an
XcI h{1l1lg,,(X)=W(X)+E( IxI2-Jl-l ) !-H1111g;:dicu hoa Lren13\{O}.
Ta c()
go:(x) ==IxI2-1I11I(X*) - h(x*) I+ E(lxI2-n -1)
==(II(x*)-h(x*)+E)lxI2-II-E
Khi cho x~O thl x*~C(), ILk d6 u(x*)~L va I
h(x*) ~ - ff(~)ds(~)
coas
Do lit) khi x~o tIll gAx) ~co, SHYra ge(X»O He'll x lIll gan O. I-)~l 111=inJ'{gAx) :xEB\{ O} } (m c6 the la -co).
(tinh chit clla tong thuc Poisson)
Tc1n L<.liday{adLrong B\{O}sao tho g,,(ak)~m khi k~co.
Day {ad n~m Lrung L~p compact B uen L6n t~i day can {bdhQi ll~ ve bE B.
Nell hE B\{ O}lhl h::l111(lieu hoa g" d~ll ghi Lr! cllc Lil5u khi x=b ,theo nguyen Iy CIt'CLieu Lhl g" la ham h~ng (luau bang m) tren B\{O},ma khi x gfin 0 thl gjx»O Hen SHYra m >O.Ne'u bE DB thl m=O do gib)=O .Ne'u b=O thl m 2::0 do g,:Cx»O khi x gftn O.Do d6 La luon c6 m 2::O,suy ra ge(x) khong am Lren B\{O) vdi nll.>iE >0, do d6 w(x) khong am tren B\{O}.
Trong hi<5u Lh(t'cclla ge(x) ta XCI ham dicu boa -w thay VI w,ly lu~n tLt'dng LL.I'La c6 ke't quit -w(x) khong am trenB\{ 0 },t((c Ia w(x) khong dlt'dng lrcnB\{O}.
Ta (la ch(rng minh w(x) vi'ra kh6ng am vaa kh6ng dlWng tren
B\{O},nen C()the SHYra w(x) bang 0 tren B\{O}.Biell Ih(t'cclla nghiC;111II lren l{'\B la Biell Ih(t'cclla nghiC;111II lren l{'\B la
I I ')-il II(X) =X- w(x*) + hex) = hex) ') =~ f'xl- -) f(~)dS(~) OJaBlx- ~III
-H{lIll sf) u c6 bieu th(t'c d tn~n th~t Sl.t'la nghi~m clla hai loan ne'li u lh6a dc Linh ch[lt (i),(ii),(iii).Ta thiy u thoa (i)va (ii),cho x~co Lhl
(jJ (,,' £' c:;w.'
,j{U,ilt Yrz,11 (5«(> (;7/((;(; Z(}(J! 62 Jf~u:!lb/? 5lirvlz4 r1i
u(x) -+~ frU;')dS(~)DB DB
do (16 u Ih6a Ilnll chat (iii) VIdang xellnfong h<;lp
L= ~ It'( ~)dS(~)DB DB . Tn((jng h(Jp Lot:L fr(~)dS(~): DB Khi x-+() lhl +aJ neu L >! ff(~)dS(~) (0 DB w(x) -+ neu L <~ ff(~)dS(~) DB
Nhl!' Ih6 ,[(in t<}i1'>0 suo cho w luon Juon dlwng huy Juan 1LJ()nam l1'l3n mi0n B(O,r), M~l khac, ham so di0u boa w lri~l lieu lren bien DB,
Thco d!nh Iy II.3-chl!'dng 4 ,ham w c6 biGu LInk lren B \{O} la W(X)=k(lxI2-1l -I) , \ixEB\{O}
-UJ
Loc dt) II co biGu llul'c LIen u.11\B la ll(X)= Ix\2-1l w(x*)+h(x)
=lxI2-1l .k ~x *12-11-1 )+h(X) =k (I -lxI2-11) +h(x)
[J H~lIn so u c6 bit311 lh(fc d lren lh~l Sl,r Iii nglli~m cila biii loan ne'll II Lh6a cac llnll chat (i),(ii),(iii).Ta lhay II lhoa (i)va (ii),cho X-+aJ IhI
ll(X) -+k+~
fr(~)dS(~) , (adsbygoogle = window.adsbygoogle || []).push({});
(0
DB
do (I() II llH')a llnh chat (iii) neu va chi neu
Lo'c I;l L=k+~ fr(~)dS(~) CD DB k=L- L fr( ~)dS(~) DB
CI) ,I' ' 0 ';:(/)
J {{(tit YrZ/t '( )(tu J'L rc !!(J() / 63 j/' - 67/ OJ/.-
J IftiJIblb <;/ fliNt/if j/ (i
V~y ngllicl1lu c() bi6u Iherc
u(x)=
[L - ~ ff(~)JS(~)DB )(1-lxI2-")+~ flxl:DBIx ~I-,: f(~)ds(i;)
Ki/t [Will: G0p hai kct qua doi v(jj hai tHrong hC,5Pclia L,ta Olrcjc kit
qui! dn chlfng minh,
0
8) Vi d{l:
Xct hui 10,ln Dirichlet doi vdi mi~n ngoai clia qucl du odn vi B trungRII (vdi n>2) RII (vdi n>2)
L\u(X) = 0, "I/x E R'\ B HeX)= rex), "I/x E as
( i) (ii) (iii)
1i1l1 u(x)=L Ixl~oo
lrung do L bang +00 hay -00 , r lien tt,lCtren bien aB va nghi~rn
1 --
UEC-(Jt"\ 13) n C (If'\B),
Hay clufng minh hai LOanIren vo nghj~rn.
C'dlng minlt
Giii sli' h~lito,1n c6 nghj~m la u,Thco d!nh ly IV-l ,nghi~m u co di;lng Thco d!nh ly IV-l ,nghi~m u co di;lng
11
"-11
u(x)= x - w(x *)+h(x)
SHY ra w(x*)= Ixl"-2 (u(x)-h(x» , "I/x E R'\ B Do lit) w(x)= IxI2-11 (u(x *)-h(x*» , "I/x E B\{ O}
Khi x~() thl w(x)ticn toj +00 hay -00,
1 pxl' -I . (adsbygoogle = window.adsbygoogle || []).push({});
v(ji h(x)=i (D I -I" l(QdS()
,neu x ER" \ B neu x ER" \ B DBx rex) ' neu XEaS
(~) ",. /) (jjt~
.'JcU;/dt Frldt' (j(?u (f[(;m 200/ 64
cA;~v;;g,? '%?/lZh cy;;
NIH!' lhC~,ldn l'-;lir>O sao cho w lu6n Juan dlfcjng hay lu6n lu6n am tren mien B(O,r), M~t klHlc, h~\ln s6 dieu hoa w tri~t lieu tren bien DB.
Thco d!nh Iy I1.3-dn((jng 4 ,ham w c6 bi€u thlic lren B\{O} la
II?-u
w(x)=k( x - -I), \ixEB\{O}
Luc do LIco biEu thu'c tren If\B la
II?-nu(x)= x - w(x*)+h(x) u(x)= x - w(x*)+h(x) =Ix12-11.k ~x *12-11- 1)+h(X) =k(I-lxI2-U) +h(x) Cho x~co lhl u(x) ~k+~ ff(~)ds(~) 2n DB
(lieu nay mall thu£ln vdi Linhchat (iii).
(huu h<.lIl),
V~y h~li loan v6 nghi~m.