định lý liouville

trình bày về định lý liouville

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I GH1I THI(~U :

Twng dH(dng nay chung Wi dl(a ra m(H chung minh ngan gQn doi vdi djnh Iy Lioll ville md rc)ng, n()i dung clla chLWng nay (1L(~ictdnh bay dl,ra theo bIli han 141

II B'NH LY LIOUVILLE:

1 VillIL l.v Liouville

GiJ sil' h~lIll di6l1 hoa u : nil -+ n Ih6a

'\Ix E nil , I u(x) I::;M vdi M la hang so

Khi d() II la ham hang,

CllIlllg l1lilllz

D!nh Iy ni'lYc<5nhi6u dch chttng minh, hiiu he-I cac sach giao khoa v6 chtWng trlnh dc,tOham rieng d6u trlnh bay tlWng tlf nhlt quy€'n [1], nQi dung chtJ'ng minh nay nlllt sau :

- Coi XII IllY Y thuQc 1<:',

Vdi i E {I, , n}, d<,loham rieng phiin clla II then bien Xi la ham di~u .

boa, then cd ng Ihuc gi,i tri trung blnh clia ham di~u boa (djnh Iy JII.l-chlWng I) Ihl vdi I11Qir> 0 ta c<5

-I

I

f

oXi

(1'

lrong d(j

a(n) Ii'! Ih€' rich clla quit cat! (hin vi trong UII,

Dn III Vi I ::; M , ta co

( :'

JU(?;t }';z/( ,74;(/ 200/ 12

I

211

~ (X ) :::;M

( )

11-1

211

u.(n)rl1 2nM

-r

Cho r -7 () Ladt(dc 1~~(xo)1 =0

ax,1 V~y II lil h~lIn hang tren Rn,

Gi~i sii' b~lln di~ll boa II : RII -7 R th6a

\txER", Ill(X)1 :::;M+Alxll1.

Lrong d(S M, A, a la cac h~ng sf) ull'dng va a < 1

Khi lh) II I~lhil m hang lren un.

Cluing lIlinlt

~JU

' ,

.

,0.(1')ling voi phfi 11cham den tren hlnh ve

O(,)i X lflY Y lhll(?CR'\{O},

Xc Lr> I x I,

,jr,yu'jlen 7/2(///<4 1//;;

[J

~/;~il' (;;;Z~( '6r{o ,7f~(; 200/ 13

e-/'?':Ye1t <>'/lWIt/l f'l?

Vel') la the rich ell a Lluii du ban kinh I'

L~(r) = ( 13(0,1')u H(x,r) ) \ ( B(O,r) n H(x,r) )

D(r) = 13(0,r+ Ix I)\ 13(0,1'-1 xl) TbeD d!llh 19 gia trj trung blnh (d!nh 19 III-I ,clUfcJng I) thl

VCr) B(x.r)

=~-r fUdV+ fUdv

l

Vel')

Ta eu ng e6

U(O) = ~

, I,

u dV

J

VCr)

Bx liBo Bo \Bx

Suy ra

Iu(x)-U(O)I=~

,

-I JUdV- JUdV

Vel')

13x\130 I3x\130

~~

[,

,.

IUldV

J

Bx \B" Bo \Bx

L',(r)

D(r)

M,H khae ta e6 : \i~ED(r), Iu(~) 1 ~ M+A I ~I'~ ~ M+A(r+ 1 x 1t

I 1 M + A(r +Ix pe"

f

Vel')

D(r)

(tmug do ~(r) c OCr»~

( £: / <=;/ '; /jp

- M+ A(1' + I x I)':t (V(r + I x I) - VCr -I x I))

=(M + A(1' + 1 x ou)(1' + I x 0° - (1'-I x 0° 1'°

(1)

Ta lhify (1' + I x I)" - (r - I xI t la da lhuc c6 b~c (n - I) lheo r, do d6 bi6u lilac (j (I) lie'n v~ 0 khi r -7 00, tuc lu(x) - u(O) I -70 khi r -7 00, suy ra

V~y u la ham h~ng tren R",

0

C!lIl tbieb:

Clu'rng minh ciia dinh 19 Liouville md rQng vftn ap dl,lng dl(QC khi ch(fng minh uinh 19 Liouville,

3 Vi d{l:

Xet ham s6u : n? -7 R vdi dinh nghTa nln( sau

Vx=(x,y,z) EU3,

( )

2

?

I-lay ch((ng ll1inh u kh6ng la ham di~u boa tfen RII ?

C'dlng lIlinb

- Tu (;() II (0, 0, 0) = 0

va Jim U (x, 0, 0) = + <X) X~OO

(2)

3

1

x

l- + Il

l

-(

2 )

(41x12 )2

( ;(' , 1/ / '-;;;i?'

Iu(,?// j/ 12// (;(to i/1J?C 20{J/ 15 c/t;;t;!IJ1t <-:n;Vlth%

~ 4~ Ixl~ + 12.(12IxI2)~

43

~(47+12.3)IXI3 =831Xl3

M~t khac u kh6ng la ham hang (do (2)), theo d!nh Iy Liouville md n1ng thiu kh611gla ham c1i~uhoa lIen RI1

0

Cllil thiGh:

PhlMng philp khae la tinh cae c1~lOham rieng phftn

, , , '), '), ')

ax ay at ax- ay-

h~n dung phlwng trinh Maple) c1~tinh va rut gQn flu.