Một số tính chất của Hàm Điiều Hòa 7

Luận văn Thạc sĩ toán học chuyên ngành Giải Tích -Chuyên đề : Một số tính chất của Hàm Điiều Hòa

~l.({(ht Yrl;/& (tJao Cf9c 200/ 36 ,//' '" (i7;' CY/':;

-DE chC(ng minh gl)n ma khong mat llnh l6ng quc1t,trong Chl(elng nay

chung loj xel mi6n D la qua diu deJn vi B={XER" :I xI < I} vdi bien la a13={

x E R": I xI< I} trong khong gian RII vdi n~ 2

-Bai loan gia lr! bien Dirichlet mi6n ngoai se c6 nhi€u nghi~m.Trongcac Sc1chgiao khoa,ngl(oi ta thl(Ong them tlnh chat CI}th~ cua u 0 vung vo q(clhl mdi xac lIinh dl(l}C bi~u thac cua u.Trong chuang nay ,chung toi l1l(a faml)t dang nghiQI11u1ng qm1t dll cIll(a 1'0tinh ch~fl cua 1I0 vo c~(c

-N(ji dung cua dll(ling nay dlCl;5Ctdnh bay d~t'a theo bai ban IS].

II lYIQTs6 DJNH NGHIA vA DJNH LY

1 Dilll, if' :

Xel mi2n 0 =13 (0,1') \ {O} ={x E nn : 0 < x < r}, vdi r> D.

Khi lIl) lc1nlc.lih~ng so c c6 cae tlnh chat sall :

. C E (0, I)

. Ml)i h2\111c1i6uhoa dlCOngu xac l1inh tren Q d€u th6a

\ix, Y E 13 (o,~), Ixi = Iy\=> clI(y) < u (x)

Clul thich:

Hang sf) c khong ph\! thuoc ham di6u boa u

Clllfllg mill"

- Gl)i K IA m~t ceiu ban ldnh ~ (tac K = ~ S) lhl K la t~p col11pacttrong

O Theo l1inh 1)/Harnack thl tl1n t~\i h~ng s6 c E (0, 1) co tinh chat

J;L(1 It 'jrt~t-(&to dt;c 200/ 37 Jfiu;}IbJt- ,9'h{Wth- %

C U(Y) <U(X)VtJi nH}ih~lIndi~ll hoa dl((jng II xac djnh tfen Q va v6i x, Y E K.

- Cui t lllYY thuoc (0, 1) va u 13ham di~u hoa dlf0ng tfen n.

H~l1n s(1 x -1- u (tx) cung 13 ham di~u hoa duong tren n, do d6 eungtho a bat dang lh(ic 0 tren, t(ic la

CU(ly) < u(tx) ,

r'\Ix, y E - S

- Ta thay ~ c6 lh~ c6 gia lrj bat k5' trong khoang (o,~) ne'u t dtiQC

chqn thich ht.ip trong (0, I) Do d6

'\Ix, Y E B (o,~), IxI = Iy I=>eu (y) < u (x).

0

2) Villh If':

.Gi~i sli II : Rli -1- Ria ham di~u hoa tren B\{O}={XERII : 0< I xI < I}.

. Ham sf) A[ullren B\{O} ducjc d1nh nghla nhl( sall:

Vdi x tuy Y thuQc B\{O}, A[uJ(x) la gia tr1 trung blnh eua ham u tren

m~t diu b,1n Idnh lxi, tuc la

trung lit) w la di~n rich cua m~t du don vi.

Khi l16 AluJ co bi~ll thue nhu sau :

.Tncc'lnghejp n = 2:

Xet ham so f : (0,1) ~ R dlC<JC xac d!nh bdi

s

Sau day ta se ttm bi~u th((c clla f(I x I), r6i suy ra bi~u th((c CI1aA[u](x)

- Ham s<1-:,~ u(r~)ur lien t~ICd~u khi r~ thuQc t?P compact trang B\{O},

11 neB ~E rlSrl

( U lAphap vecld drin vi hudng ngoai cua an ).

./'>U= 0 (do u la ham di~u boa)

Tl'( dJng lhere li~n tren, ta suy fa

'Vr E (0, I), f ~.(Vu)(~)ds(~) =h~ng sO' (h~ng sO'nay c1u~:!cgQi la kl)

r rS

- 1'1:(bi~u th(fc clh 1"(r) (j tren, ta SHYra

Gic1slt ham so th~(eu co cae tinh eha't salt:

ii) Lila ham lien t~le tren :B\{O}={XE RI1:o<1xl S';1}

iii) u(x)=O, '\IxE8B={XERI1:lxl=1}

iv) u(x)2:0, '\Ix E B(O,r)={xE RI1:0<1x I <r} (vdi r la h~ng sothuQc(0, I ) )

Khi do u co bi~u thue nillt salt

G()i Alul(x) la gia trj trung binh eua u tfen m~t du ban kinh 1 x I.

