Post on 03-Apr-2018
7/28/2019 Mt s vn v khng gian Sobolev
1/64
- 1 -
Li cm t
Thi gian thm thot thoi a, chp mt m em hon
thnh bn nm i hc. Nh ngy no, u kha hc, ba cn
a n trng gp thy c mi, bn b mi vi bao bng lo
lng. Vy m cui cng em cng tri qua bn nm hc. Bn
nm hc tp vi bit bao kh khn, vt v, c nhng lc vp
ng em tng nh mnh khng th vt qua. Nhng mong
mun c lm lun vn khi tt nghip thc y em phn
u nhiu trong hc tp. Cui cng vi kt qu t c trong
cc nm u, em c b mn phn cng lm lun vn di s
hng dn ca thy Phm Gia Khnh. c lm lun vn l
mt nim vui, nim vinh d ln i vi em. Nhng bn cnh
cng c khng t ni lo v gp nhiu kh khn, no l khan
him ti liu, thi gian hn hp, m kin thc th mi v tng
i kh Nhng vi kin thc m em c thy c b mn
trang b trong cc nm qua cng vi s hng dn nhit tnh
ca thy Phm Gia Khnh cng nh s ng vin gip ca
gia nh, bn b, cui cng cun lun vn cng c hon
thnh. Em xin gi li cm n chn thnh nht n thy hng
dn, cc thy c khc trong b mn, cng gia nh v bn b.
Cn Th, thng 5 nm 2009
Ngi vitSinh vin. Phm Trn Nguyt Tho
7/28/2019 Mt s vn v khng gian Sobolev
2/64
- 2 -
PHN M U
1. L DO CHN TI
Nh bit, vic nghin cu qu trnh ng trong t nhin cng nh trong
x hi thng dn n vic kho st mt hay nhiu phng trnh o hm ring
bng vic nh lng ha cc c trng ca i tng nghin cu bng cc i
lng ton hc. Nhng ta cng d nhn thy rng cc quy lut t nhin thng
dn n cc h thc phi tuyn gia cc tham bin nn cn phi xt phng trnh
vi phn phi tuyn. Tuy nhin, khi xut hin nhng kh khn ton hc thc s.
Bi vy, khi xy dng m hnh ton hc chng ta buc phi bt tnh chnh xc v
b qua nhng phn thm phi tuyn b hoc chuyn sang tuyn tnh ha trong mtln cn ca nghim cho bng cch a bi ton v bi ton tuyn tnh. Vn
cha , gii bi ton ny ta li c nhng s thay i nht nh i vi gi
thit ca bi ton tng ng nghim ca n cng c nhng thay i nht nh.
Khi , vic tm nghim c in ca bi ton mi vn cn rt phc tp, v th,
utin ngi ta xy dng nghim suy rng ca n, sau thit lp trn ca
chng v chng minh n l nghim c in ca bi ton. Ni nh vy thy
rng, khng gian nghim ca bi ton c gii c nhiu thay i so vikhng gian nghim ca bi ton thc t ban u. V vy, vic chn cc khng
gian hm cho nghim ca cc bi ton c mt vai tr quan trng m bo tnh
t ng ca bi ton. Mt khng gian phim hm tuyn tnh c s dng rng
ri trong l thuyt phng trnh o hm ring l khng gian Sobolev.
C phn yu thch ton hc ng dng v c s hng dn gi ca
thy Phm Gia Khnh hon thnh lun vn tt nghip kha hc cng nh
bc u lm quen vi Phng trnh o hm ring hin i em quyt nh
chn ti Mt s vn v khng gian Sobolev .
2. MC CH NGHIN CU
Mc ch nghin cu nhm nm c cc nh ngha, nh l, tnh cht
lin quan n khng gian Sobolev. c bit quan trng v cng l mc tiu chnh
l xy dng h thng v d minh ha v gii cc bi tp c lin quan n n.
7/28/2019 Mt s vn v khng gian Sobolev
3/64
- 3 -
Qua , gip cng c cc kin thc c hc trong sut 4 nm i hc nh:
gii tch 1, 2, khng gian tp, o v tch phn Lebesgue, gii tch hm
3. PHNG PHP NGHIN CU
Qu trnh lm lun vn s dng kt hp nhiu phng php nghin
cu, nhng ch yu l phng php tng kt kinh nghim.
C th, kt hp phng php tng hp, so snh, phn tch, nhn xt trong
qu trnh nghin cu l thuyt. u tin, sau khi tm c ngun ti liu tham
kho th tng hp cc kin thc trong vi cc kin thc sn c. Sau , tin
hnh so snh, phn tch chng, chn ra nhng kin thc trng tm, ng ghi nh,
t a ra nhng nhn xt ring. Cui cng, tng hp, trnh by li theo hiu
mt cch r rng.
Phng php tng hp, phn tch, so snh, nh gi cng c kt hp sdng trong gii v sp xp cc bi tp. Sau khi tng hp cc bi tp t cc ngun
t liu khc nhau, s tin hnh phn tch, so snh, chn lc cc bi tp hay ph
hp vi ni dung l thuyt sp xp chng theo mt trnh t hp l. Cui cng, tin
hnh phn tch, nh gi gii mt cch chi tit, trnh by mt cch r rang.
4. NI DUNG LUN VN
Ni dung ca lun vn gm 2 chng.
Chng 1. Gm 4 , trong 1. Khng gianp
L ,2. Bin i Fourier,3. Hm suy rng, 4. Khng gian Sobolev. y, ch tp trung gii thiu mt
s nh ngha, nh l, tnh cht v cc v d c lin quan m khng quan tm n
vic chng minh cc nh l, tnh cht .
Chng 2. Gii chi tit v sp xp mt cch tng i hp l cc bi tp
c lin quan n phn l thuyt gii thiu chng 1.
7/28/2019 Mt s vn v khng gian Sobolev
4/64
- 4 -
PHN NI DUNG
CHNG 1
KIN THC CHUN B1. Khng gian pL
1.1 Khng gian pL
Cho ,, S l mt khng gian o, trong l mt tp con m ca
khng gian Euclide n chiu Rn, Sl -i s trn tp o c Lebesgue v l
o Lebesgue. Cho p1 , ta nh ngha khng gian pL nh sau
Vi p1 , ta nh ngha
pL { ff : l hm o c v xdxfp
}
v
p
pp
p
pdfxdxff
/1/1
Vi p , ta nh ngha
L { ff : l hm o c v kxf hu khp ni 0, k }
v
f inf { KxfK :0 hu khp ni}
Ch . Ni kxf hu khp ni tng ng vi ni rng
0: Kxfx .
Nu gf, l hai hm o c tha xgxf hu khp ni th f v g
c xem l ging nhau. Do , 0p
f khi v ch khi 0xf hu khp ni,
vi p1 .
Cho p1 , ch s q tha 111
qp
c gi l s m lin hp cap.
Ta thy, 1p th q . Ngc li, p th 1q .
7/28/2019 Mt s vn v khng gian Sobolev
5/64
- 5 -
1.2 Mt s nh lv bt ng thc
1.2.1B
Cho a, b l hai s thc khng m,p, q l cp s m lin hp. Khi ,
q
b
p
aab
qp
.
1.2.2Bt ng thc Hoder.Nu qp LgvLf th
1Lfg v qp gffg 1
Du = xy ra khi v ch khi BA, R+ sao cho qp xgBxfA .
1.2.3Bt ng thc Minkowski.Nu pLgf , th
pppgfgf , vi p1 .
Du = xy ra khi v ch khi BA, R+, 022 BA sao cho BgAf .
1.2.4nh l. pL l khng gian Banach.
1.2.5nh l. pL l khng gian phn x, vi p1 .
1.3 Tch chp
Cho 1, Lgf , tch chp ca f v gc nh ngha l
dyygyxfxgf
1.4 Gi ca hm
1.4.1nh ngha. Cho f l mt hm lin tc trn Rn. Gi ca f , k hiu
l supp f , l bao ng ca tp 0: xfx .
K hiu cC (Rn) l k hiu tp tt c cc hm lin tc vi gi compact.
cC (Rn) thng c vit l D (Rn).
1.4.2 V d
Cho :f R Rc xc nh
0,0
0,2/1
x
xexf
x
Khi , Cf .
7/28/2019 Mt s vn v khng gian Sobolev
6/64
- 6 -
Cho :f Rn Rcxc nh
ax
axexf
xaa
,0
,))/((222
, vi 2212
... nxxx
Khi , xf D (Rn
) v supp f axxaB :),0( . Cho 0 v nh ngha /xx n , vi 1L (Rn), 11 v
0x khi 11
. Tht vy,
1/ nnn RRn
Rdyydxxdxx , vi /xy .
Cho 0 v nh ngha /xCx n , vi nR dxxC 1 v: Rn R c cho bi hm
1,0
1,))1/(1( 2
x
xex
x
Khi , x D (Rn) v supp ),0( B
2. Bin i Fourier
2.1 K hiu
nxxxx ,...,, 21 (Rn),
n
j
jjxx
1
. , vi ,x (Rn).
nndxdxdxxdm ...
2
1212/
o c Lebesgue trn Rn.
yxfxfy , vi y thay i trn Rn,
/1
xfxfn
, 0 ,
;11
ffy
11
ff .
Cho 1, Lgf (Rn), tch chp nR dyygyxfxgf ,111 gfgf .
a ch s jn a,,...,, 21 N,
n
j
ja
1
. Cho Rn,
n
n
...21 21 vij
jx
, v nnD
...21 21 .
7/28/2019 Mt s vn v khng gian Sobolev
7/64
- 7 -
Cho 1Lf (Rn), bin i Fourier ca f c nh ngha l nR
xi xdmexff .
