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LE CO
LY T PO
HN EC
UE IQ
OLE C
LY T PO
N CH E
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
H 1 () 1 H0 () H m () H(div) W m,p ()
OLE C
EC LY T PO
NIQ H
UE
C
LE O
LY T PO
HN EC
UE IQ
N = 1 P1 P2 N 2
PO LE CO
TEC LY
HN
OLE C
LY PO
CH TE
UE IQ
N
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
PO LE CO
TEC LY
HN
UE IQ
OLE C
EC LY T PO
NIQ H
UE
C
LE O
LY T PO
HN EC
UE IQ
PO LE CO
TEC LY
HN
UE IQ
C
OLE
LY PO
CH TE
N
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
OLE C
LY PO
CH TE
N
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
OLE C
LY PO
CH TE
N
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
OLE C
LY PO
CH TE
N
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
OLE C
LY PO
CH TE
N
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
OLE C
LY PO
N C H TE
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
OLE C
LY PO
CH TE
N
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
OLE C
LY PO
CH TE
N
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
C
U E H N I Q C
LE O
LY T PO
HN EC
UE IQ
C
OLE
YTE POL
C
LE O
LY T PO
HN EC
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
N RN N = 1, 2 x t f (x, t) (x, t) c c V
OLE C
LY PO
CH TE
N
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
d dt
PO LE CO
V V ds n V q q n ds =V V
TEC LYc dx =V V
HNV
UE IQ
f dx
q n ds
divq dx.
V + divq = f c t x t N qi divq = q = (q1 , ..., qN )t . xi i=1
k =
OLE C
LY PO
CH TE t
N
UE IQ
q = k,
.
, ..., x1 xN
c k = f, t
= div
=
C
LE O
LY T POi=1
N
2 . x2 i
HN EC
UE IQ
n
C
LE O
LY T PO
HN EC
UE IQ
t = 0 (t = 0, x) = 0 (x),
0
OLE C
LY T PO
N CH E
UE IQ
(t, x) = 0 x t > 0.
(t, x) n(x) (t, x) = 0 x t > 0, n
n
(t, x) + (t, x) = 0 x , t > 0 n
C
LE O
LY T PO
HN EC
UE IQ
c k = f (x, t) R+ t (x, t) R+ (t, x) = 0 (t = 0, x) = (x) x 0
C
LE O
LY T PO
HN EC
UE IQ
K c J/(kg K) k Jm2 /(kg K s) c k
D N F D , N , F
(f, 0 ) (f, 0 ) k q
OLE C
LY PO
CH TE
N
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
q k c
C
LE O
LY T PO
HN EC
UE IQ
x u k T u 2u u ru + 1/2rx + 1/2 2 x2 2 = 0 (x, t) R (0, T ) t x x u(t = T, x) = max(x k, 0) x R
u(0, x) t = 0 k T > 0 x t = 0 r V (x, t) RN V c + cV k = f R+ t =0 R+ (t = 0, x) = (x) 0
OLE C
LY PO
CH TE
N
UE IQ
C
LE O
LY T POPe = cV L , k
HN EC
UE IQ
C
L c + cV = f R+ t (x, t) R+ V (x) n(x) < 0 (t, x) = 0 (t = 0, x) = (x) 0
LE O
LY T PO
HN EC
UE IQ
V = R f V 2 +V 2 = 0 (x, t) R R+ t x x (t = 0, x) = 0 (x) x R
OLE C = k/c
LY PO
CH TE
N
UE IQ
+
1 (t, x) = 4t
0 (y) exp
(x V t y)2 4t
dy.
V = 0
R
0
+V =0 (x, t) R R+ t x (t = 0, x) = (x) x R 0
C
LE O
LY T PO
HN EC
UE IQ
C
0 R
LE O
LY T PO
(t, x) = 0 (x V t)
HN EC
UE IQ
f = 0 t t V V = 0
OLE C
LY PO
CH TE
N
UE IQ
f = 0 (x, t) > 0 ( x , t2 ) V (x, t) x t ( , ) (x, t) R t > 0 R
C
LE O
LY T PO
HN EC
UE IQ
V min 0 (x) (x, t) max 0 (x) (x, t) R R+ ,xR xR
C
LE O
LY T PO
HN EC
UE IQ
OL E C
LY PO
CH TE
N
UE IQ
u u = f R+ t + R u=0 u(t = 0) = u 0
= (0, 1)
C
LE O
LY T PO
HN EC
UE IQ
f = 0 u(t, x) x 1 d 2 dt1
C
v(x) 1 0
LE O
LY T PO0
u2 (t, x) dx
HN EC=1 0 1 0
u u (t, x) dx x2
UE IQ
[0, 1] v(0) = 0dv (x) dx. dx1 0 2
v 2 (x) dx
u2 (t, x) dx
f(x) OLE C
LY PO
CH u(x) TE
x
N
UE IQ
f u u 2 u u = f R+ t2 u=0 + R u(t = 0) = u0 u (t = 0) = u1 t u
C
LE O
LY T PO
HN EC
UE IQ
N = 1 u0 u1 f = 0 = R U1 u1
C
(x, t) (x, t) t t
LE O
u(t, x) =
LY T PO
1 1 (u0 (x + t) + u0 (x t)) + (U1 (x + t) U1 (x t)) , 2 2
HN EC
UE IQ
(x, t) u0 u1 [x t, x + t] u(t, x) R u1 (x)
OLE C
LY POt
CH TE
N
UE IQ
(x,t)
xt
x+t
x
= (0, 1) f = 0 u(t, x)
C
LE O
LY T PO
HN EC
UE IQ
d dt1 0
C
LE O
TEC O LY Pu (t, x) dx + t2 1 0
u t
u (t, x) dx x2
HN
UE IQ= 0.