Ta se eh(tng minh u = A[u] tren B \ to}.

Xel mi~n B(O, f), rhea dint ly 11.1 (ehudng 4) ton tc~ih~ng so eE(O, I) saG cho vt'ii nH,>iham oi~u boa kh6ng am u tren B (0, f) o~u thoa

'\Ix, y ~ B (o,~) va I xI = I y1

.cu(y) < U (x)

D~l W=u- e.Alu]

Chltllg ll1inh dW1e chiu thanh cae bttde nlll! salt

Ihioc 1: (Chung minh w kh6ng am lfen B \ {O})

Coi x E B ( O,~)

'\Iy E B (o,~), Iy I = I x I => eu (y) < u (x)

, I .

,'fWill f/tZl? (.;YfO ekeD 200/ 41

LA~~:yZlt 9'h(l//bti 'f/;i

SHY I'd cAlul(x):::; u (x)

lI(X) - cAlul(x) ~0

w(x) ~ 0

Do ell)w khC>ugam lren B (D,~)

.G(.>iL==inf {w(x): x E B\{O}}

l'{)n lai day {ad lwng B \{O} saD cho w (ad -7L khi k -7 00

Day [a"J BallI trung t~p compact B nen t6n t~1iday con {bd hOi l~1v~ b

E D

Tn('1ug IH.fpbEB\{O}:

Ilam di~u boa w d'.ll clfc li6u khi x ==b, lheo nguyen Iy c~tcti6u

lhl w la ham hang lren B\{O}, w c6 gia ld khong am trang13(o,~) nen suy ra L ~ O

Tn('jng IH.fpbEGB : w(b)=D nen L = D.

Tn(Ojugh(}pb ==0 : w khC>ngam lren B(o,f) nen L ~ O.

Ta 11IC>ud) L ~ 0, do d6 w khong am lren B\{O}.

V~y la Cl) lI(X) - cAlul(x) ~ 0, '\Ix E B \{O}

Ihide 2: (ChCrng minh HeX) - Alul(x) ~ 0, '\Ix E B \ {O})

Bang ljUY IH,'P, la c6 till E (0, I) vdi ll1l)i Il1 EN

Ta U) tlll - tlll-J == C (1 - till-I) > 0, do d6 day {tm}tang, d6ng thai clay nay

hi ch~n biji I Hen hoi tl,lv~ gidi IH,lnma ta gqi la 1.

Cho 111. )-00, ll( (5.2) SHYra

I ==c + t (l - c)

,~jtf(/( fUll '(Jew ,)1oc : !{,i(J/ 42

/(y; ~yg1t ?-ZUltlt 'Yci

l = 1

- Xel day ham {wm} yoi Will=U -tm A[ul '

Ta lh,l'y Wm - cA\wm] = u-tm Alul cA [u -lmAlu]!

=II - tmAIuI- cAIll! + rImAIIII

=u-lc+tm(l-c)IAlunl

=U-tlllt-1A\Llml

= Wm+l

- Thcl) kel qLl21(I) d bt(dc I thl ham di~Ll hoa Wo khong am tren B\{ ()}

Ap dl,lI1gkc't LlU21(I) Jai vdi ham W()thl Ol((,iCtinh chflt khong am clla ham WItren13\{O:,

Bang Lluy n"Lp, ta suy ra ham Wm kh6ng am tren B\{O} vdi mQi so'nguycn dll\!ng 111,tLtCla

W11l(X)= u(x) - tmAlul(x) ~ 0, \imE N ,\ixE B\{O}

Clio Il) -} 00 thl tm -)0 1 (uo (3)), suy ra

u(x)-AllIj(X)~O, \ix E 13\{0}

(4)

GiilsLoeIt'\n ti;li Xu E B\ {O} san cho u(xo) > Alu](xo)

Do u lien t~IC[(.IiXonen t6n li;li mOt Ian c~n V clla Xosao cho

HeX)> A[u](xo) , \iXEV,

M~l kllac, theo ket qua (4) d bt(dc 2 ta co

lI(X)~ Alul(xo) , \ixEoB(O,l xol)

