpa~ne piBnanna 2-ro pony
s " s - g t ( x )+ ~ ( y ) ( B a ( y ) c o ) ( x - y ) d S = F ~ ( y ) o ~ ( x - y ) d S , x ~ S. (31)
s s
I3 306pa.,'KeHH~t
3
I . . . . . x - y ) ) e ,
8,~2( 3 2~
,3 J
e, ttri.ni,.isae, mo alp.,.<lxemt~ ~o .,.<i~;pa (.D~ (y)o~)(x-y) ~ (B~ (x)~)(x-y), "roMy clip.<i-
~enHM Z~o (31) e inTerpa.nbHe piBHIIHHII
= o, (32)
~ e r~ae e~nnHg niHiiano aeaa.aer~nHA pO3B'aaoK gO (Y) Taxala, mo
I go(x)m(x-y)dS - C O = const ~ 0, x e ~2.
S
Y~osa poaB'aaHoeri p immnna (31) 3a6eane~xyeaa,ca yUOBOm (28). OTace, BipRa T ~ a
TeopeMa.
TeopeMa 3. Hexa~ eu~onyem~ca (28). Ich3'e eOunu~ S mo~nicmn) Oo Oo~inbno~
aaumusnoi" cmanoi" poso' ~ o r 3aOa~i muna Hea~lana (1), (6). Bin 6usna~aembca
3ziOno 3 (29), (31).
AnanoriqHo BasqamT~Ca rpannqrli aa~atli/area piBaJmHJt (1) y npocTopax yaa-
ran~uennx qbyrmttia.
6. Poan'aaox yaa ra~ .neno$ aa~la,fi ]Iipix~e. Bi~Io~o, mo icHye e~mnnlt y
E'(R 3) poaa'aaoK pianmma (7) fi (x) = co * (B~t FI - F2). /Ina/IoniJvorloi ~ e D (R 3)
(,.n.a) (q~, "" D* = ~ , ( B , 6 - ~ ) ) - - - (@;~, , , 4 - ~ ) --
ISSN 0041-6053. Yh'p. ~tam. ~.'vpn., 1999, m. 51, N ~ 1
OCHOBHI FPAHHqHI 3A]],AHI fl, JI~ O/~HOFO PIBH~IHHg B ]IPOBOBHX FIOXI]2I-IHX 57
= F : > =
= ( f q)(x)(Ba(y)o))(x-y)dx, F I ) - < f ~p(x)co(x-y)dx, F2).
R 3 R ~
YMOBa fi = 0 B R 3 \ ~ nar (OCKiJmKH Ba (y)CZ MaC TO~-Xo~y oco6,rmnic'n,)
f q)(x)[<<Bct(y)r.0)(x-y), FII) - <co(x-y), Fz)]dx = 0,
R 3 X~
TO6TO
<(Bct(y)~)(x-y), F1>-<co(x-y), F2) = 0, xeR3\-~. (33)
3ni~cH 3HaA~e~o HeBi~OMy y3araaabHeny qbyHxtlim F~. I3 (33) ~L~ ~XOBiJ~SHOI ~ e
r D(S)
lira f ~p(x_~)Ba(x)((O)(x_~-y), F:)-<(Ba(y)co)(x_~-y) , F I ) ) d S = O,
r S_~
TO6TO
< lim .~ q)(x_~) (BaCx)co) (x_~-y) dS, F 2 ) =
~...->0 S_,
= <lim f (p(x_e)Bct(x)(Ba(y)co)(x_e-y)dS, Fz). (34)
r s_r
BHKOpHeTaCMO qbopMylr/(18). Toni 3a zeMom i3 (13) 3 icnysanaz
x ~S It(y) (Ba(Y)(o)(x-y)dS BHn~HBar
lim Bct (Y) ~ (P(x-e) (Ba (x) co) (x_e- y) dS = V l (y,~) e D(S).
P.-.~O
S_r
Tenep (34) MO>XHa 3anacaTH y BHrJzA~i
1 r~(y)+fr~(x)(Ba(x)o , ) ) (x-y)dS , F2) = <Vl(y, rp) , Fz), q) e D(S).
$
Iwrerpa~He piBHWaHa
~ 0 ( y ) + f (p(x)(Ba(x)co)(x- y)dS = g(y) (35)
S
cIIpH~KeHHM ~0 irrrerpa~bHoro piBHJtHHR (25), a TOMy 3a yOBe/~eHHbl BHII_~e OyHO-
3HaqHO pO3B'~I3He. OT~Ke, IIepeTBOpCHH$1/,I
<g,F:> = <Vl(y,~o,), Fl>, g e D(S), (36)
/~e ~pg - - pO3B'~ZOX piBH~IHHJI (35), O~HO3HaqHO BH3HatlaeTl~$[ F 2 E D ' (S ) .