x
f u(t, x) u (x) t f (x) u = f u=0
,
f
u V u(t, x) R+ RN C |u|2 (t, x) V (x) u u + u V u = 0 i RN R+ t u(t = 0) = u N R0
OLE C
LY PO
CH TE
N
UE IQ
u
C
LE O
u(t, x)
LY T PO
HN EC
UE IQ
v(t)
PO LE COR
R v v u x R
TEC LYR v v t = |u(t, x)|2 dx =R u t
1 |v|2 , 2 t
HNu x
|x| +
UE IQ
|u0 (x)|2 dx.
u (t, x) x2
2
+ V (x) |u(t, x)|
dx =R
u0 (x) x
2
+ V (x) |u0 (x)|
2
RN f x x+ u(x) f (x) RN u(x)
OLE C
LY PO
CH TE
N
UE IQ
dx.
u ( + )(divu) = f u=0
> 0 2 + N > 0 u fi ui 1 i N f u RN ui ( + ) (divu) = fi xi ui = 0 1 i N ( + ) = 0 ui N = 1
C
LE O
LY T PO
HN EC
UE IQ
C
f (x) RN u(x) p(x)
LE O
LY T PO
HN EC
UE IQ
p u = f divu = 0 u=0
> 0 N p u = f divu = 0 N = 1 N 2
f (x) R u(x) (u) = f u=0 u n = 0
OLE C
LY O P
CH TE
N
UE IQ
u n = un n
C
LE O
LY T PO
HN EC
UE IQ
PO LE CO
TEC LY
HN
UE IQ
OLE C
LY POt nt
CH TE
N
UE IQ
(tn, x j)
x j x
x > 0 t > 0
C
LE O
LY T PO
HN EC
UE IQ
C
un (tn , xj ) u(t, x) j
LE O
LY T PO
(tn , xj ) = (nt, jx) n 0, j Z.
HN EC
UE IQ
un + 2un un 2u j1 j j+1 (tn , xj ) x2 (x)2
(x)2 2u (t, x) x2
u(t, x x) + 2u(t, x) u(t, x + x) =
(x)4 4 u (t, x) + O (x)6 12 x4
x j
C
OLE
LY POu (tn , xj ) V x
CH TEun j1 2x
N
UE IQ
u 2u u +V 2 =0 t x x
V
un j+1
un+1 un1 u j j (tn , xj ) t 2t
un+1 un1 un un un + 2un un j j j+1 j1 j1 j j+1 +V + = 0. 2t 2x (x)2
PO LE CO
LY T
HN EC
n j
UE IQ
C
LE O
un un1 u j j (tn , xj ) t t
LY T PO
HN EC
UE IQ
un un1 un un un + 2un un j j j+1 j1 j1 j j+1 +V + = 0. t 2x (x)2
(un )jZ (un1 )jZ j j (un+1 )jZ (un )jZ j j n
OLE C
un+1 un un un un + 2un un j j j+1 j1 j1 j j+1 +V + = 0. t 2x (x)2
EC LY T PO
un+1 un u j j (tn , xj ) t t
NIQ H
UE
un1 un1 un1 + 2un1 un1 un un1 j j j+1 j1 j1 j j+1 +V + = 0. t 2x (x)2
n (u0 )jZ j 0 uj = u0 (jx) u0 n = 1 (u1 )jZ j
C
LE O
LY T PO
HN EC
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
= 1 u0 (x) = max(1 x2 , 0).
H C T E Y POL OLE C
t = 0.1(x)2
N
UE IQ V =0
= R n 0 (un )jZ j R = (10, +10) x = 0.05 (un )200j+200 j un j
C
LE O
LY T PO
HN EC
UE IQ
C
t x t t x
LE O
LY T PO
HN EC
UE IQ
OLE C
LY PO
CH TE
N
UE IQ
t = 2(x)2
t t x t t x 2t (x)2
PO LE CO
LY T
HN EC
UE IQ
PO LE CO
TEC LY
HN
UE IQ
OLE C
LY PO
CH TE
N
UE IQ
t = 0.4(x)2
t = 0.51(x)2
un+1 = j t n t u + 12 (x)2 j1 (x)2
un+1 j un , un , un j1 j j+1
C
LE O
LY T POun + j
HN ECt n u . (x)2 j+1
UE IQ
u0 m M
C
LE O
LY T PO
m u0 M j Z, j
HN EC
UE IQ
m un M j Z n 0. j
2t > (x)2 .