SHY ra gia trj trung blnh cua u tren m~t dtu 8B(0,l xol) iOn h(in A[lIl(xo),llrc la

.Lw( It Fltlto O{tO (7i9c 20{)/ 43

Jf:f~VJlb;to L9'hCl/lth- %;

Ihifie 4 : (ke'llu~n)

Alul c() bi~u lhlrc nhl( lrong dinh 19 1I.2(d1lWng 4), nen u Lung y~y

- Tn((Jng hl}P Il co:2:

Ta Ct) u(x) co:be-In I x I)+ c, \lXE B\{O}

Khi Ix I-~ I lhlu (x) -) 0 nen suy ra C= o.

Khi x E 13 (0, r) lhlu (x) ~ 0, suy fa b ~ 0

V~y u(x) =b (-In I x I) ydi b ~ 0

- Tnf(ing hl.ip n > 2:

Tau')u(x)co:blxI2-n+c, \Ix E 8\{0}

Khi I x 1-) I lhl u (x) -) 0 nen SHYfa c co:- b,

Llu lk) u(x) =b (I X 12-11-I)

Khi x E 13 (0, r) lhlu (x) ~ 0, SHY ra b ~ O

Cui a IllYy IhuQc D

T6n 1<.li qua cau B (a, f) c Q

'I'a s0 chl"rngminh u la ham ui~u hoa lren 8(a, f).

Do phep Ijnh Lie'nkh6ng lam lbay d6i Hnb di~u boa cua ham so nen ta c() Ih~ gd sera= 0 ma kh6ng mill tinh l6ng quat.

Thc!) c()ng lh(rc Poisson (dinh 19 II.l-chlWng 1) ta c6

Um (x )co: f H(~, x)ul1l (~)dS(~),

1~I=r

'\lXE 13(0, f), \1m E N

Ta Ihfiy

,: '

f ' co '-,,/'

.Jtt(ilt /tl/t C:7ao Clt~)C 200/ 44 Jt;;~vyJ1b 3ha~th- CP;i

Ulll (x) - fH(~, x)u(~)dS(~) S;; fH(~, X)IUm(~) - u(~)ldS(~)

Do lit) U ui~lI boa tren B (0, r)

Suy ra u ui~lI boa tren B (a, r)

V~y LIdieu boa Iren Q

2 - T =x

x-'

Ixl2

c) Suy 1'a tu tint ChEllb)

d) Cui x toy YthuQcE*.T6n t~i YEEthoa x=y*

Suy ra x*=y** ,ma LheDtinh chat (b) thl y**=y,nen x*=y

VGy x* LhuQcE

e) Suy fa lu ojnh nghTaclia x* va tinh chat (a).

0

3) Hill II IIghia.

Ghl Sl( L~p E e R'\{O} va ham so u : E-7 R

Ham s6 KI ul au\ic djnh nghTa nhL(salt

Kluj:E*-7R

\;Ix E E*, K[u](x):= Ix12-11 u(x *)

(j:' '/,"/) ~~/'

,cL,{ui/l ;/(1/I" Ciao O~~c :::!()O/ 46 ,//~ - 6,- CY/,~

JlJtttJlbll- ,!3h{b714 f/ c'i

4) /Jill" /y.

Gi~i su' Ii c IC\{ O} va ham so LI:E~ R

Khi lh\bien d6i K cua ham so Kluj chinh la u ,tL1'cla KIKluj]=u

Clll~ng minh

Do E**=E nen mien Xclcdint clla KIK[ull ehinh la E.

'IIx E E, KIKIu II(x):= Ixl2-n Klu](x *)

Gia si'i'w=KI gJ thl w :E*~R

Khi Lh)Kiwi c6 mi~n xac dinh la E**=E

Vdi x lllYylhuQc E,la c6

2-11( ) I 1

Gi~lSlt'u dieu hoa tren qua cau oejn V!B c R'\lien t~lCtren B.

Khi (It) KIu I dl(JC xac dinh tren RU\B b(~ic6ng thue

~ f Ix12_1KllIl(x)= ~ "' IQ=I!x - ~I" u(~)dS(~)

f IxI 2-11_lxI2 u(~)dS(~) (0

~Ixl- - = Ix - ~IIxl

2X

.1Wilt j/~ht' bao (j~(JC 2001 48 JJ:f;':fIblt ,-3Y;;a7t~%

7) J)i"h Ii

Gia sl'rt~p m0 E c IC\{O} va ham so u : E-t R

Khi Ll6,ham so 1Idieu hoa (ren E ne'u va chl ne'u K[uj dieu hoa tren E*

Cillfllg millh

-BliO'e 1:

Giii sl'r1Ila ham so dieu hoa tren n.11.TachCrngmint Klu] la ham so dieu boa tren R'\{ O) nhLrsau.