Vi mi Rn, xiex l mt hm c trng trong Rn.
2.2 Tnh cht c bn
1Lf (Rn),1
ff
.
1Lf (Rn), 0 Cf ( Rn). 1, Lgf (Rn), gfgf ^ . fef yiy ^ , fxfe xi ^ 00 v ff . Nu 1Lf (Rn) v 1Lfj ( Rn) th fif jj ^ .
Nu 1Lf (Rn) v 1LfD (Rn), k , th .^ fifD
Nu 1Lf (Rn) v 1Lxfxj (Rn) th f kh vi n j v ^ xfixf jj
Nu 1Lf (Rn), 1Lfx (Rn) v fD tn ti, th
^ xfixfD
2.3 V d
(Gauss) 2/2xex ; 2/2 e ; 2/12
n
ij
jxx .
(Poisson)
2/121/
n
nxCx , vi nC lm cho 11 th
e .
(Fejer)
n
j
j
j
nx
xCxK
1 2
2
2/
2/sin;
n
jK
11 .
(de la Vallie Pousin) (cho 1n ) xKxKxV 22 . Khi ,
2,0
2,2
,1
V .
7/28/2019 Mt s vn v khng gian Sobolev
8/64
- 8 -
2.4 nh l o cabin iFourier
Nu 1Lf (Rn) v 1 Lf (Rn) th dmefxf xiRn
hu khp ni.
2.5 nh l Plancherel
Nu21
LLf (R
n
) th2
Lf (R
n
), 22
ff v nh x
:F 21 LL (Rn) 2L (Rn) c cho bi fFf c thc trin thnh mt ng
c 2L (Rn) 2L (Rn).
2.6 Khng gian Schwartz S
Khng gian Schwartz S l khng gian ca cc hm tiu chun m bt bin
i vi cc php ton bin i, php nhn cn bng, m rng, nhn bi hm c
trng, tch nh x, php tnh tch phn v php bin i Fourier. Khng gian
Schwartz S c m t
S { C (Rn): ,,supnR
xDxx
}
y , l a ch s.
Mt vi ch
Ch SD nn S tr mt trong pL (Rn), p1 . Mt hm
2x
ex
, 0 thuc Snhng khng thucD.
Cho mt a thc P v S , SxxP v SDP . S khi v ch khi vi mi s nguyn 0k v vi mi a ch s ta
c xDx k 21 gii ni.
l song nh t S vo S. Khi , ta c cc kt qui. ^ xixD .
ii. ^ iD .
7/28/2019 Mt s vn v khng gian Sobolev
9/64
- 9 -
2.7 Hm suy rng iu ha
2.7.1nh ngha. Khng gian tp i ngu 'S ca khng gian
Schwartz Sc gi l khng gian ca nhng hm suy rng iu ha.
2.7.2 V d
Cho pLf (Rn), p1 , nh ngha CSTf : c xc nh bi nRf dxxxffT ,
Khi , 'ppf
fT do fT l lin tc.
Nu M (Rn) (Khng gian ca cc o gii ni thng thng, ingu ca 0C (R
n)), xt
n
R
xdxT
Khi 'ST .
Chof l mt hm o c trn Rn sao cho vi mi s nguyn khngm kta c pk Lfx 21 (Rn), vi p1 . Khi ,
nRf dxxxfT
xc nh mt hm trong 'S , do
fT
nRkk
dxxxxfx
22
11
nn hm cho l hm iu ha.
Nu l mt o thng thng trn Rn sao cho Mx k 21 (Rn), theo cch xc nh trn 'ST . o cho c gi l o iu ha.
2.7.3nh l. Mt hm tuyn tnhL trn Sl mt hm iu ha nu v ch
nu tn ti mt hng s 0
C v s nguynl,
msao cho
SCLml
,,
, .
7/28/2019 Mt s vn v khng gian Sobolev
10/64
- 10 -
2.7.4 Ton t trong S . Cho T S.
Php tnh tin. Nu hRn, nh ngha STT hh , th'STh .
Php nhn vi mt phn t ca S. Cho S , nh ngha TT . Khi , 'ST . Nu P l mt a thc trn Rn, PT c nh
ngha ging nh trn cng l mt hm iu ha.
Php phn x. ~~ TT . Khi , ST~ . Php tnh vi phn. Cho mt a ch s , nh ngha
SDTTD ,1
(Cng thc trn cho ta mt php tnh tch phn). Do , 'STD .
Tch chp. Cho S nh ngha TT . Khi ,'ST .
Lc , ta xt hm xTxF . Khi CF (Rn) v . TdxxTdxxTdxxxF nnn R xR xR
iu ny ch ra rng, tch chp vi mt phn t ca Sl mt qu trnh trn. Vi
mi hm iu ha T, th CT .
Bin i Fourier.nh ngha , TT S . Khi , ' ST .Kt hp php bin i Fourier v php vi tch phn, ta c
i. TxiTD
ii. TiTD
2.7.5 V d. Cho T tha 0 T . Khi
xx
)0('
j
0,1 x = x
jj , S , vi phn l mt trng hp c bit ca tch chp.
7/28/2019 Mt s vn v khng gian Sobolev
11/64
- 11 -
3. Hm suy rng
3.1 Khng gian cc hm chun D
3.1.1 nh ngha. Mt hm tiu chun nxxxx ,...,, 21 trn Rn l
mt hm kh vi v hn trn
v trit tiu bn ngoi min gii hn, min giihn c th ph thuc vo hm tiu chun. Khng gian tt c cc hm tiu chun
trn c k hiu l D .
3.1.2 V d.
Cho : R R c xc nh bi
1,0
1,1/12
x
xex
x
D dng kim tra l mt hm v hn kh vi, tr trng hp 1x . V l hm chn, ta ch cn kim tra tnh kh vi ti 1x .
Ta c,
0lim 11
1
2
x
xe
Suy ra lin tc ti 1x
Hn na,
0lim 11
11
1
22
x
xx
em
Do , tt c o hm ca bng 0 ti 1x
Mt hm tiu chun i xng cu trn Rnc cho bi
1,0
1,1/12
x
xex
x
Trong , x l khong cch t tm nx.
3.1.4 Mt s tnh cht
Nu D21 , th Dcc 2211 vi mi s thc 21 cvc . Nu thuc D v a kh vi v hn trn th .a cng thuc
D .
7/28/2019 Mt s vn v khng gian Sobolev
12/64
- 12 -
Nu thuc D th mi o hm ring ca cng thuc D . Cho hm nh trong v d hm tiu chun i xng cu, khi
0
xxcng l mt hm tiu chun trn Rn trit tiu ngoi hnh cu tmx0 bn
knh .
Cho Dxxxm ,...,, 21 (R
m) v Dxx nm ,...,1 (Rn-m). Nu
nxxx ,...,,. 21 mxxx ,...,, 21 . nm xx ,...,1 th D. (Rn).
3.2nh ngha v dy rng
Chng ta ni mt dy Dm l mt dy rng trong D nu
0m , trong D tn ti mt tp con compact c nh K sao cho
supp Km vi tt c m, m v tt c o hm ca n u hi t u n 0
trn K.
3.3 nh ngha v hm suy rng
Mt hm tuyn tnh T trn D c gi l mt hm suy rng trn
nu 0mT vi mi dy rng m trong . Khng gian cc hm suy rng
c k hiu l 'D .
3.4 nh l. Cho f l mt hm kh tch a phng trn mt tpcon m Rn. nh ngha
dxxxfTf
Khi , fT thuc 'D .
Nhn xt. Cho pLf , 1p . Khi , fT 'D .
V d
Hm suy rng Dirac.Cho x Rn, nh ngha
Dxx , (Rn)
D dng chng minh rng 'Dx (Rn).
Trng hp 0 c gi l hm suy rng Dirac.
7/28/2019 Mt s vn v khng gian Sobolev
13/64
- 13 -
Cho Tc nh ngha bi 0nnT , D (R), n=1,2,...
Khi , 'DTn (R).
Trng hp 1n , 1T c gi l hm suy rng lng cc.
Nhn xt. khng c sinh ra bi bt c hm kh tch a phng no.
Tht vy, nu tn ti mt hm kh tch a phng f sao cho fT , khi
00 BR B dxxfdxxxfdxxxfn
vi D sao cho supp 0 B , 10 , 1 trn 02/B . Do ,
0 khi 0 .
Mt khc, 1 , vi mi . V vy, 0 khi 0 l muthun.
3.5 Tnh cht ca hm suy rng
Nhc li. Cho n
,...,, 21 , trong i , ni ,...,2,1 , l cc s
nguyn dng, khi c gi l mt a ch s.
K hiu mt s php ton lin quan n a ch s
n
ii
1
ni i1 !! xxx ni i i ,1 Rn
Cho hai a ch s nn ,...,,,,...,, 2121 . Khi , khi
v ch khiii , vi mi n,2,1,i .
l mt a ch s, ta nh ngha php ton vi phn D l
n
nxxD
...11
.
Tnh cht 1. Cho 'DT , vi l tp m con ca Rn, v a ch s .
Khi ,
DDTTD ,1 .
7/28/2019 Mt s vn v khng gian Sobolev
14/64
- 14 -
Tnh cht 2. Cho 'DT , Rn l tp m v C . Khi ,
DTT , .
Tnh cht 3. Cho 'DT , R l mt tp con m, D , v mt
a ch s , ta c cng thc Leibniz
TDDTD
!!
!.
V d. Cho :H R R l hm Heaviside c cho bi
0,0
0,1
x
xxH
Ta c
0''' 0 dxxTT HH
Khi , HT' .