u0 0 u0 = (1)j j
t n 1 4 (x)2 < 1
C
OLE
LY PO
t (x)2n
CH TE
N
UE IQ
un = (1)j 1 4 j
V = 0 1 j J un = un = 0 n N m 0 M 0 J+1 m u0 M 1 j J j n+1 n uj uj n 0 t x m un M 1 j J j
C
LE O
LY T PO
HN EC
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
t = 0.4(x)2
x 0.5 0.1 0.05 t t/(x)2 t = 1 t
OLE C
LY T PO
N CH E
UE IQx
V = 1 t = 0.4(x)2 = 1 = 0.01 = 0.1 V
C
LE O
LY T PO
HN EC
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
OLE C
LY PO
CH TE
N
UE IQ
0.4(x)2 V = 1 = 1 = 0.1 = 0.01
C
LE O
LY T PO
HN EC
UE IQt =
= 0 = 0 t x
C
LE O
LY T PO
HN EC
UE IQ
V = 1 = 0
C
OLE
LY PO
CH TE
N
UE IQt = 0.9x
= 0
un+1 = j
V t n V t n u u . + un j 2x j1 2x j+1 un+1 j t un un un j1 j j+1 V V > 0 V < 0 V > 0 un un u j+1 j (tn , xj ) V V x x
C
LE O
LY T PO
HN EC
UE IQ
PO LE CO
V > 0 V
TEC LY
un+1 un un un j j j+1 j +V =0 t x
HN
UE IQ
un un u j j1 (tn , xj ) V x x
un+1 un un un j j j j1 +V =0 t x
OLE C
LY PO
CH TE
N
UE IQ
0.9x V = 1
C
LE Oun+1 = j
V t n V t u + 1 x j1 x
LY T PO
|V |t x.
HN ECun , j
UE IQ
t =
C
un un j1 j un+1 j
LE O
LY T PO
HN EC
UE IQ
u0 = (1)j j
OLE C
LY T PO
N CH E
t x
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
C
LE O
LY T POdy dt
HN EC
UE IQ
= f (t, y) 0 < t < T y(t = 0) = y0
0 < T +
(0, T ) t = 0 t = T
t t = 0 t = T x t x
f A u u A u A(u) = f
OLE C
EC LY T PO
HN
UE IQ
f u f
C
LE O
LY T PO
HN EC
UE IQ
u
C
LE O
LY T PO
HN EC
UE IQ
OLE C
LY PO
CH TE
N
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
= (0, 1) (0, 2)
C
u(x, y) = e n sin(ny)sh(nx) x = 0 x > 0 u(x, y) n
LE O
x y 2 2 u u = 0 x2 y 2 0 < x < 1 u(x, 0) = u(x, 2) = 0 u n u(0, y) = 0, (0, y) = e sin(ny) 0 < y < 2 x
LY T PO n
HN EC
UE IQ
t x u(x, y)
OLE Ca
LY PO
CH TE
N
UE IQ
2u u 2u 2u u +c 2 +d +e + f u = g. +b 2 x xy y x y
b2 4ac = 0
a, b, c, d, e, f
b2 4ac < 0 b2 4ac > 0 ax2 + bxy + cy 2 + dx + ey + f = 0
b 4ac < 0 b2 4ac = 0 b2 4ac > 0 (x, y) (t, x) 2
C
LE O
LY T PO
HN EC
UE I Q
U E I Q N C H E Y T O L P L E CO Zz A 2u u 2u 2u u +C +E + F u = G, +B +D X 2 XY Y 2 X Y
(x, y) (X, Y ) J = Xx Yy Xy Yx Z z
X + Y Y X Y X2
2 C = aYx
A
2 2 = aXx + bXx Xy + cXy B = 2aXx Yx + b(Xx Yy + Xy Yx ) + 2cXy Yy + bYx Yy + cYy2 B 2 4AC = J 2 (b2 4ac)
u = 1 a = 1 x b = c = d = e = f = 0 b2 4ac = 0 y 2 2
OLE C2
2
2
LY PO2 2
CH TE
N2
UE IQ2 2
2
C
LE O
LY T PO
HN EC
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
OLE C
LY PO
CH TE
N
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
C
(0, 1) u 2u 2 = 0 (x, t) (0, 1) R+ t x u(0, x) = u (x) x (0, 1). 0
LE O
O LY P
N H TEC
UE IQ
(0, 1)R+ t > 0 0 N
x = 1/(N +1) >
(tn , xj ) = (nt, jx) n 0, j {0, 1, ..., N + 1}. un j
(tn , xj ) u(t, x) u0 = u0 (xj ) j {0, 1, ..., N + 1}. j
u(t, 0) = u(t, 1) = 0 t R+ un = un +1 = 0 n > 0. 0 N
OLE C
LY PO
CH TE
N
UE IQ
(un )1jN j N R N xj 1 j N un j un+1 un un + 2un un j j j1 j j+1 + =0 t (x)2
n 0 j {1, ..., N }
PO LE CO
un+1 + 2un+1 un+1 un+1 un j j j1 j j+1 + = 0. t (x)2
TEC LY
HN
UE IQ
C
1 + 2c c c 1 + 2c 0
LE O
LY T PO
un+1 un j j N 0 c t , c= (x)2 c 1 + 2c c c 1 + 2c
HN EC
UE IQ
0 1
un+1 + 2un+1 un+1 un+1 un un + 2un un j j j1 j j+1 j1 j j+1 + + (1 ) = 0. t (x)2 (x)2
=
= 0 1 = 0 = 1/2 5(un+1 un ) un+1 un un+1 un j+1 j j1 j+1 j j1 + + 12t 6t 12t
= 1/2 (x)2 /12t
OLE C
un+1 + 2un+1 un+1 un + 2un un j1 j j+1 j1 j j+1 + + = 0. 2(x)2 2(x)2
LY PO
CH TE
N
UE IQ
un + un+1 + un1 un un+1 un1 j1 j+1 j j j j + = 0, 2t (x)2
un+1 + 2un+1 un+1 3un+1 4un + un1 j j j j1 j j+1 + = 0. 2t (x)2
C
LE O
LY T PO
HN EC
UE IQ
f (t, x) f (t, x) (tn , xj ) f (tn , xj )
C
LE O
LY T POun+1 j
HN EC
UE IQ
un un + 2un un j j1 j j+1 + = f (tn , xj ). t (x)2
un j (n , j ) (n, j) (n , j )
OLE C
u u (t, 0) = 0 (t, 1) = 0. x x
LY PO
CH TE
N
UE IQ
un un un un N 1 0 = 0 N +1 =0 x x
un un +1 N 0 N (un )1jN j un un un un N 1 1 = 0 N +2 =0 2x 2x
x1 xN +2 un un +2 N +2 1 N (un )0jN +1 j u(t, x + 1) = u(t, x) x [0, 1], t 0.