Coi x lLIYy thuQCqua du don vi B.

Ta cti u(x) ~ ~ J I-lxi' u(i;)dS(~)~ J H(~.x)u(C)dS(~)

=0 (theo chd thieh 0 Il-2-d1LWng I)

Do LIt)Klujla h~lJl1dieu hoa lren n!\B .

Theo dint nghla,ta c6

'IXE R"IB K[u](x):~ Ixl2-n{lx~2 J Suy ra Klulla ham giai rich tren W\{O} (do rich va hlJp cae h~\111 giiii rich la ham giai rich).

Do d6 ham ~(Klu}) la ham giili tieh tren UII\{0 },dong thai ham nay lrit;l tieulren R'\B,suy ra D.(Klu}) lri~t lieu lren eelR'\{O}, tue ham Klu} aiel! boa lren R'\{O}.

-Ihide 2 :

XGt u 1[1ham dieu hoa tren E c W\{ O}.

Cui b tLIYYthuQc E*.

Ta c6 a=b* thuQc E thoa b=a *

Theo djnh Iy IIl.6-clutong 1,t6n t~\imQt Ian c~n Va clla a saD cho u e6 the khai lri2n lhanh mQt ehu6i hQi t~1dell tren vung e~n nay.

Ham s6 Klulla tC1ngcua mQLchlloi cae ham K[ulll},ma Cell:ham

KllIlIl}dicu boa lren E*c R'\{O} (kel qua bl(OCI), nen lheo dinh Iy II.6

(chuoung4) Lac6 Klu J la h~\Jndi€u boa lren R1\ {O}.

IhiO'c 3 :

Gj~1Sll'KllllIA ham diell hoa ,do u=KIKlulJ nen then ket qui1clla buoc

2 La c6 1I I,} hAm c1i€lI hoa ,

LJ

CJlIi lhich:

th('o K[u I(x) ~ [x12 -n 1I( [X~2) di! tjnh cae l1~n ham dong philn cua K[u] then cac d~lo h~\Jl1rieng phgn Clla u,sau d6 tinh i'1K[u] va chang minh bj~u th((c nay bhng (),

[V-NGHI(~M CUA HAl TOAN DIRICHLET MIEN NGOAI.

lIBjllh Ii :

Xet bAi Lm1nDirichlet d6i voi mien ngoai coa qU(1diu don vi B trang RI1

(vdj n> I)

i'1u(x) =0, '\Ix E R"\ BHeX)= rex), '\Ix E oB

Lrongl1{)r lien t~lCtren bien oB va nghi~m u E C2 (R"\ B) n C (R'\B).

MOLnghi~m d~c bi~Lella bai loan nay la

Theo L1jnhIy llI.6-cht(dng 4 ,K[v] co bi€u th(rc sall

{ HlIll v di@uhoa tfen t~p md B\ {O}nen theo djnh Iy 111.7-chuang 4 la

c6 Kl v Il~1 h~lIn di@u hoa lfen R"\ B

-D€ chang minh KI u I lien l~lc lfen R"\B, ta chI dn ki€m tinh lien t~IC

cib Klvll<.li 013.

Cui ~lllY 5' lhuQc oB

Neu x -7 ~trong R"\B lhi x':::;-;- j-~:::; ~ :::;~, hk do v(x*)-7f(~).

v ~y KI v I lien t~ICtren I~"'\B

Ham sf{ Klvl thoa cae tlnh cha't ciia b[ti loan nen la mQI nghi~m eila b[li

trong d6 r lien l~IClren bien 3B.

Khi d6,mqi nghi~m uEC\R"\B )nC(R'\B) clla bAi loan d~u c6 d<.tng lren nll\ B nilL(sau

Ta lh{{yh IA II1Qlnghi~m clla bAi tOi:ln(lheo djnh 19 IV.l d lren)

Gia Sl( II lA mQt nghi~m ba't ky cua bili loan dang xet

-Gqi h(x)= CD1I:::llx - 111

'"

Ixl::: 1

.L(trilt F Ii", lDaD?(9C 200/ 52 Jt;~~JJt- 3hlVlb~%

(:;l)i w=Klgllhl w X,1c((jnh trcn B.