Nhn xt. T v d trn, r rng '' ff TT .
3.6 Tch chp ca hm suy rng
Cho hm u bt k trn Rn v x Rn, k hiu
)( xyuyux v yuyu
.
Suy ra
uuxx v yxyx .
Vi 'DT (Rn), D (Rn), v x Rn, nh ngha
xx
TT
D thy 'DTx (Rn).
3.6.1nh ngha. Cho 'DT (Rn), v D (Rn), :T Rn Rnc cho
bi
x
TxT , vi mi x Rn.
7/28/2019 Mt s vn v khng gian Sobolev
15/64
- 15 -
3.6.2 nh ngha. Cho ', DST (Rn) Ta nh ngha hm suy rng ST
trn D (Rn) l
0
STST , D (Rn).
nh ngha 3.6.2 tng ng vi iu kin
STST , vi mi D (Rn).
4. Khng gian Sobolev
4.1 Khng gian Sobolev
Cho l mt tp m con ca Rn c bin l . Ta bt u vi nh
ngha.
4.1.1 nh ngha. Cho s nguyn m>0 v p1 . Khng gian
Sobolev c nh ngha
mLuDLuW pppm ),()()(, pmW , l tp hp tt c cc hm thuc )(pL c o hm suy rng n m
cng thuc )(pL .
Ta c )(D , khng gian ca tt c cc hm kh vi v hn vi gi compact
trong , th tr mt trong )(pL , vi p1 . Nu )(D th )(DD ,
vi mi a ch s . Nh vy,
)()()( , ppm LWD , vi p1 .
)(, pmW l mt khng gian vct.
Trn )(, pmW ta trang b mt chun,,
.pm
, nh sau
Vi p1 , ta nh ngha
p
m
p
Lpmp
uD
/1
0)(,,
.
.
Vi p , ta nh ngha
)(0,,max
Lmm
uDu
.
7/28/2019 Mt s vn v khng gian Sobolev
16/64
- 16 -
Trng hp c bit 2p , ta k hiu )()(2, mm HW , cho )( mHu ,
khi
,2,, mmuu
Vi 0
m , ta c )()(
,0 ppLW , chun trn
p
L ca hm )( p
Lu ck hiu l
)(pLu .
Khng gian )(mH c mt php ton nhn trong t nhin c nh
ngha
m
m vuDDvu
,),( , vi )(,
mHvu
Php ton nhn trong ny sinh ra,
.m
.
Trong trng hp Rn
, (mH Rn
)
c mt s m t khc qua bin iFourier.
Cho (1Lu Rn),
nR
x dxxfeu )()( 2
l s bin i Fourier ca u.
Ch . (1L Rn) (2L Rn) th tr mt trong (2L Rn), nhng hm trong
(2L Rn) c s bin i Fourier thch hp, nh l Plancherel khng nh rng
)R()R( n2n2
LLuu .
Cho mHu (Rn), theo nh ngha ta c (2LuD Rn), vi mi
m , nh vy )( uD c xc nh tt. Hn na, Ta c uiuD )2()^( .
Do , ( 2Lu Rn), vi mi m .
Ngc li, nu (2Lu Rn) sao cho ( 2Lu Rn), vi mi m ,
th (2LuD Rn), vi mi m . V th (mHu Rn).
4.1.2 B . Tn ti hng s 00 21 MvM ch ph thuc m v n sao
cho vi mi Rn,
m
m
m MM )1()1(2
2
22
1
7/28/2019 Mt s vn v khng gian Sobolev
17/64
- 17 -
T b , chng ta c nh ngha ca (mH Rn).
4.1.3 nh ngha
(mH Rn)
)()(1)( 2
2/22 nm
nRLuRLu
Kt hp vi chun
222
)()()1( uu m
RRHnnm .
T nh ngha trn cho php chng ta nh ngha (sH Rn), vi mi 0s .
4.1.4 nh ngha. Cho 0s , nh ngha
(sH Rn)
)()(1)( 2
2/22 ns
nRLuRLu .
Kt hp vi chun222
)()()1( uu s
RRHnns .
4.1.5 nh l. Vi mi p, p1 , )(, pmW l mt khng gian Banach.
* Xt khng gian tch: )1((,...1 nLLL ppnp ln)
Vi chunp
n
i
p
Lipuu
/11
1)(
, vi 111 ),...,(
np
nLuuu
Khi , nh x 11
, ,...,,)(
np
n
pm Lxu
xuuWu l mt php
ng c. Ta c mt s tnh cht
)(, pmW l khng gian phn x, vi p1 . )(, pmW l khng gian tch c, vi p1 . )(mH l khng gian Hilbert tch c, vi p1 .4.1.6 nh ngha. Cho p1 , t )(,0
pmW bng bao ng ca
)(D trong )(, pmW .
)(,0 pmW l mt khng gian con ng ca )(, pmW .
Phn t ca )(,0 pmW gn ging trong khng gian nh chun
)(, pmW bng nhng hm thuc C c gi compact trn .
7/28/2019 Mt s vn v khng gian Sobolev
18/64
- 18 -
)(,0 pm
W l khng gian con thc s ca )(, pmW , tr trng hp Rn.
4.1.7 nh l. Cho p1 , khi (,1 pW Rn) = (,10pW Rn).
4.1.8 nh l. Cho p1 , vi mi s nguyn 0m th
(,pmW Rn) = (,0 pmW Rn).
Trng hp c bit mH (Rn)= mH0 (Rn).
Ta c th ni rng pL l mt tp gm cc lp hm. Nh vy, khi ni u
l mt hm lin tc trong pL ngha l ta ang ni ti mt lp hm m c i
din l hm u lin tc.
Kt qu sau c trng cho pW ,1 khi I R l mt khong m.
4.1.9 nh l. Cho I R l mt khong m, nu IWu p,1 th u l hmlin tc tuyt i.
4.1.10 Ch . Cho I R l mt khong m gii ni, v d )1,0(I . Khi
, nu IWu p,1 th ta c th vit
dttuuxux
0 '0
Nh vy,
q
IL
qpx px
xuxuxdttuxudttuxuu p/1/1/1
00''')0(
Ly tch phn trn 1,0 , ta c
IpILIL
q
ILucuucdxxudxxuu ppp ,,111
1
0
/11
0'')0(
trong 1c khng ph thuc u. Cng nh vy ta c
IpIpIpIpucuucuuxu
,,13,,0,,12,,0'')0()(
Trong ,2
c v 3c c lp vi u.
4.1.11 Ch . Khng gian ln hn cha khng gian ca nhng hm trn
hn.
7/28/2019 Mt s vn v khng gian Sobolev
19/64
- 19 -
Ly B(0,1) 1:,,1
,1 Ip
p uIWu l hnh cu n v trong IW p,1 . Khi
, nh x )(: ,1 ICIWi p lin tc. Do , B(0,1)=i(B(0,1)) l mt tp gii ni
u trong )(IC .
Mt khc, cho Iyx , , ta c
q
Ip
q
ILyxuyxuyuxu p
/1
,,1
/1'
suy ra B(0,1) lin tc u trong IC , t nh l Ascoli-Arzela suy ra B(0,1) l tp
compact tng i trong IC . Hay ni cch khc, )(: ,1 ICIWi p l mt ton
t compact.
Trn khng gian )(, pmW , ta nh ngha na chun
p
ma
p
Lpm puDu
/1
,,
c xem l o hm cao nht ca khng gian nh chun pL .
4.1.12B (Bt ng thc Poincare). Cho l mt tp m gii ni
trong Rn. Khi , tn ti mt s nguyn dng pCC , sao cho
,,1 pLuCu p , )(
,10
pWu .
,,1 puCu nh ngha mt chun trn )(
,1
0p
W tng ng vi chun
,,1.
p. T 0
,,1
pu theo bt ng thc Poincare suy ra 0u . Do , n l mt
chun.
Ta c,
p
p
p
p
p
p
p
L
p
L
p
puCuCuuuDu pp
,,1,,1,,11
,,1)1(
v
p
p
p
Lp
p uuDu p
,,11,,1 .
T hai bt ng thc trn, ta c
p
p
pp
p
p
puCuu
,,1
/1
,,1,,1)1( .
7/28/2019 Mt s vn v khng gian Sobolev
20/64
- 20 -
4.1.13 V d. Bt ng thcPoincare khng ng vi min khng gii ni.
V d, nu ly Rn v D (Rn), xc nh bi
2x,0
1,1 xx , 10
t kxxk
/ , th
0
/1
1)(,,1
p
p
RLkRpk np
n D
, khi k .
Trong khi )),0(()(
kBnp RLk , khi k .
4.2 Khng gian i ngu ca khng gian Sobolev
phn ny ta xt khng gian sobolev ca s nguyn m cng nh phns.
4.2.1 nh ngha. Cho p1 , q l s m lin hp ca p. Khng gian
i ngu ca khng gian )(,0 pmW , vi m l s nguyn ln hn 1, c k hiu l
)(, qmW .
Nh vy, mH l khng gian i ngu ca mH0
.
4.2.2 nh l. Cho q
WF,1
, khi , tn ti q
n Lfff ,...,, 10 sao cho
p
i
n
i
iWv
x
vfvfvF ,10
10 , (2.1)
V
q
LiniqfF
0,,1max .
Khi l tp gii ni, ta c th gi s rng 00 f .
Gi s nh l trn l ng. V D tr mt trong
pW ,1
0, hm tuyn
tnh th xc nh duy nht nn n c nh trong D .