un = un +1+j j N
C
LE O
un = un +1 n 0 0 N
LY T PO
HN EC
UE IQ
C
F (u) = 0 F (u) u (t, x) n, j
LE O
LY T PO
HN EC
UE IQ
n+m Ft,x {uj+k }m mm+ ,
k kk+
=0
m , m+ , k , k +
LY O E P O L C Ft,x {u(t + mt, x + kx)}m mm+ , (t, x) p x
F (u) = 0 u(t, x)
CH TEt
N
UE IQ,
k kk+
x q t
O (x)p + (t)q
t x n+m Ft,x ({uj+k }) = 0 u(t, x) u(t + mt, x + kx)
t/(x)2 = 1/6
C
LE O
LY T PO
HN EC
n+m uj+k
UE IQ
t x t x
C
(t, x)
v(t, x)
LE O
LY T PO
HN EC
UE IQ
C 6
v(t, x x) + 2v(t, x) v(t, x + x) v(t + t, x) v(t, x) + t (x)2 t (x)2 vtt vxxxx + O (t)2 + (x)4 , = vt vxx + 2 12
vt , vx v v t/(x)2 = 1/6 t (x)2 vtt = vtxx = 2 vxxxx
OLE C = 1/2
LY PO O t + (x)2 O t + (x)2 O t + (x)2
CH TE L2
N
UE IQ
L2 L 2t (x)2 L2 L
O (t)2 + (x)2
= 1/2
L2 2(1 2)t (x)2 L2 L2 L2
O (t)2 + (x)4t O ( x )2 + (x)2
O (t)2 + (x)2
C
LE O
LY T PO
HN EC
UE IQ
C
un = (un )1jN RN j x 1/p
LE O
LY T POunp
HN EC
UE IQ
=
N
x|un |p j
1 p +,
j=1
p = + un = max1jN |un | j x N x = 1/(N +1) x un p Lp (0, 1) [0, 1] Lp [xj , xj+1 [ p = 2, +
N CH E T Y O L EP O L C K>0 t x un K u0 u0 n 0,
UE IQ
t
x
RN x x n+m Ft,x ({uj+k }) = 0
C
UE Q N I ECH LY T PO LE O n+m uj+k
un+1 = Aun ,
C
RN RN A A 1 2c c 0 c 1 2c c t , c= (x)2 c 1 2c c 0 c 1 2c
LE O
LY T PO
HN EC
UE IQ
An n An u0 K
A un = An u0 A u0 n 0, u0 RN .
M = sup
Mu , u
uRN ,u=0
C
L
OLEmin 0,
LY POAn K
n 0,
CH TE
N
UE IQ
A
L
n 0 0jN +1
1 j N un max 0, j
u0 j
min
u0 j
0jN +1
max
u0
u0
u0
C
LE O
LY T PO
HN EC
UE IQ
U E IQ N C H E LY T PO LE CO L L 2t (x)2
t
x
L t (x)2 L 2t (x)2 L2
= 1/2
L L2 u(t, x+ 1) = u(t, x) x [0, 1] t 0 un = un +1 n 0 un = un +1+j 0 j N N N + 1 un j un (x) un = (un )0jN j [0, 1]
xj+1/2 = (j + 1/2)x 0 j N x1/2 = 0 xN +1+1/2 = 1 un (x) L2 (0, 1) L2 (0, 1) un (x) =kZ
OLE Cun (k) = 1 0
un (x) = un xj1/2 < x < xj+1/2 j
LY PO
CH TE
N
UE IQ
un (k) exp(2ikx),
un (x) exp(2ikx) dx 1 0
|un (x)|2 dx =kZ
|n (k)|2 . u
un un (k) v n (x) = un (x + x) v n (k) = un (k) exp(2ikx) 0 x 1un+1
u (x x) + 2un (x) un (x + x) (x) u (x) + = 0. t (x)2
C
LE O
n
n
LY T PO
HN EC
UE IQ
1
un+1 (k) =
C
k Z
LE O
un+1 (k) = A(k)n (k) = A(k)n+1 u0 (k) u
LY T PO
t ( exp(2ikx) + 2 exp(2ikx)) un (k). (x)2
HN EC
UE IQ
A(k) = 1
4t (sin(kx))2 . (x)2
un (k) n |A(k)| 1 2t(sin(kx))2 (x)2 .