I-Hling di~ll boa tn~n t~p mo' R"\B nen w la ham di~ll boa tfen B (do

dinh Iy IIl.7-cllltdng 4).Ham g b~ng 0 tfen aB nen ham W cung b~ng 0 tren

NI1lev~Y w th6a cae Hnh cha't Hell trang dinh Iy

-Ta Cl) w=Klgl nen g=Klwl(theo dinh Iy llI.5-chlWng 4) ,SHYfa

0

.Yi~'~ht 'l(i~l (am (jac 200/ 53 .-,A:f~~!lJJb.9h{UJt~%

3) Vi d{l:

Xcl bAi loan Dirichlel d6i vdi mi€n ngoai ciia dla troll don vi B trang u?

') L\u(x) =0, '\Ix E R-\ Bu(x) =rex), '\Ix E 8B

-(i)(ii)lim HeX)==L

IXI-7oo

(hfj'uh~ln ) (iii)

lrung dt') r lien [~ICtren bien 8B va nghi~m llEC2(U?\ B )nC(R\B).

tHy chang rninh kef qll(1sall:

Cllllllg mill"

Gia sLfbai loan co nghi~m la u

Th~o c1jl1hIy IV.2-chlfc1ng4 ,nghi~m u co d<;lng

u(x)=w(x *)+h(x)

v(1i h(x)=

~

f lxl2_12It

Suy ra w(x*)=u(x)-h(x) , '\Ix E R-\B

Dod6 w(x)=u(x*)-h(x*) , '\Ix E B\{O}

Khi clH> X-7() trong B thl X*-7oo , u(x*) -7L (gi~l thief) va

.(:I<;~ldL'f(?~{ '(,;~o ~c 200/ 54

,jJ:j; 'j/b/L '%V/btf %

h(x"') ~.~211:f f(~)us(~)

aB

Do lit> khi x~O trung B thl

(tinh chftt clia cong thltc Poisson)

(ta gqi gidi h',111nay 1£1L' )

-Sau day ta se dllYng minh w bang 0 tren B\{O},

Xct ham g,lx)=w(x)+E(-lnl xl) IHlIl1gt;ui~u boa tren B\{O}.

Khi X~O lhl gix) ~oo, suy ra g,,(x»O nell x (hi gan O.

l-)~t m=inf'{ g,,(x) :xEB\{O}} (m c6 thi la -.00).

T6n t~liday{adlrong B\{O}sao cho gt;(ak)~m khi k~oo.

Day {ad Ham trong t~p compaGt B nen t6n t<;1i day con {bdhQi

tu v6 hE B,

Neu hEB\{Ollhl ham di~u hoa g" d(,lt gia tr! clfe tiiu khi x=b,

lhcn nguyen Iy Cl,retieu lhl g" 1£1ham hang (luon bang m) tren

B\{0 I,ma khi x gan 0 thl g,,(x»O nen suy ra m >O.Neli bE aB thl m=O

dn go;{h)=O.Neu h=O thl III;?0 do g,,(x»O khi x gaB O.Do d6 ta luon c6 III ;:::O,suy ra g,,(x) khong am tren B\{O} vdi mQi E >0, do d6 w(x) khtHlg [1mtren B\{OI,

Trong biiu lluk cua gix) LaxcL ham di~lI boa -w thay VIw,ly

lu~11lu'dng H,rla c6 ket qua -w(x) khong am trenB\{O Lt(rc fa w(x) khol1g llLrdngLren B\{OI,

Ta ua ch(rng Illinh w(x) VITakhong am VITakhong dlrOng tren B\{O},l1cn C()the suy ra w(x) bang 0 Lren B\{O}.

Dn (It>va bi6u th((c clia u LrenU,z\B la

au

Ham s6 u c6 biill th(rc d Ln3n th~t slf la nghi~m clla bai loan nell uIhl1a de linh chftL (i),(ii),(iii).Ta Lhfty 1Ithda (i)va (ii),cho x~oo Lhl

u(x) ~~ ff(~)dS(~)

211:

an

oLU-{?-/t Frz,/t C)(tu en t?c !to{N 55

.11bi ch~n d vung voCl,fC, ILtcIii :