Cho D , (2.1) c vit li nh sau
n
i i
i
i
n
i
ix
ff
xffF
10
10
7/28/2019 Mt s vn v khng gian Sobolev
21/64
- 21 -
Nh vy, F c th c xc nh vi hm suy rng
n
i i
i
x
ff
10 . Mt
hm trn pW ,1 th c xc nh vi mt hm suy rng, l o hm suy rng
ca mt phn t trong qL . Do , khng gian i ngu ca pW ,10 c k
hiu l qW ,1 .
nh l trn cng ng vi khng gian i ngu ca pW ,1 (tr trng
hp ta khng gi s 00 f ngay c khi gii ni), nhng s xc nh vi hm
suy rng th khng th. Tht vy, khng gian i ngu ca pW ,1 cng bao
gm s m rng ca hm suy rng trn pW ,1 , nhng s m rng ny khng
duy nht.
Cho pnm / , khi CW pm,
Do , gi tr ca im c nh ngha tt.
Nu 0x , D , th
00 xx
v
,,00 pmLx Cx .
Cho khng gian nh chun pmW , ,0x
lin tc trn D v tnh lin
tc c thc trin n pmW ,0 . Suy ra qmx W,
0 , vi pnm / .
Vi mi min xc nh , hm suy rng Dirac thuc khng gian Sobolev
ca mt s m ln no .
By gi, ta nh ngha khng gian Sobolev cho mt s thc s bt k.
nh ngha. Cho p1 , th
ppnsp
ps Lyx
yuxuLuW
/, : .
Cho ms , 10,0 m ,
mWuDWuW ppmps ,: ,,, .
7/28/2019 Mt s vn v khng gian Sobolev
22/64
- 22 -
psW ,0 l bao ng ca D trong ps
W, v qsW ,0 l i ngu ca
psW ,0 .
Khi Rn v 0s ,
s
H (Rn
)={ 2Lu (Rn
) : 22/2
1 Lus
(Rn
)}v
2/1
22
)(1
nns R
s
RHduu
Khng gian i ngu ca sH (Rn) l sH (Rn), khi s>0.
4.2.3 nh l. Cho 0s , khi
sH (Rn) 'Su (Rn): 22/2 1 Lus (Rn)}
4.2.4 Ch . Nu l hm suy rng Dirac th
1
Do ,
dxxnR 0
V
sH (Rn) 22/21 Ls (Rn).
iu ny ng cho 2/ns , v tch phn
0 2
1
1dr
r
rs
n
ch hu hn khi
2/ns .
Trng hp c bit, sH (R), 2/1s .
7/28/2019 Mt s vn v khng gian Sobolev
23/64
- 23 -
CHNG 2
BI TP
Bi 1
a. Chng minh rng nu gf, l hm lin tc c gi l tp compact th
supp gf supp f + supp g .
Gii
a. GiA, B ln lt l gi ca f v g. Gi s
BzAyzyBAx ,:
Xt
.dyygyxfxgf D thy tch phn khc khng ch khi By v Ayx . Nhng nu
BAx th cy vx-y ln lt khng thuc voB vA. V vy,
0 xgfBAx
V cA vBu l compact nn A+B l compact. V vy,
supp gf supp f + supp g .
Bi 2. Cho pLfp
1 . Chng minh rng
1,:sup qq
pgLgfgdf .
Gii
Trng hp 1. p1 . Theo bt ng thc Holder ta c
qpgffgd
Do , v phi lun ln hn hoc bng v tri. Trng hp ng thc xy ra, gi
s 0p
f . ( 0p
f ng thc hin nhin ng). t fffgpp
psgn
11 . Khi
, qLg v 1q
g . Hn na,
p
pp
pfdfffgd
1
7/28/2019 Mt s vn v khng gian Sobolev
24/64
- 24 -
Trng hp 2. 1p . Khi ,
gffgd 1 . Nh vy mt chiu ca
bt ng thc xy ra. Gi s 01
f . t fg sgn . Khi , 1
g v
1ffgd .
Trng hp 3. p . Mt chiu ca bt ng thc hin nhin ng nh
trc. Gi s 0
f . Cho
f0 . Chn mt tp o c A sao cho
A0 v xf , Ax . nh ngha
AfA
g
.sgn1
trong , A
l k hiu hm c trng caA. Khi , 1Lg v 11
g . Khi ,
A dfAfgd
1
Do , phn trn l ng cho mi (
f0 ).
Bi 3. Cho hm baLf loc ;1 , ta ni hm baLg loc ;
1 l o hm suy rng
ca f nu
b
a
b
adxgdxf ' vi mi baCc ;
K hiudx
dfgfg ;' . Hy chng minh cc tnh cht sau cao hm
suy rnga.o hm suy rng l duy nht hu khp ni
b. ''' 2121 ffff
c. '' cfcf .
Gii
a. Vi mi baCc ; , gi s c 21, gg tha
b
a
b
adxgdxf 1' v
b
a
b
adxgdxf 2'
Suy ra
021 b
adxgg , .
Do , 21 gg hu khp ni. Hay o hm suy rng l duy nht hu khp
ni.
7/28/2019 Mt s vn v khng gian Sobolev
25/64
- 25 -
b. Vi mi baCc ; , ta c
b
a
b
a
b
a
b
a
b
a
b
adxffdxfdxfdxfdxfdxff ''''''' 21212121
M
b
a
b
adxffdxff '' 2121
Suy ra
b
a
b
adxffdxff ''' 2121
Vy ''' 2121 ffff .
c. Vi mi baCc ; , ta c
b
a
b
a
b
a
b
adxcfdxfcdxfcdxcf ''''
M
b
a
b
adxcfdxcf ''
Suy ra
b
a
b
adxcfdxcf ''
Vy '' cfcf .
Bi 4. Gi s x l hm Heaviside
0,0
0,1
x
xx
v cc hm suy rngx
1v x c nh ngha: vi mi hm th Dx (R)
x
dxx
xx
x 0lim,
1v 0, xx
Hy chng minh cc ng thc sau
a. xxdxd
b. xxdxd 1
ln
c. xxdx
dsgn , trong xxx sgn
d. xxdx
d , trong xxx .
7/28/2019 Mt s vn v khng gian Sobolev
26/64
- 26 -
Gii
Gi s Dx (R) tha supp aa,
a. Ta c
xx ,' xx ',
a
a
dxxx '
a
dxx0
'
ax0
0 a 0 xx ,
Vy xx ' .
b. Ta c
xx ,ln' xx ',ln a
adxxx 'ln
a
adxxxdxxx
'ln'lnlim
0
aa
aadx
x
xxxdx
x
xxx
lnlnlim
0
a
adx
x
xdx
x
x
lnlim
0
x
xdx
x
x
x
,
1lim
0
Vyx
x1
ln' .
c. Ta c
xx ,' xx ', a
adxxx '
a
adxxxdxxx
''lim
0
aa
aadxxxxdxxxx
0lim
a
adxxdxxaaa
0lim
a
adxxdxx
11lim
0
a
adxxxsigndxxxsign
0lim
a
adxxxsign xxsign ,
Vy xsignx ' .
7/28/2019 Mt s vn v khng gian Sobolev
27/64
- 27 -
d. '' xxx xxxx '' xxx ' xxx
Ta chng minh 0xx .
Tht vy,
xxx , xxx ,
t xxx 1 . Khi , x1 cng l mt hm th v
xxx , xxx , xx 1, 01 00 0
Vy xx ' .
Bi 5. Cho Cf v g l hm lin tc vi gi compact. Chng minh
rng Cgf .
Gii
chng minh Cgf ta ch cn chng minh
g
x
f
x
gf
ii
, ni 1 .
u tin ta chng minh gf l hm lin tc. Gi s supp Kg . Chn
mt tp compact Csao cho CKx va CKhx , vi h nh. Khi , f
l lin tc u trn C. Khi ,
K
dyygyxfyhxfxgfhxgf .
Cho 0 . Chn 0 sao cho CKhx v
yxfyhxf vi h . Do , xgfhxgf 1
g hay
gf l hm lin tc. Xt 0,...,0,1,...,0ie , trong 1 xut hin v tr th i.
Khi ,
h
xgfhexgfi
K i dyygyxfheyxfh
1
Ki
i
dyygheyxx
f (1)
vi mi , 10 . Vix
f
l lin tc trn C v b gii ni nn theo nh l v s
hi tb chn ca Lebesgue v phi ca (1) hi t v xgx
f
i
.
7/28/2019 Mt s vn v khng gian Sobolev
28/64
- 28 -
V vy
gx
f
x
gf
ii
.
Bi 6. Cho p1 , pLgLf ,1 , chng minh rng
pLgf vpp
gfgf1
.
Gii
+ Vi 1p hoc p ta d dng c c iu cn chng minh.
+ Vi p1 , cho qLh . Xt hm s
)()()(, xhygyxfyx
Khi hm ny l o c. Hn na,
dxxhdttxgtfdxdyxhygyxf )()()()()()(
dtdxtxgxhtf )()()(
.1
fhgqp
Do , nh x hgfh l mt hm tuyn tnh lin tc trnqL . Do
, pLgf v .1 pp
gfgf
Bi 7. Cho hm suy rng bt k 'DT (Rn), cc hm tiu chun
D21,, (Rn), x Rnl im bt k v l a ch s bt k. Khi ,
a. xxx TTT
b. DTTDTD
c. 2121 TT
d. Nu 0T vi mi D (Rn) th 0T .