2t (x)2 k Z un2 2 1
=0
|un (x)|2 dx =kZ
|n (k)|2 u
kZ
L2 k0 u0 (k0 ) = 0 |A(k0 )| > 1
OL E C
LY PO
CH TE0
|0 (k)|2 = u
1
|u0 (x)|2 dx = u0 2 , 2
x k0 x /2 L2
N
UE IQ
2t (x)2
L2
L2 L 0 x 1
un+1 (x x) + 2un+1 (x) un+1 (x + x) un+1 (x) un (x) + = 0, t (x)2
un+1 (k) 1 +
t ( exp(2ikx) + 2 exp(2ikx)) (x)2
C
LE O
LY T PO
HN EC
UE IQ
= un (k).
un+1 (k) = A(k)n (k) = A(k)n+1 u0 (k) u
|A(k)| 1 L2
C
[0, 1] R R un j n L2 un n+1 u
LE O
LY T PO
HN ECA(k) =
UE IQ
1+
4t (sin(kx))2 (x)2
1
.
k
A(k) A(k) C |A(k)| 1 k Z.
C
OLE
un = A(k)n exp(2ikxj ) j
LY PO
CH TE
N
UE IQ
L2
xj = jx,
t x L2
1/2 1
C
LE O
L2 2(1 2)t (x)2 0 < 1/2
LY T PO
HN EC
UE IQ
L2
C
T > 0 K(T ) > 0 t x un K(T ) u0 0 n T /t, u0
LE O
LY T PO
HN EC
UE IQ
2u u 2 = cu t x
(t, x) R+ R,
v(t, x) = ectu(t, x) c > 0 u |A(k)| 1 + Ct k Z.
OLE C
LY PO
CH TE
N
UE IQ
u(t, x) un j u0 = u0 (xj ) j
en
LYT PO E OL C T > 0,t,x0
lim
tn T
sup en
= 0,
HN EC
UE IQ
en = un u(tn , xj ) j j
p T > 0 CT > 0
PO LE CO
tn T
TEC LY
HN
U E IQ q
sup en CT (x)p + (t)q .
un+1 = Aun ,
A N u un = (n )1jN uj n uj = u(tn , xj ) n
n 0 tn T p q n C((x)p + (t)q ) n n ej = uj u(tn , xj )
C
OLE
un+1 = An + t u
LY POn
CH TEt,x0
N
UE IQ
lim
n
= 0,
en+1 = Aen t
n
en = An e0 t
n
Ankk=1
k1
.
un = An u0 K u0 An K K n e0 = 0 n
en tk=1
Ank
C
LE O
LY T POk1
tnKC (x)p + (t)q ,
HN EC
UE IQ
CT = T KC
C
[0, T ] N
LE O
LY T PO
HN EC
UE IQ
un+1 un un+1 un un1 un+1 n n1 u u un Un = , un1
OLE C
LY PO
CH TEA2 0 U n,
N
UE IQ
N A1 A2 A1 Id
U n+1 = A U n =
U n = An U 1
A An = sup
2N An U 1 K U1
U 1 R2N ,U 1 =0
n 1.
n+1
n1
n
u (x) u 2t
C
LE O+
(x)
u (x x) + 2un (x) un (x + x) = 0, (x)2
LY T PO
x [0, 1]
HN ECL2
UE IQ
un+1 (k) +
C
U n+1 (k) = A(k)n U 1 (k) A(k) k Z un (k) n A(k)n2
LE O U n+1 (k) =
LY T POun+1 (k) un (k) =
8t (sin(kx))2 un (k) un1 (k) = 0. (x)2
HN EC
UE IQ1 0
8t (x)2 (sin(kx))2 1
U n (k) = A(k)U n (k),
U n (k)
=
supUR2 ,U=0
A(k)n U U 2
2
K
n 1,
U 2 un2 2
R2 k Z
=kZ
|n (k)|2 K u
L2 k0 A(k0 )n n u0 (k0 ) u1 (k0 ) L2 A(k) A(k) 2 = (A(k)) A(k)n 2 = A(k) n (M ) 2 M (A(k)) 1 A(k) 8t 2 + (sin(kx))2 1 = 0 (x)2 1 (A(k)) > 1 L2
OLE C
LY POkZ
|0 (k)|2 + |1 (k)|2 = u0 u u
CH TE
N
2 2
UE IQ+ u1 2 , 2
C
LE O
un+1 j un j
= A(k)n
LY T POu1 j u0 j
HN EC
UE IQ
exp(2ikxj )
A(k)
(A(k)) 1
PO LE CO
A(k) (A(k)) B B (B) B n (B)n , A(k) L2 A(k) 2 = (A(k)) A(k)n 2 = A(k) n 2 A(k) A(k)
LY T
HN EC
k Z,
UE IQ
L2
OLE C
LY PO
CH TE
N
UE IQ
L2
L2 t x t/(x)2
= (0, 1) (0, L) 2 u 2u u 2 2 = 0 (x, y, t) R+ t x y u(t = 0, x, y) = u0 (x, y) (x, y) u(t, x, y) = 0 t R+ , (x, y) .