I"IoKa~KeMO, RIO 3Halt~eHa yzara.rmHeHa qbyHKai~ F 2 3a~oBo~Hae yMony (33).
PozraasaeMo dpyHXttim W(x) = <O~(x-- y), V2> - <(Ba(y)co)(x- y), FI>, x e
e s162 To~ai
Lw(x) = ( L ( o ( x - y ) , F 2 ) - (L(Ba(Y)(O)(x-y) , Fl> = O, x e n ,
I SSN 0041-6053. Yxp. ~am. .~.'ypn., 1999. m. 5 J , N ~ 1
lira B a (x) x
c , - . ) x 0 E ~
58 F. EL YlOEIYLI.IAHCbKA
lira w(x) = ~hm.lxl_+**~o(x-" y), F2) - ( l l i m**(/~a(y)to)(x-y), FI) = 0,
Ixl-~**
lim ~ to(x_e)(Bot(x)w)(x_e)dS_ e = O, t o ~ D(S) , (37)
G~O &t
~riz~no ~ (18) i (36).
Or~e, ~ya~t~ia w (x) e poaa'za~o~ aoB.ittm~oi yaaran~nenoi aattam rrma He~-
~a.a aazp iaaa .na (1). FIo~ax~e~o, ttlo w(x) ~- 0 , x ~ f~ ~. Bcepeaani o6am~ri
fZeG, po~Mimenoi no~a no~epxne~ S_ e, BipHe ~o6pa~xen~ poas'~a~y piBrlarrH~l (1)
I.te(ye)co(z-ye)dS, z. e ~ee, to(~) (38)
S_,
/~e P-e (x-e) ~ pO3B'Z3OK peryzzpnoro i.verpasmnoro piB.znna
1
-~ l.te (x_e) + ! I.te (y_e) (Bo~ (x)o~ ) (x_e -Y-e) dS = Ba wCx_e).
Hexa~ 17 (x_ e, Y-G) ~ p e z o ~ e n r a at~pa u~oro inTerpaa~aoro piunznaa. Toai
~tG (Y-e) = ~ F (Y-G, x_~) (B a w) (x_ e) dS + Bc~ w (y_~).
s_,
l'Ii~c'ra~sla~o,~n ueR ~npa3 ~ a ~t e (Y-e) Y (38), o/Iep~KycMo
w(x) = f c~e(x_ e, z)(Bo~w)(x_~)dS, z ~ f~eG, (39)
S_~
~e
ei,~(x_e, z) = co(x- z_O + ~ F(y_e, x_~)o~(x- y_e)dS.
S_~
Icuyr lira OG(x_~, Z) = ~ (x , z ) . 3aaeMo~ [1, c. 65]
e--~0
lim [ ~e(x_e, z)(B~xw)(x_G)dS = lim ~ (lim ~ ( x _ e, z))(Baw)(x_e)dS,
ToMy i3 (39) o~aep~cyeMo
w(z) = lim [ O(x,z)(B,.w)(x_G)dS, z e ~e,
a Tozi ia (37) tO(Z) = O, Z ~ ~e"
Ma/~OBea-lrl e~mfica~ pO3B'$taKy 3OBnillIFIOi y3ara~sneaoi 3a~aqi wnny Hei~ana i
noKaza~l, mo SHaft/lena zri~HO 3 c~opMyaar, trl (35), (36) y3arayl~nena ~y'tllcd.ti~t F 2 e
e D'(S) ~aaOBOZbnae yMo~y (33).