Gii
a. Cho y bt k thuc Rn
xyTyTx
xyT
yxT
yx
T yTx
Ngoi ra, ta c
yTTTyTxxyxyx
7/28/2019 Mt s vn v khng gian Sobolev
29/64
- 29 -
b. Ta s chng minh cho trng hp 0,...,1,...,0,0 ie vi 1 v tr
th i. Trng hp bt k c chng minh tng t.
xx
Th
T
xTh
xTxTh
xTxTh
hexTxTh
xTx
i
x
he
xh
heh
heh
heh
ih
i
i
ii
i
0
00
00
lim
1lim1lim
1lim
1lim
Ngoi ra,
xxT
xT
Th
xTh
xTx
i
x
i
xheh
heh
iii
1
lim1
lim00
c. Ta phi chng minh rng
xTxT 2121
vi mi x Rn. Do
0 TxT x
T cu a suy ra
00 2121 TT
khi
2121 0 TT
m
dyyxdyyx
dyyyxxx
py
Ry
R
n
n
2sup2121
212121
Tch phn cui trn tp compact supp
2 c th c xem nh gii hn
khi 0 ca tng Riemann
p
ppx 21 trong tng c m rng trn
tt c im nt tch phn trn Rn.
7/28/2019 Mt s vn v khng gian Sobolev
30/64
- 30 -
Do ,
021lim
p
ppx 21
trong D (Rn). Do ,
0
lim0
21sup 21
210
2121
2
TdyyyT
pTTT
p
p
p
d. Ta phi chng minh rng 0T , vi mi D (Rn). Ta c
0
TT
M D (Rn) suy ra
D (Rn). Do ,
0
T
Suy ra
00
T
Vy 0T .
Bi 8. Cho 'T (Rn) v (Rn), ta nh ngha
x
TxT .
Trong , l mt khng gian tp c sinh ra bi mt dng hi t c
bit trn C v ' l lp cc hm suy rng vi gi compact. Chng minh
rng
a. TTT xxx b. DTTDTD , vi l a ch s bt k.
c. Cho T ' (Rn) v D (Rn) th DT (Rn) v
T = T = T .
7/28/2019 Mt s vn v khng gian Sobolev
31/64
- 31 -
Gii
ng thc a v b c th c chng minh tng t ng thc a v b trong
bi 7. chng minh cu c,t K supp(T) v H supp(). Khi , KvHl
compact. T nh ngha,
x
TxT
Ta c supp
x Hx , 0 xT nu KHx suy ra
HKx . Do ,
supp HKT supp(T)+supp()
V supp T l mt tp ng trong tp compact K+H nn n l tp
compact. iu ny chng minh rng DT (Rn). Tr li vn , chng
minh cu c. ta cn chng minh ti hm gc. Xt D0 (Rn) sao cho
HK
0
trong DHK
(Rn) l mt hm ngng ca K+H.
Khi ,
00
000
0
TT
dyyyTdyyyTT HKHK (1)
Khi , xt Hh v Kk th
khkhkkhh
00
Do , 0 TT trn H. T ,
0
0
00
TdyyyT
dyyyTT
H
H
(2)
T (1) v (2) ta c
T = T (3)
7/28/2019 Mt s vn v khng gian Sobolev
32/64
- 32 -
Ly 1 bt k thuc D (Rn). Khi , s dng cc tnh cht ca tch chp v
(3), ta c
1111 TTTT
S dng cu dcabi 7, ta c TT .
Bi 9.Cho 'DTi (Rn), 3,2,1i . Khi ,
a. Nu 1T hoc 2T thuc ' (Rn) th 1221 TTTT .
b. Nu t nht hai trong ba iT ' (Rn), 3,2,1i th
321321 TTTTTT
c. Vi bt k a ch s ta c 212121 TDTTTDTTD .
Gii
a. Cho D21 , (Rn), ta c
1221212121212121 TTTTTTTT
Nu 1T ' (Rn) th ta c th s dng cu c ca bi 8 v nu 2T ' (R
n)
th ta c th s dng cu dcabi 7 c 12212121 TTTT (1)
Ngoi ra, t (1) ta cng c
21121212 TTTT (2)
Do tch chp c tnh giao hon nn v phi ca (1) bng v phi ca (2).
Do , ta c
2121 TT 2112 TT
Do ,
2121 TT 2112 TT
v 2 l ty , ta c
121 TT 112 TT
7/28/2019 Mt s vn v khng gian Sobolev
33/64
- 33 -
Ta li c, 1 l ty , ta c
21 TT 12 TT .
b. Nu 21,TT ' (Rn) th 21 TT ' (R
n). Do , nu t nht c 2
'DTi (Rn), 3,2,1i , thuc vo ' (Rn) th c 321 TTT v 321 TTT c
xc nh.
Xt 3T ' (Rn), ta c
321321321 TTTTTTTTT
v
321321321 TTTTTTTTT
v DT 3 (Rn). Do l ty , ta chng minh c b.
By gi, nu 3T ' (Rn) th c 21,TT ' (R
n). Do ,
123231321 TTTTTTTTT
V 1T ' (Rn) nn
321123123 TTTTTTTTT
c. Cho D (Rn) v a ch s , ta c
2121212121 TDTTDTDTTDTTTTD
do l ty , ta chng minh c c.
Bi 10. Cho 'DT (Rn), khi ,
TTT .
Mt cch tng qut, vi a ch s bt k ta c
TDTD
Gii
Cho D (Rn). Khi ,
xxxxx
0 .
7/28/2019 Mt s vn v khng gian Sobolev
34/64
- 34 -
Do . Ngoi ra,
TTTT
Do D (Rn) l ty nn
TTT .Vi a ch s bt k ta c
TDDTTDTD .
Bi 11. Cho 1L (Rn), 0 v 11
. Chng minh rng 0 , c
nh ngha l /xxn , vi 0 , l mt xp x ng nht thc, ngha l
ff ,1Lf (Rn), khi 0 .
GiiDo
dyxfyxfyxfxfnR
,
Ta c
dxdyxfyxfyffn n
R Rn /
11
yny yn
dyyf
ffy ./2
/1 1
1
Cho 0 , chn 0 sao cho vi mi ta c 1ff .
Bi 12. Cho p1 . Chng minh rng D (Rn) tr mt trong pL .
Gii
Xt S l lp cc hm kh tch n gin (Tc l 0: xx ). Khi
, trit tiu bn ngoi tp c o v hn nn pL .
Xt pLf , 0f . Khi , tn ti mt dy n S sao cho nn f ,0
hi t theo o v f. Hn na,pp
nff . Do , p
n Lf v
0 pnf , khi n . Suy ra, tp cc hm n gin khng m tr mt trong
pL .
7/28/2019 Mt s vn v khng gian Sobolev
35/64
- 35 -
Cho S v 0 . Theo nh l Lusin, tn ti mt hm cCg (Rn) sao
cho g hu khp ni, tr mt tp c o b hn tha
g . Do ,
pp
p
pdgg
/1/1
2 .
Suy racC (R
n) tr mt trong pL . ChocCf (R
n), d thy
KBfpfp 1;0sup*sup
Trong , K l tp compact v ff * . Do ,
0,0*** Kffdffffpp
K
p
p.
Bi 13.
a.i vi khng gian S(Rn) ton t Fourier F: S(Rn) S(Rn) l mt ng
cu tp. Hy chng minh cc cng thc sau
i) gfgf ii) ff ..
iii) dxgfgdxf
iv)
dxgfdgfn
2
v) gfgf . vi) gfgf n 2.
b. Cc cng thc i), ii), iii), iv), v), vi) trong cu a s nh th no, nu
dng php bin i Fourier theo cng thc
?2 .2/ n
R
ixn dxxfef
c. Chng minh cc cng thc v), vi) trong cu a i vi trng hp
1Lf (Rn) hoc 2Lf (Rn) v g S.
Gii
a.
i) Ta c
gfdxexgdxexfdxexgfgfnnn
R
xi
R
xi
R
xi
Vy gfgf .
ii) Ta c
fdxexfdxexff xiR
xi
Rnn
...
Vy ff .. .
7/28/2019 Mt s vn v khng gian Sobolev
36/64
- 36 -
iii) Ta c
dxxgxf dxdxgfe xi
t x . Suy ra ddx Khi ,
dxxgxf
dxdgxfe
xi
dxdgexfxi
dxxgxf Vy dxgfgdxf
.
iv) Ta c
dgf dxdgxfe xi dxdgexf xi
dxdgexf xi
dxxgxf
n2
Suy ra
dxgfdgfn
2 .
v) Ta c
gf dxxgfe xi dydxyxfeyg xi
t yxX . Suy ra yXx v dxdX . Khi ,
gf dydxyxfeyg xi dydXXfeyg yXi
dydXXfeyge Xiyi dyfyge yi
dyygef yi gf
Vy gfgf . .
vi) Ta c
gfdyygyxf
dydfeygdydygfe
ddyygefedgfexgxf
nn
yxinyxin
yinxixi
22
22
2.
Suy ra gfgfn
2.
.b. Ta c
nR
xin dxxfef 2/2
Suy ra
nR
xin dfexf 2 2/
7/28/2019 Mt s vn v khng gian Sobolev
37/64
- 37 -
i) Ta c
gfdxexgdxexf
dxexgdxexfdxexgfgf
nn
nnn
R
xin
R
xin
R
xi
R
xin
R
xin
22
22
2/2/
2/2/
Vy gfgf
.ii) Ta c
fdxexfdxexff xiR
nxi
R
n
nn
.2.2. 2/2/
Vy ff .. .
iii) Ta c
dxxgxf dxdxgfe xin
2/2
t x . Suy ra ddx Khi ,
dxxgxf dxdgxfe xin 2/2
dxdgexf xin
2/2 dxxgxf
Vy dxgfgdxf
.
iv) Ta c
dgf dxdgxfe xin 2 2/ dxdgexf xin
2 2/
dxdgexf xin
2 2/ dxxgxf
Suy ra
dxgfdgf .
v) Ta c
gf
dxxgfe xin
2/2 dxdyygyxfe xin
2/2
dydxyxfeyg xin
2/2
t yxX . Suy ra yXx v dxdX . Khi , dydXXfeygedydXXfeyggf XinyiyXin
2/2/ 22
dyfyge yi dyygef yi
gfn 2 2/
Vy gfgf n .2 2/ .