3M>O,3 [>0, \Ix E U2\13,1xl >1'~ I HeX)I<M

lrong de) r licn LI.letrcn bicn 013vii ghi~m UEC2(U2\B)nC(n?\B),

Hay eh(fng minh nghi~m clia biii loan Jii

(Icing IhlJi lu(x *) 1< M khi I x*1 dti /dn (x* E n?\B),

j{{~i/{ Yrt/t' '(tv JL~C j!{j() / 56 J1:;;ljIblt 5YicWth tfi

do d6 I w(x) I hi d}~n hdi M+I~ ff(~)dS(~)1 khi x gan 0,

DB

XCL ha m g,lx)= w(x) -/-1=;(-Inlxl)

Ham g"t,li6u boa Iren B\{O},

Khi x~() ,Ihl g,,(x) ~CX),SHYra g,,(x»O ntll x dLi gftn O

f)[,il m=inf{ g,,(x) :xEB\{OI} (m c6 Iht3Ia -CX).

'I'(inL"li day{adLrong B\{Olsao cho g,,(ak)~m khi k~CX),

Day {ad nam Imng I~p compacL 13 Ben tan t'.li day con {hdhQi t~1v€hEB

Ncli hEB\{O}lhl ham di€u hoa g~ d<;ll gia Iri Cl,fCtit3l1 khi x=h ,tbeo

nguyen Iy cl,fCLieu Lhl g~ la ham hang (luan bang m) lren B\{0 },ma khi x

gin 0 Lhl g,,(x»O nen SHYra m >O,N6u hE 8B thlm=O do g~(b)=O ,Ne'u b=OLhl m ;:::() do gjx»O khi x gan O.l)o dt) La luan c6 m ;:::O,suy ra g,,(x) khangal1llren H\{OI vdi mqi E >0, do d6 w(x) khang am lfen B\{O},

Trung hit3u Iinte cLia g~(x) ta xet ham di€u boa -w thay VI w,ly lu~nll(ling llf Ll c() kc't L/u.i -w(x) khong am IrenB\{ 0 },t(rc Iii w(x) khong dl(Ong

Ire n B \ ( () I,

B\{0 !,ncn c() the SHYra w(x) bang 0 tren B\{O},

T6m l'.li, nghi~11lclla bai loan Iren Iii dllY nhat va c6 bit3u thl'i'c tTen n2\ B la

( /J ," i /J ,.ij';;

cy/,-./ 'jI-a'pc-'ll- :7h£Mth j/ /;;

trong U()r lien l~IClren bien aB va nghit%m UE C2 (R2\B) n C (U2\B)

IHy clurng l11inh nghi~111 CUi.!bai loan la

Cluj thieh :Vi cl~l3 (Iu(x) I bi ch~n khi Ix I kha 16n )la lru'ong h(.jp d~c

bi~l CUi.!vi cll.14 (I u(x) Ico th6 lien ra 00 khi x.~ oo).Chung toi v~n tdnh bay vicl~\3 vll11116n phan bit%lr5 tnc()ng hc;iPu bi ch~n va khong bi ch~n

CluIng lIlillh

Gd Sl~btli locin co nhgit%1111a1I

Thel> dinh Iy IV 2-chlcung 4 , nghit%m l\ c6 d<;tng

nell xER2\B

nell x EaBSuy ra

3B

(tinh cha't cua cong th(tc Poisson),

Xet ham gJx)= w(x)+£(-lnlxl) ,vdi XE B\{O}

Ham gt;di~lI hoa tren B\{ O}

D~t l1l=inf{ gt;(x) :xEB\{O}} (m c6 th61a -00)

T6n l<.liJay{adtrong B\{O}sao cho gt;(ak)~m khi k~oo

j{a}lt Frt-It 6au {/'['!c 200/ ss

am Lren B\{O} vdi mqi E >0, do d6 w(x) khong am Lren B\{O},

Twng hi6u LhC(cclla g,,(x) la xet h~lIn di€u boa -w thay VI w,ly lu~n

Il(dng It,r la Cl) kCI qua -w(x) khong am trenI3\{O},tC(cIii w(x) khong dl(OngLren Ii \ {° I

Ta da cIlll'ng minh w(x) vua khong am vua khong dlWng tren B\{O},ncn c() Ih6 SHYra w(x) bang 0 lren B\{O}.

Tl)l111',li,nghi~m dla bili Loan tren Iii duy nha't va c6 bitSu tllll'Clren

")-,u(x) = rex), '\Ix E aB

(i) (ii)

Hay chCrng minh ke'lquLl saIl :

Khi L hun tH~nIh1nghi~ m Ii}<.Iuynha't va c6 bi€u Ihuc tren n?\ B la