C
LE O
LY T PO
HN EC
UE IQ
x = 1/(Nx + 1) > 0 y = L/(Ny + 1) > 0 Nx Ny t > 0 (tn , xj , yk ) = (nt, jx, ky) n 0, 0 j Nx + 1, 0 k Ny + 1.
C
un j,k u(t, x, y)
LE Ok y
LY T PO
HN EC
UE IQ
(tn , xj , yk )
y
(x j, y k )
OLE Cun 0,k =
LY PO u0 j,k = 0, k,
CH TEj xun j,0
N
UE IQx
un x +1,k N
n > 0 = un y +1 = 0, j. j,N
= u0 (xj , yk ) j, k.
un+1 un j,k j,k t +
n n n n un un j1,k + 2uj,k uj+1,k j,k1 + 2uj,k uj,k+1 + = 0 (x)2 (y)2
n 0 j {1, ..., Nx} k {1, ..., Ny }
t t 1 + . (x)2 (y)2 2
C
LE O
LY T PO
HN EC
UE IQ
L
0.01 (1., 0.)
C
LE O
LY T PO
HN EC
UE IQ
OLE C
LY PO
CH TE
N
UE IQ
un+1 un j,k j,k t +
+
un+1 + 2un+1 un+1 j1,k j,k j+1,k (x)2
un+1 un
PO LE CO
LY T
un+1 + 2un+1 un+1 j,k1 j,k j,k+1 (y)2
HN EC
UE IQ
= 0.
un j,k j k un un j,k
C
Nx
LE O
LY T PO
HN EC
UE IQ
un = (un , ..., un y , un , ..., un y , ..., un x ,1 , ..., un x ,Ny ). 1,1 1,N 2,1 2,N N N
j k Ny D1 E1 0 E1 D2 E2 M = ENx 2 DNx 1 ENx 1 0 ENx 1 DN x
Dj Ny 1 + 2(cy + cx ) cy 0 cy 1 + 2(cy + cx ) cy Dj = cy 1 + 2(cy + cx ) cy 0 cy 1 + 2(cy + cx )
C
OLE
LY PO
CH TE
N
UE IQ
t cx = (x)2
t cy = (y)2 Ej = (Ej )t Ny cx 0 0 0 cx 0 Ej = . 0 cx 0 0 0 cx
M M
C
LE O
LY T PO
HN EC
UE IQ
un j,kn+1/2
uj,kn+1/2
t un+1 uj,k j,k
+
C
LE On+1/2
t
+
TEC O LY Puj1,k + 2uj,kn+1/2 n+1/2
uj+1,k
n+1/2
HN + +
UE IQ2(y)2 2(y)2
n n un j,k1 + 2uj,k uj,k+1
2(x)2
=0
uj1,k + 2uj,k
n+1/2
uj+1,k
n+1/2
un+1 + 2un+1 un+1 j,k1 j,k j,k+1
2(x)2
= 0.
L2
2u u 2 2 = 0 t x
uj,k
un j,k t +
/2
n+1/2
un+1 uj,k j,k t
C
OLE+
uj1,k + 2uj,k
n+1/2
LY POn+1/2
CH TE+n+1/2
2u u 2 2 = 0 t y
N
UE IQ
uj+1,k
n+1/2
2(x)2
n n un j1,k + 2uj,k uj+1,k =0 2(x)2 n+1/2
n+1/2
un+1 + 2un+1 un+1 j,k1 j,k j,k+1 2(y)2
+
uj,k1 + 2uj,k
uj,k+1
n+1/2
2(y)2
=0
L2
C
LE O
LY T PO
HN EC
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
zone o lon veut de la prcision
x y x y
OLE C
LY PO
CH TE
N
UE IQ
(0, 1) V > 0 u + V u = 0 (x, t) (0, 1) R+ t x + u(t, x + 1) = u(t, x) (x, t) (0, 1) R u(0, x) = u0 (x) x (0, 1).