Cni~airtnotuennzra (33) Bnanaaena ctxnaa y3arasmneaa dpym~uia F2. ~I~cno, zK-
m o 6 icnyBaan ~ i y~racn, neni ~yaKttji F21, F22, ~mi ~a~tOSOSmn~OTb (33), TO F 2
= F21 - F22 ~ROBO.rlbH$1Jla 6 ysosy
(O~(x--y), FZ) = O, x e f l e. (40)
,~a, mctfist w~(x) = ( ~ ( x - y), F~) r Lwl (x )=O, x e ~ ,
ISSN 0041-6053. YKp. 1,fam. ~.'vpn., 1999, m. 5 i , N e }
OCHOBHI FPAHHHHI 3A/]Attl B2Dt O~HOFO PIBH~IHH~I B/IPOBOBHX HOXI/2I-IHX 59
l i m f tp(x_~)Ba(X)Wl(X_~)dS = O, q~eD(S ) , w~(x)-~0, I x l ~ ,
r JS-~
a TObly i3 r p03B'.q3Ky 30BHi/.Ilaboi yzara . r lbaeHoi 3a/Iaai THny Het iMaHa SH-
na rmar J~g 6 y ~ o NOKa3aHO BHIRr III0 yMo~a (40) ez~i~aneHTHa y ~ o ~ i
l im [ epCx_e )Ba(x ) ( co (x_e -y ) , F2>dS = "0, ~pe D(S) ,
~ o ~ ,
TO6TO
I +f o ( ; cpfy) q~fx) (Ba(x)co) ( x - y ) =
s
3 a OgHO3HaqHO~O po3B'J~3HiCTaO i H T e r p a ~ s H o r o piBHJmHJ~ (35) o /~epmyeMo, m o
(g , F 2 ) = 0 / ~ a g o B i a S H O i g e D ( S ) , TO6TO F 2 = 0 y D ' ( S ) .
Tard~M qHHOM,/lOBe/leHO TaKy TeopeMy.
T e o p e M a 4. Hexaa F 1 e D ' ( S ) . Icnye eaunua pos#aaotc u ( x ) ysaeanbnenoi"
saDa~i ]lipixne a/za pisn~nn~ (1). Bin 8uana,~aem~a sa dpop~tynoto
u (x ) = ( ( B a ( y ) c o ) ( x - y ) , FI) - < c o ( x - y ) , F 2 ) , x e f~,
a y3aeanbnena qbyngt~ia F 2 ousna~ taem~ seiano s dpopzo, naatu (35) , (36) .
A n a ~ o r i q n o BHBHalOTbCJt y a a r a ~ b H e n a 3 a ~ a q a TImy He~MaHa, a T a K O ~ 30BniIiIHi
y~ra2IbHeHi rpaHI~Hi 3a/Iaqi.
1. Aepanoou,t M. C. ~.aa,n'ra~tecKHe crmry.aapH~e Hwrerpo-aaqbqbepeHtmam,H~r onepaTop~ //
YcnexH MaT. tlayK. -- 1965. -- 20. N g 5. - C . 3 - 120.
2. Butautc M. H., ~ctcuu F. 14. ~2~2IHnTHqeeKHr ypaBHeaaa B cnepaxax B orpaHaqenaoti O6JIaCTH 14
HX npH~O>KeHHa // TaM. ~ge. -- 1967. -- 22, N g 1. - C. 15 - 76.
3. Buu~wc M. 14.. 9ctcu, F. 14. YpaBHeaaa B cnelrrgax nepcMeHHoro nopa~tga // Tp. MOCK. MaT.
o-ha. - 1967. - 1 6 . - C . 25 -50 .
4. Boneou,~ .IL P. Fanoz~JmnTaqecgHe ypanHeHHa ~ C~pTKaX///IOga. AH CCCP. - 1966. - 168,
N* 6. - C . 1232 - 1235.
5. Xep~auDep 31. Flce~/~o/1HqbqbepeaaHa~,a~r onepaTop~. - M.: Map, 1987. - 690 c.
6. 9flDenb~tan C. ,ll., ~puub 5[. M. HocTpOeHrle H Heeaeao~aa~e K.rlaccaqecgrlx qbyH/laMenTa21bHbiX
petuenntl aa~aq~ KOmH pammMepHo napa6o.qrtqecgnx nCr ypa~nennR H
Mar. aceae~to~aHaa.- 1971.-Bbm. 6 3 . - C. 18 -33 .
7. ,l~pinb 5[. M., El'icge/tb~tatt C. ~. II)yn/laMeaTa.rlbHi ~a ' rp , t t i po3n'~13xiB nCVB/gO/:(rldpepeHllia.rlbHHX
napa6o.aiqHaX eue'reM 3 tlerJIalI~KHMH CHMBOJlaMH H Kpa~toai ~ a q i ~ pi3HHMH BHpOII2Ke:HHJIMH i
OCO6.rlrlBOCT.qblH: 36. llayK, npaub. - LlepniBRi, 1990. - C. 21 - 31.
8. Ko~.vSeii A. M. 3a~a,~a Korea ~a.qa zno.mounoHmaX ypaaHeH,tt ~apo6noro nopat~Ka///~.qbd/~-
pemt. ypaaaeHHa.- 1989.- 25, N ~ 8 . - C. 1359- 1368.