7/28/2019 Mt s vn v khng gian Sobolev
38/64
- 38 -
vi) Ta c
xgxf . dgfe xin 2/2
ddyygefe yinxin 22 2/2/
dydygfeyxin
2
dydfeyg yxinn 2/2/ 22
dyygyxfn 2 2/
gfn 2 2/
Suy ra gfgf n 2. 2/ .
c.Trc tin, ta chng minh vi 1Lf (Rn) hoc 2Lf (Rn) v g S th
gf u c xc nh tt. Tht vy,
+ Vi 1Lf (Rn), Ta c
dyygyxfxgf dyygyxf
Suy ra 1Lgf v
1L
gf dxxgf dxdyygyxf
dydxyxfyg 11 . LL gf
+ Vi 2Lf (Rn), Ta c
dyygyxfxgf dyygyxf
Suy ra 2Lgf
V
2L
gf 2/12dxxgf 2/12
dxdyygyxf
2/1
22.
dxdyygdyyxf
2/122 dydxyxfgL 22 . LL gf
Nh vy vi 1Lf (Rn) hoc 2Lf (Rn) v g S th gf u c xc
nh tt.
7/28/2019 Mt s vn v khng gian Sobolev
39/64
- 39 -
By gi, ta s chng minh cc cng thc gfgf . ,
gfgf n 2. cho trng hp 1Lf (Rn). Do S tr mt trong 1L (Rn) nn
f 1L (Rn) th kf S sao cho ffk . Tc l,
Nk 0,0 sao cho 0kk th 1Lk ff .
+ Khi , ta c
dxxgfxgfgfgf kLk 1
dxdyygyxfyxfk
dydxyxfyxfyg k
dyygffLk 1 01 Lg
v
dffff kL
k 1 ddxxfxfe kxi
ddxxfxfe kxi
ddxxfxfk
dffLk
1 0 d
Do ,
gfgf k
klim gfgfk
k..lim
Vy gfgf . .
V
gfgf
kklim. gfgf n
k
n
k22lim
Vy gfgf n 2. .
+ By gi, ta s chng minh cc cng thc gfgf . ,
gfgf n 2. cho trng hp 2Lf (Rn). Do S tr mt trong 2L (Rn) nn
f 2L (Rn) th kf S sao cho ffk . Tc l,
Nk 0,0 sao cho 0kk th 2Lk ff .
7/28/2019 Mt s vn v khng gian Sobolev
40/64
- 40 -
+ Khi , ta c
2/12
2
dxxgfxgfgfgf kLk
2/12
dxdyygyxfyxfk
2/122 dxdyygdyyxfyxfk
2/122 dydxyxfyxfg kL
22LkL
ffg 02 L
g
V
2/12
2
dffff k
Lk
2/12
ddxxfxfek
xi
2/12
ddxxfxfek
xi 2/1
2
ddxxfxfk
02 Lk
ff
T ,
gfgfgfgfn
nn
n..limlim
.
Vy gfgf . .
V
gfgf k
klim. gfgf nk
n
k22lim
.
Vy gfgf n 2. .
Bi 14.
a. Vi f S(Rn), hy chng minh cng thc
fifD
b. Hy thit lp cng thc i vi fD vi f S(Rn)
c.Dng nh ngha ca php bin i Fourier ca f S nh sau
,,, ff S(Rn)
tm fDfD , , vi f S.
7/28/2019 Mt s vn v khng gian Sobolev
41/64
- 41 -
Gii
a. Trc tin ta chng minh rng vi mi i ta c fifx
i
i
.
Tht vy,
dxxfx
efx
i
xi
i
dxxfe
x
xi
i
dxxfei xii fi i
Khi , vi mi i ta c
f
x i
i
i
dxxf
xe
i
i
i
xi
dxxfe
x
xi
ii
i
i
1
dxxfeixi
i
i
fii
i
Suy ra
fifD .
b.Trc tin ta i tm cng thc tnh
fi
, vi mi i.
Ta c
dxxfedxxfef si
i
si
ii
dxxf
edxxfe
i
si
i
si
dxxfixe ixi xfixi
Khi , vi mi i ta c
dxxfedxxfef si
i
si
ii
i
i
i
i
i
i
dxxfedxxfei
i
i
i
i
si
i
si
dxxfixe iixi xfix ii
Suy ra
xfixfD .
7/28/2019 Mt s vn v khng gian Sobolev
42/64
- 42 -
c.
+ Tm fD
DffD ,1, Df,1 ,1 if
,fi , fi
Vy fifD .
+ Tm fD
,fD ,fD ,1 Df ixf,1
ixf ,1 ,fix
Vy fD fix .
Bi 15. Gi s 1Lf (R). Chng minh rng
a. Lf (R), 1 LLff
,
b.f lin tc u trn R,
c. Nu ta cng c 1' Lf (R), chng minh rng 0 f khi .
Khng c thm gi thit 1' Lf (R), iu khng nh trn c cn ng khng?
Gii
a. Ta c
1LRR
xi
R
xi fdxxfdxxfedxxfef
T nh ngha chun ca L (R) suy ra1
LL
ff
.
b. Ta c
0' ''
R
xixi
R
xixidxxfeedxxfeeff
, khi '
Suy ra f lin tc u trn R.
c.Do 1' Lf (R) nn 1' Lf (R) v R df .
Ta c
fif '
7/28/2019 Mt s vn v khng gian Sobolev
43/64
- 43 -
Do ,
'' fi
i
ff
Cho , ta c
0'
limlim
fif
Ta khng cn n iu kin 1' Lf (R). Tht vy, ta c th chn
1', Lgg (R). (Khi , 0 g khi .) sao cho 1
1 LL
gfgf . Khi
,
khig
ggfggfggffL
,0
1
Bi 16. Tm bini Fourier ca cc hm s sau
a.
,1
22x
xf R, 0 b. 0,2
1 4/2 aea
xf ax
c.
,
sin1
x
xxf R d.
khcx
cxbexf
x
,0
,
e.
khcx
cxbxf
,0
,1 f. xaa;
g. ,xa a>0 hng s, x R.
Gii
a.
,1
22x
xf R, 0 . Ta c
dx
x
ef
xi
22
Xt
xexg .
Ta c
220
0
0
0 2
ie
i
edxedxedxeeg
xixixixxixxix
7/28/2019 Mt s vn v khng gian Sobolev
44/64
- 44 -
M
degxg xi
2
1
Suy ra
dee xix
222
21
t y , ta c
dye
ye
iyxx
22
2
2
1
t
x
xy, ta c
de
xe ix2222
1
Suy ra
edex
ix
22
1.
b. 0,2
1 4/2 aea
xf ax
.
Ta c
dxea
e
dxea
dxeea
f
a
xaia
aaxxiaiaxxi
22
2222
2
4/4/
2
2
1
2
1
ta
dxdtai
a
xt
22 .
Khi ,
Iedteefa
ta
2
2
2
By gi ta i tnhI. Ta c
0
2
0
2
0 0
2
22
2222
2
1
.
rr
utut
edrdre
dtdueduedteI
7/28/2019 Mt s vn v khng gian Sobolev
45/64
- 45 -
Suy ra
I
Do ,
2
2
a
a
e
e
f
Vy 2 aef .
c.
,
sin1
x
xxf R. Ta c
0 0
00
0
0
sinsin1
sincos21sin1
sinsin1sin1
dxx
xdx
x
x
dxx
xxdx
x
xee
dxx
xedx
x
xedx
x
xef
xixi
xixixi
By gi ta tnh tch phn
0
sindx
x
xI
Theo bin i Laplace ta c
0
1
dtex
xt
Do ,
dtxdxedxdtexIxtxt
0 00 0sinsin
Xt
0sin xdxeK xt
t
t
evdxedv
xdxcoxduxuxt
xt
sin
Khi ,
00
0
sinxdxcoxe
txdxcoxe
tt
xeK xtxt
xt
7/28/2019 Mt s vn v khng gian Sobolev
46/64
- 46 -
t
t
evdxedv
xdxduxu
xtxt
sincos
Suy ra
0
2
2 sin xdxett
Kxt
Do ,
22t
K
Nn
dttI
022
t duutgdttgut 21 . Khi
21
12/
022
22
utg
duutgI
T ,
1sinsin1
0 0
dxx
xdx
x
xf
Vy 1 f .
d.
khcx
cxbexf
x
,0
,.
Ta c
i
eee
i
dxedxeedxxfef
bicic
b
xi
c
b
xic
b
xxixi
1
Vy
i
eef
bici
.
7/28/2019 Mt s vn v khng gian Sobolev
47/64
- 47 -
e.
khcx
cxbxf
,0
,1.
Ta c
i
eee
idxedxxfef
cibic
b
xic
b
xixi
1
Vy
i
eef
cibi .
f. Ta c
adxx
dxxixdxedxe
a
a
a
a
a
xixi
aaaa
sin2cos2
sincos
0
;;
Vy aaa sin2 ; .
g.
ax
axxa
,0
,1 , vi a l hng s ln hn 0.