C
LE O
LY T PO
HN EC
UE IQ
C
x = 1/(N +1) > 0 N t > 0 (tn , xj ) = (nt, jx) n 0, j {0, 1, ..., N + 1} un (tn , xj ) u(t, x) j un = un +1 n 0 un = un +1+j 0 j N N un = (un )0jN RN +1 j un+1 un un un j j j+1 j1 +V =0 t 2x n 0 j {0, ..., N }
LE O
LY T PO
HN EC
UE IQ
L2 (tn , xj ) L2 un (k) un
OLE Cun+1 (k) =
V t sin(2kx) un (k) = A(k)n (k). 1i u x
LY PO
CH TE2
N
UE IQ
1,
|A(k)|2 = 1 +
V t sin(2kx) x
2kx
un+1 un+1 un+1 un j j j+1 j1 +V = 0. t 2x
L2 un j+1
2un+1 j
2t
C
LE O
LY T POun j1 +V un j+1 2x
un j1
HN EC =0
UE IQ
U E I Q N EH T C O LY P E L CO L2 |V |t x. t/x t x
un+1 (k) =
cos(2kx) i
V t sin(2kx) un (k) = A(k)n (k). u x
V t x2
|A(k)|2 = cos2 (2kx) +
sin2 (2kx).
|A(k)| 1 k |V |t x k |A(k)| > 1 (tn , xj ) u
(ut + V ux ) (tn , xj )
OLE C
u(tn , xj+1 ) u(tn , xj1 ) 2u(tn+1 , xj ) u(tn , xj+1 ) u(tn , xj1 ) +V = 2t 2x (x)2 2t 1 (V t)2 (x)2 uxx (tn , xj ) + O (x)2 + (x)4 . t
LY PO
CH TE
N
UE IQ
O (x)2 /t 2
t (x) t/x en en tnKC (x)2 + t . t
x/t
t
C
LE O
LY T PO
HN EC
UE IQ t
un+1 un un un j j j+1 j1 +V t 2x
C
LE O
TEC O LY PV 2 t 2
un 2un + un j1 j j+1 = 0. (x)2
HN
UE IQ
(t)2 utt (tn , xj ) + O (t)3 . 2
u(tn+1 , xj ) = u(tn , xj ) + (t)ut (tn , xj ) +
u(tn+1 , xj ) = u(tn , xj ) (V t)ux (tn , xj ) +
(V t)2 uxx (tn , xj ) + O (t)3 . 2
u(tn+1 , xj ) = u(tn , xj ) V t +
u(tn , xj+1 ) u(tn , xj1 ) 2x
(V t)2 u(tn , xj+1 ) 2u(tn , xj ) + u(tn , xj1 ) + O (t)3 + t(x)2 . 2 (x)2
OLE C
u(tn , xj ) un j L2 |V |t x
LY PO
CH TE
N
UE IQ
L2 |V |t x
|V |t x V t/x 1, 0, 1
un+1 = un + un + un , j1 j j+1 j
, ,
C
LE O
V t/x
LY T PO
HN EC
UE IQ
C
LE O
TEC O LY P
HN
UE IQV >0 V < 0.
un+1 un un un j j j j1 +V =0 t x un+1 un un un j j j+1 j +V =0 t x
L |V |t x L 2 L
L2 |V |t x V u
OLE C
LY PO L2
CH TE
N
UE IQ
O t + (x)2 O t + (x)2 O t +(x)2 t
L2 L |V |t x L2 |V |t x L2 L |V |t x
O (t)2 + (x)2 O t + x
C
LE O
LY T PO
HN EC
UE IQ
UE Q I HN TEC O LY EP L CO
OLE C
LY PO
CH TE
N
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
(x)2 (V t)2 2t 1 (x)2 uxx
C
x = |V |t
LE O
u 2u u +V 2 =0 t x x
LY T PO
HN EC= (x)2 2t
UE IQ1 (V t)2 (x)2 .
x = |V |t x x = 0.01 V = 1 T = 5 t = 0.9x t = 0.45x
OLE C
u |V | 2u u +V (x |V |t) 2 = 0. t x 2 x
LY PO
CH TE
N
UE IQ
u V (x)2 u +V + t x 6 1
(V t)2 (x)2
C
LE O
LY T PO
HN EC
3u = 0. x3
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
OLE C
LY PO
CH TE
N
UE IQ
x = 0.01 t = 0.9 x V = 1 T = 5
C
LE O
LY T PO
HN EC
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
2 3 u + V u u u = 0 (x, t) R R+ 2 t x x x3 u(t = 0, x) = sin(x + ) x R,
V, , , , R u(t, x) = exp( 2 t) sin (x (V + 2 )t) +
un1 j
un+1 j 2t
+V un j+1 2x un j1 = 0.
|V |t M x M < 1
OLE C un+1 j t
LY POun+1 j+1 4x +V
CH TEun j+1 +V
N
UE IQ
un j
un+1 j1
un j1 = 0. 4x
(0, 1) 2 u 2u + t2 x2 = 0 (x, t) (0, 1) R u(t, x + 1) = u(t, x) (x, t) (0, 1) R+ u(t = 0, x) = u0 (x) x (0, 1) u (t = 0, x) = u1 (x) x (0, 1). t
PO LE CO
LY T
HN EC
UE IQ
C
un = (un )0jN RN +1 j un = un +1 n 0 0 N un = un +1+j u j N (0, 1) u u0 0 u1 C (0, 1) u(t, x) = Ct
LE O
LY T PO1 0
HN EC
UE IQ
u1 (x) dx = 0.
n 1 j {0, ..., N }un+1 2un + un1 j j j (t)2 un + 2un un j1 j j+1 +(1 2) (x)2
un+1 + 2un+1 un+1 j1 j j+1 (x)2 un1 + 2un1 un1 j1 j j+1 + =0 (x)2 +
0 1/2 = 0 = 0
C
OLE
u0 = u0 (xj ) j
LY PO
CH TExj+1/2 xj1/2
N
UE IQ
u1 u0 j j = t
u1 (x) dx,
1/4 1/2 L2 0 < 1/4 t < x
1 , 1 4
t/x > 1/ 1 4
un+1 (k) 2n (k) + un1 (k) + (k) n+1 (k) + (1 2)n (k) + n1 (k) = 0, u u u u
C
LE O
LY T PO
HN EC
UE IQ
U n+1 (k) =
C
U n+1 (k) = A(k)n U 1 (k) (1 , 2 ) 2 2 (1 2)(k) + 1 = 0. 1 + (k)
LE O
LY T POun+1 (k) un (k) =
(k) = 4
HN ECt x2
sin2 (kx).