9. OeDoptog M. B. ACHMnTOTaKa qby~tgRtm Fpaaa ncen~to~taqbdpepemtaanbaoro napa6o.a.,~ecgoro
ypanHeHr~a // TaM ~ e . - 1978. - 14, N ~ 7. - C . 1296 - 1301.
10. Grubb G. Boundery problems for systems of partial differential operators of mixed o r d e r / / J .
Func. Anal . - 1977.- 2 6 . - P . 131 - 165.
11.///u~oo F.E. MaTeMaTHqecKrffi aaa21H3. BTOpOtt CHeRHa21bHld[! gypc. -- M.: H3/I-BO MOCK. yH-Ta,
1984. -- 207 C.
12. BnaDu~mpoa f C. YpaBHeHHH Ma'r~MaTHtleCKOfl C]~H3HKH, -- M.: Hayxa, 1981. - 512 c.
13..llonyutanctca,s F. H. 0 lt~KOTOpblg CBO~ICTBaX pelllr HCJ1OKa,/lbHldX ~,rlJIHnTHqeCKHX 3a,/laq B
npocrpaaerae o6o6meaa~x qbyHgttail//YKp. ~aT. a~ypn. - 1989. --41, N ~ 11. - C. 1487 - 1494.
O~tepxaHo 21.10.96
ISSN 0041-6053. Y~p. ~u~m. ~pn., 1999, ra. 51, I~ I
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| id | umjimathkievua-article-4582 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T03:01:38Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/dc/17a6c49aff13dd24af60e3d9395e29dc.pdf |
| spelling | umjimathkievua-article-45822020-03-18T21:09:14Z Basic boundary-value problems for one equation with fractional derivatives Основні граничні задачі для одного рівняння в дробових похідних Lopushanskaya, G. P. Лопушанська, Г. П. We prove some properties of solutions of an equation $\cfrac{\partial^{2\alpha}u}{\partial x_1^{2\alpha}} + \cfrac{\partial^{2\alpha}u}{\partial x_2^{2\alpha}} + \cfrac{\partial^{2\alpha}u}{\partial x_3^{2\alpha}} = 0, \quad \alpha \in \left( \cfrac 12\, ; 1 \right ]$, in a domain $\Omega \subset R^3$ which are similar to the properties of harmonic functions. By using the potential method, we investigate principal boundary-value problems for this equation. Доведені деякі властивості розв язків рівняння $\cfrac{\partial^{2\alpha}u}{\partial x_1^{2\alpha}} + \cfrac{\partial^{2\alpha}u}{\partial x_2^{2\alpha}} + \cfrac{\partial^{2\alpha}u}{\partial x_3^{2\alpha}} = 0, \quad \alpha \in \left( \cfrac 12\, ; 1 \right ]$ в області $Ω ⊂ R^3$, аналогічні властивостям гармонійних функцій. Методом потенціалу досліджено основні граничні задачі для цього рівняння. Institute of Mathematics, NAS of Ukraine 1999-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4582 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 1 (1999); 48–59 Український математичний журнал; Том 51 № 1 (1999); 48–59 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/4582/5868 https://umj.imath.kiev.ua/index.php/umj/article/view/4582/5869 Copyright (c) 1999 Lopushanskaya G. P. |
| spellingShingle | Lopushanskaya, G. P. Лопушанська, Г. П. Basic boundary-value problems for one equation with fractional derivatives |
| title | Basic boundary-value problems for one equation with fractional derivatives |
| title_alt | Основні граничні задачі для одного рівняння в дробових похідних |
| title_full | Basic boundary-value problems for one equation with fractional derivatives |
| title_fullStr | Basic boundary-value problems for one equation with fractional derivatives |
| title_full_unstemmed | Basic boundary-value problems for one equation with fractional derivatives |
| title_short | Basic boundary-value problems for one equation with fractional derivatives |
| title_sort | basic boundary-value problems for one equation with fractional derivatives |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4582 |
| work_keys_str_mv | AT lopushanskayagp basicboundaryvalueproblemsforoneequationwithfractionalderivatives AT lopušansʹkagp basicboundaryvalueproblemsforoneequationwithfractionalderivatives AT lopushanskayagp osnovnígraničnízadačídlâodnogorívnânnâvdrobovihpohídnih AT lopušansʹkagp osnovnígraničnízadačídlâodnogorívnânnâvdrobovihpohídnih |