Ta c
a
i
ee
i
edxedxxae
aiaia
a
xia
a
xixi sin2
Vy
asin2
.
Bi 17. Tnh ff vi
a. 1;0xf b. x
xxf
sin
Gii
Ta p dng tnh cht 2. fffff
a. Ta c
111 101
01,0
ixixixixi e
ie
idxedxexdxexff
Suy ra
222 1
1
iefff
Vy 22 11
ieff .
7/28/2019 Mt s vn v khng gian Sobolev
48/64
- 48 -
b. Ta c
aaa
sin2 ;
Do
defxf xi
21
Nn
dea
xxi
aa
sin1
,
Suy ra
dea
x xiaa
sin1,
t dxdx . Khi , ta c
dxe
x
xa xiaa
sin1,
Ta c
,
sin
dxexx
dxexff xixi
Do
,
22
f
Vy
,
2
ff .
Bi 18. Cho 2
xexg
. Tnh gg .
Gii
Ta c
degdeggxgg
xixi
22
1*
2
1*
M
4/2
eg Nn
2/22
eg
7/28/2019 Mt s vn v khng gian Sobolev
49/64
- 49 -
Suy ra
221
.42
2
2
2
2
2.
2
12
1
2
2.
2
1*
x
xixi edeedteexgg
Vy 22
22*
x
exgg .
Bi 19. Cho 0,2
22
xxf . Vi 0, , tnh ff .
Gii
Ta c
e
x22
1
Theo nh l o ca php bin i Fourier ta c
22
4
2
14
22.22
1
.2
1
2
1
xdee
deedeee
deffdeffxff
ixb
ixbix
ixix
Vy 22
4
xxff .
Bi 20. Tm hmf tha
22221
bxayx
dyyf
, ba 0 .
Gii
t
22
1
axxg
a
Khi
a
ae
ag
7/28/2019 Mt s vn v khng gian Sobolev
50/64
- 50 -
V
22 ayx
dyyfdyyfyxgxfg
aa
Biu thc cho tr thnh
xgxfg ba
Suy ra
ba gfg
Hay
ba gfg .
T
ba
ebfea
Ta c
abeb
af
M
2221
2
12
1
abxbabadee
abbaba
deeb
adefxf
xiab
xiabxi
Vy
22 abxbaba
xf
.
Bi 21. Cho 1Lf (R). t
dxxxfa
cos1
v
dxxxfb
sin1
Nu f lin tc, chng minh rng
dxbxaxf
0
sincos
p dng tm hm f tha:
2,0
21,2
10,1
sin0
dxxxf .
7/28/2019 Mt s vn v khng gian Sobolev
51/64
- 51 -
Gii
Ta c
dxbxa
ddyyyfxdyyyfx
dydyxyfdyxyf
dydyxyf
dydyxyfidyxyf
dydyxiyxyfdydeyf
dedyeyfdefxf
yxi
xiyixi
0
0
0 0
0
sincos
sinsincoscos1
sinsincoscos1
cos1
sincos2
1
sincos21
21
2
12
1
p dng, t
0,
0,
xxf
xxfxF
Theo chng minh trn ta c
dxbxaxF
0
sincos
Ta c
2,0
21,4
10,2
sin2
sin1
0cos1
cos1
cos1
0
0
0
dxxxfdxxxFb
dxxxfdxxxfdxxxFa
7/28/2019 Mt s vn v khng gian Sobolev
52/64
- 52 -
Khi
x
xx
x
x
x
x
dxdxdxbxFxfx
4cos2cos8cos4cos2
sin4
sin2
sin
22
1
1
0
2
1
1
000
Vy 0,4cos2cos8 2
xx
xxxf
.
Bi 22. p dng nh l Plancherel tnh cc tch phn sau
a.
dx
x
x2
sin
b.
221 x
dx
c.
dx
x
xxx6
2sincos.
Gii
nh l Plancherel cho ta22
22
2LL
ff .
a. Ta c
sin2 1;1
Theo nh l Plancherel ta c
2
1;1
2
1;1 22 2LL
Hay
dxxd
2
1;1
2
1;1 2
Do
1
1
2
42sin2
dxd
Suy ra
dxx
xI
2sin
7/28/2019 Mt s vn v khng gian Sobolev
53/64
- 53 -
b.t
21
1
xxf
Khi
ef
Theo nh l Plancherel ta c
22
222
LLff
Hay
dxxfdf
22
2
Do
dxx
de
22
2
1
12
Suy ra
211
0
2
22
dedxx
.
c. t
1;12 1 xxf
Khi
1
0
21
1
2 cos12sincos1 dxxxdxxixxdxexff xi
t
xvxdxdv
xdxduxu
sincos
212
. Ta c
1
0
1
0
1
0
2
sin
4sin
2
sin
12
xdxxdx
x
x
x
xf
t
xvxdxdv
dxduxu
cossin
.
7/28/2019 Mt s vn v khng gian Sobolev
54/64
- 54 -
Ta c
3
1
0
1
0
sincos4coscos4
dx
xxxf
Theo nh l Plancherel ta c
dxxfdf
22
2
Hay
dxxd
1
1
226
2
12sincos16
Suy ra
.15
21
3
2
5
1
43
2
54
124
14
sincos
1
0
35
1
0
241
0
22
6
2
xxx
dxxxdxxd
Bi 23. Gi s AdfnR . Chng minh rng
ddbaf 22 LL baA .
Gii
Ta c
ddbaf ddbaf
ddaffb 21
.22/12/1
ddafdfb 2/12
ddafbA 2/122/1
2
1.22/12
2/1 ddafbA
2/1222/1 ddafdbA
2/122/1 2 ddafbAL
2/122/1 2 ddfabA L 22 LL baA .
7/28/2019 Mt s vn v khng gian Sobolev
55/64
- 55 -
Bi 24. Chng minh rng vi mi hm s x S(Rn), vi 12/ nk ta
lun c
xn
R
max nk RHC. .
Gii
Vi 12/ nk th
dnR
k22/2
1 .
Do , 22/2 1 Lk (Rn). Suy ra kH (Rn), v
2/1
221
dnnk R
k
RH
Khi , ta c
x
nR
xin
de
2
nR
xin
de
2
nR
n
d
2
21
.22/22/2 1.12
nR
kkn
d
2/1
222/1
21.12
nnR
k
R
kn
dd
nk RHC. , vi mi x Rn.
Suy ra xn
R
max nk RHC. .
Bi 25. Xt dy cc hm s xxfm , R
mx
mxm
xfm
2
1,0
2
1,
Chng minh rng xfm
l dy c bn trong H-1(R).
Gii
Ta c
m
mxm
dxmdxxixmdxemdxxfef
m
m
m
m
m
m
m
m
xi
Rm
xi
m
2
2sin
sin
cossincos
2/1
2/1
2/1
2/1
2/1
2/1
2/1
2/1
7/28/2019 Mt s vn v khng gian Sobolev
56/64
- 56 -
Do
m
mf
m
2
2sin
nn 1lim
m
mf , tc l
0,02 n
sao cho 0nm
th 21
mf v
0',02n
sao cho 0'nn th 2
1
n
f
Khi 00 ';max,0 nnN sao cho Nm v Nn th
nmnmnm ffffff 1111
T ,
Nnm
dd
ff
dff
ffff
RR
nm
R
nm
RLnmRHnm
,,011
11
2/1
2
2/12
2
2/12
2
2/12
21
Vy mf l dy Cauchy trong H-1(R).
Bi 26. Cho xf H-s(Rn), x Hs+1(Rn). Chng minh cng thc tch
phn tng phn
nn
Rj
Rj
dxx
xxfdxx
x
xf .
Gii
Cch 1. Chng minh bngphng php quy np.
+ Ta chng minh cng thc ng vi 1n . Tht vy,
R dxxxf ' R xfdx RR dxxxfxxf ' R dxxxf '
+ Gi s cng thc ng vi kn . Tc l
kk
Rk
jR
k
j
dxdxdxx
xxfdxdxdxx
x
xf...... 2121
, vi .,...,1 kj
7/28/2019 Mt s vn v khng gian Sobolev
57/64
- 57 -
Ta phi chng minh cng thc ng vi 1 kn . Tc l phi chng minh
11 121121 ...... kk
Rk
jR
k
j
dxdxdxx
xxfdxdxdxx
x
xf , vi .1,...,1 kj
i) Trng hp .,...,1 kj
121121 ......1
kR R
k
jR
k
j
dxdxdxdxxx
xfdxdxdxx
x
xfkk
121 ...
k
R Rk
j
dxdxdxdxx
xxf
k
1 121 ...k
Rkk
j
dxdxdxdxx
xxf
ii) Trng hp 1 kj
1
...... 2111
1211
k kR Rk
Rk
k
k
k
dxdxdxdxxx
xfdxdxdxx
x
xf
k
R Rk
kRkk
dxdxdxdxx
xxfxxf
k...211
111
1 1211
...k
Rk
k
dxdxdxx
xxf
Suy ra cng thc ng vi 1 kn .
Vy ta chng minh c cng thc tch phn tng phn.Cch 2
Xt 0,...,0,1,0,...,0je , 1 xut hin v tr th j.
t xfhexff j v xhex j . Ta c
xfxfxfxf xfxfxf
ejhxfxfxf
ejhxfxfxf
fejhxfxfxf
fxxfejhxhexfxf j
7/28/2019 Mt s vn v khng gian Sobolev
58/64
- 58 -
Khi
nn R
jhR
j
dxxxfhexfh
dxxx
xf
1lim
0 nR h dxxfh
1lim
0
n
R
j
h
dxfxxfejhxhexfxf
h
1
lim0
nRj
hhdxxxfejhxhexf
hxf
h
1lim
1lim
00
nRjj
dxx
xxf
x
xxf
nnR
jR
j
dxx
xxfdx
x
xxf
1 111...nn R njjRjjR j dxdxdxdxxxfdxx
x
xf
n
Rj
dxx
xxf
Vy
nn
Rj
Rj
dxx
xxfdxx
x
xf .
Bi 27. Cho x Hs(Rn), x Hs(Rn), xx Hs(Rn),
j
xxx Hs-1(Rn),
jx
xx Hs-1(Rn). Chng minh cng thc Leibnitz
jjjx
xxx
x
xxx
x
Gii
Xt 0,...,0,1,0,...,0je , 1 xut hin v tr th j.
t xhex j v xhex j
Suy ra xhex j v xhex j
Khi
hexhex jj xx
xxxx
7/28/2019 Mt s vn v khng gian Sobolev
59/64
- 59 -
Ta c
xxxj
xxhexhexh
jjh
1lim
0
xxhh
1lim
0
xxhexxxhexh
jjh
1lim
0
xxhexh
xxhexh
jh
jh
1lim
1lim
00
xhex
hxxxhex
hj
hj
h
1lim
1lim
00
jj
x
xxx
x
x
Ch . 01
lim0
hh
. Tht vy,
xhexxhexhh
jjhh
1lim
1lim
00
xhex
hxhex
hh
jjh
11
lim0
xhex
hxhex
hh
jh
jhh
1
lim1
limlim000
00
jjx
x
x
x
Vy ta chng minh c cng thc Leibnitz
jjjx
xxx
x
xxx
x
.
Bi 28. Ta nh ngha hm suy rng xax , Rn, 0a bng ng
thc sau
ax xdxxax ,
trong xd l phn t din tch mt cu bn knh a trong Rn.
a. Tm bin i Fourier ca hm ax b. Chng minh rng ax 12/ nH (Rn).
7/28/2019 Mt s vn v khng gian Sobolev
60/64
- 60 -
Gii
a. Ta thy ax S(Rn) v supp ax axx , . Nh vy,
ax c gi compact. Do ,
xi
eaxax
,
ax xxi
de
axxdxixcox sin
ax xax x dxidxcox sin
V xsin l hm l theo x , v mt cu axx , i xng qua tm, nn
ta c
0sin ax xdx
Do ,
ax ax xdxcox .
b. Do 1xcox v mt cu axx , l tp compact, nn tn ti hng s
C sao cho
Cax
Khi , ta c
daxn
R
n 212/21
dCnR
n 12/22 1
Suy ra 22/12/21 Laxn (Rn)
Vy ax 12/ nH (Rn).
Bi 29. Cho X l mt khng gian Banach. Ta ni hnh cun v
1: xXxS l li ngt nu yx , 1 yx th , 10 ta c
Syx 1 . Xt xem hnh cu n v trong QC , QL1 , QL2 c li ngt
khng, Q l mt min b chn trong Rn.
Gii
+ Ta chng minh hnh cu n v trong QC khng li ngt.
Tht vy, chn 0tx , 1tx , 1max txx
QtQC, 1ty , vi mi
t QC .
7/28/2019 Mt s vn v khng gian Sobolev
61/64
- 61 -
Khi , vi mi tha 10 , ta c
tytxyxQtQC
1max1
1max txQt
11
Suy ra yx 1 khng thuc hnh cu n v trong QC .
Hay hnh cu n v trong QC khng li ngt.
+ Ta cng chng minh c hnh cu n v trong QL1 khng li ngt.
Tht vy, chn 0tx , QL dttxx 11 , Qty1
, vi mi Qt .
Khi , vi mi tha 10 , ta c
QL dttytxyx 11 1
Q dtQtx
1
1
QQ dtQdttx1
1 11
Suy ra yx 1 khng thuc hnh cu n v trong QL1 .
Hay hnh cu n v trong QL1 khng li ngt.
+ Hnh cu n v trong QL2 li ngt.
Tht vy, yx , 122 LL
yx v tha 10 , ta c
2/1
211 2
QL dttytxyx
2/12
1
Q dttytx
2/1
2222 112
Q dttytytxtx
2/1
2222 112
QQ Q dttydttytxdttx
2/1
2222 112
LQL ydttytxx
2/122 22121LL
yx
1121 2/122
Suy ra yx 1 thuc hnh cu n v trong QL2 .
Vy hnh cu n v trong QL2 li ngt.
7/28/2019 Mt s vn v khng gian Sobolev
62/64
- 62 -
Bi 30. Cho f l mt hm kh tch a phng trn mt tp con m
Rn. nh ngha dxxxfTf . Chng minh rng fT thuc 'D .
Gii
D thy fT l mt hm tuyn tnh trn D . Ta chng minh n lin tc.Cho m l mt dy rng trong D . Xt K l tp compact c nh sao
cho supp Km , vi mi m. Khi ,
dxxfxT mKxmf max .
V m
l mt dy rng nn 0maxlim
xm
Kxm
Do 0mfT khi m .
Bi 31. Gi s dy ...,2,1, mxfm cc hm thuc QCk hi t yu trong
QL2 n mt hm f, cn dy ...,2,1, mfD m , vi kn ),,...,( 1 , b
chn trong QL2 , Q l mt min b chn trong Rn. Chng minh rng hm fc
o hm suy rng fD .
Gii
V m
fD b chn trong QL2 nn theo nh l Bolzano-Weierstrass tn
ti dy con kmfD ca mfD hi t trong QL2 . Gi s ufD Lmk 2
.
Vi mi hm mf QL2 xc nh mt hm suy rng QDT
mf' . Khi ,
mTT fL
fm,
2
Tht vy,
QLdxxxfxfdxxxfxfTTQ
mQ
mffm
2,0
Tng t, ta c mTTu
L
fDkm
,2
.
Theo cng thc Ostrogradski, ta c
fDf
k
fm
k
fDmu
TDTDTTTkm
kkm
k
1lim1lim
Do ufD theo ngha ca o hm suy rng.
Vy tn ti fD .
7/28/2019 Mt s vn v khng gian Sobolev
63/64
- 63 -
PHN KT LUN
Phng trnh o hm ring ra i vo khong th k th XVII do nhu cu
ca c hc v cc ngnh khoa hc khc. N ngy cng c vai tr quan trng,
c ng dng rng ri trong khoa hc v cng ngh. Ngy nay, phng trnh
o hm ring tr thnh mt b mn ton hc c bn va mang tnh l thuyt cao
va mang tnh ng dng rng. Trc s pht trin nh v bo ca khoa hc cng
ngh, chc chn rngphng trnh o hm ring s cn pht trin mnh m hn
na trong tng lai, m ra mt con ng cho nhng ai yu thch nghin cu
ton hc ng dng. Trong qu trnh hc tp, c thy c gii thiu, em cm
thy rt c hng th vi mn hc ny. Cho nn, khi c lm lun vn em xinc nghin cu vphng trnh o hm ring. Tuy nhin, khi bt tay vo lm
em mi thy rng, tuy trn th gii phng trnh o hm ring v ang pht
trin mnh, nhng nc ta vn cn rt t sch ni v ti ny, nu c th cng
i su nghin cu lm sng t l thuyt, cn bi tp ch l gi n gin. V th,
c s gi ca thy hng dn, bc u lm quen vi phng trnh o
hm ring, trong lun vn ca mnh em tm hiu v khng gian Sobolev -
c nh ton hc Sobolev S.L gii thiu vo gia th k XX, n nhanh chngtr thnh mt cng c t lc trong vic gii phng trnh o hm ring, do ,
n c nhiu nh ton hc khc tip tc m rng v pht trin nhm nghin cu
cc phng trnh o hm ring ngy cng kh khn phc tp - Nhng khc
ch, trong lun vn ca mnh, em i su chn lc mt h thng cc v d minh
ha, gii v sp xp tng i hp l cc bi tp lin quan n khng gian
Sobolev. Chnh v th, c th xem cun lun vn nh mt t liu, hnh trang cho
em sau ny nu c iu kin s tip tc nghin cu v phng trnh o hm
ring.
Do y l ln u tin thc hin nghin cu khoa hc, nn em khng th
trnh khi nhng thiu st nht nh. Em rt mong nhn c s quan tm, ng
gp kin ca qu thy c v cc bn lun vn ca em c hon chnh hn.
7/28/2019 Mt s vn v khng gian Sobolev
64/64
TI LIU THAM KHO
[1] Nguyn Minh Chng, Phng trnh o hm ring, Nh xut bn Gio dc, 2000.[2] D. Bahuguna V. Raghavendra B.V. Rathish Kumar, Sobolev Space and
Applications, Alpha science.
[3] Nguyn Minh Chng, L thuyt phng trnh o hm ring, Nh xut bn Khoa
hc v k thut, 1995.
[4] ng nh ng, Nhp mn gii tch, Nh xut bn Gio dc, 1998.
[5] ng nh ng, Bin i tch phn, Nh xut bn Gio dc, 2001.
[6] Nguyn Xun Lim, Gii tch hm, Nh xut bn gio dc.
[7] GS.TSKH Phan Quc Khnh, Ton chuyn , Nh xut bn i hc Quc gia TP.H Ch Minh, 2000.
[8] GS.TSKH Nguyn Duy Tin, Bi ging gii tch, i hc Nng, 2004.
[9] M.A Trn Th Thanh Thy, Gio trnh Tp i cng, i hc Cn Th, 2004.
[10] M.A Trn Th Thanh Thy, Gio trnh o v tch phn Lebesgue, i hc Cn
Th, 2007.