UE IQ U n (k) = A(k)U n (k),
2(12)(k) 1+(k)
1
1 0
A(k)
= (k)(4 (1 4)(k)) . (1 + (k))2
A(k) A(k)n 2 = (A(k))n (A(k)) = max(|1 |, |2 |) A(k) (A(k)) 1 t/x > 1/ 1 4 k sin2 (kx) 1 > 0 1 2 (A(k)) > 1 t/x < 1/ 1 4 0 k (A(k)) = 1
OLE C
LY PO
CH TE
N
UE IQ
A(k)n
t/x = 1/ 1 4 0 < 1/4 un = (1)n+j (2n 1) j un
= 0.25 x = 0.01 t = 0.9 x T = 5 u0 u1
C
LE O
LY T PO
HN EC
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
t > 0
N
Cn+1
E(t) = E(0)
OLE
LY PO1
CH TE2 1 0
N
UE IQ2
E(t) =0
u (t, x) dx + t
u (t, x) dx. x
2
E
=j=0
un+1 un j j t
+ ax (un+1 , un ) + ax (un+1 un , un+1 un )
ax (u, v) =
N j=0
uj+1 uj x
vj+1 vj x
.
E n+1 E(tn+1 )
E n = E 0 n 0
C
LE O
LY T PO
O(x + t)
HN EC
UE IQ
v = u t
w = u x t v w
C
LE O
v(t, x + 1) = v(t, x), w(t, x + 1) = w(t, x) (x, t) (0, 1) R+ u0 (x) x (0, 1) w(t = 0, x) = x v(t = 0, x) = u1 (x) x (0, 1).
LY T PO= 0 1 1 0 x
HN EC
UE IQ
v w
(x, t) (0, 1) R+
u w 1 2tn+1 n n 2vj vj+1 vj1 n+1 n n 2wj wj+1 wj1
v
1 t
n+1 n vj vj n+1 n wj wj
OLE C+ t 2(x)2 L2
EC LY T PO 1 2x 0 1 1 0 0 1 1 02
1 2x
0 1 1 0
n n vj+1 vj1 n n wj+1 wj1
HN
UE IQ= 0,
n n vj+1 vj1 n n wj+1 wj1
n n n vj1 + 2vj vj+1 n n n wj1 + 2wj wj+1
= 0.
t x t/x t x L2
t x
J=
C
LE O
LY T PO0 1 1 0
HN EC
UE IQ
J 1 1
C
LE O
LY T PO
HN EC
UE IQ
OLE C
LY PO
CH TE
N
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
C
U E EC LY T O EP OL C
LE O
LY T PO
HN EC
UE IQ
NIQ H
C
LE O
u = f u=0
LY T PO
HN EC
UE IQ
RN f u C 1 () H 1 ()
C
LE O
LY T PO
HN EC
UE IQ
u
RN C() C() k 0 C k () C k () k u C 2 () C() f C() f C() C 2 N 2 C 2 f C() N = 1
OLE C
LY PO
CH TE
N
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
C
LE O
N = 1 = (0, 1) d2 u 2 =f 0 < x < 1 dx u(0) = u(1) = 0.
LY T PO1
HN EC
UE IQ
f [0, 1] C 2 ([0, 1]) u(x) = x0
f (s)(1 s)ds 0
x
f (s)(x s)ds x [0, 1].
[0, 1] m R d2 u 2 = f 0 < x < 1 dx u(0) = 0, du (0) = m. dx
OLE C
LY PO
CH TE
N
UE IQ
u(1) = 0 m m u(1) m
C
LE O
LY T PO
HN EC
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
RN C 1 N 1 n = (ni )1iN RN dx N ds N 1
OLE CC ()1
LY PO
CH TE
N
UE IQ
w
ni
C 1 w (x) dx = xi w(x)ni (x) ds,
i
w w w w
C
LE O
LY T PO
HN EC
UE IQ
0 O EP OL
LY Ti
HN ECi
UE IQyN Q+ y Q
C
i+1 i1
C 1 u v C 1 () u(x)
OLE C
LY POv(x)
CH TE
N
UE IQ
v (x) dx = xi
u (x) dx + xi
u(x)v(x)ni (x) ds.
w = uv
C 1 u C 2 () v C 1 () u(x)v(x) dx =
u(x) v(x) dx +
u =
u xi
1iN
C
LE O
LY T PO v
u
u n
HN EC
u (x)v(x) ds, n
= u n
UE IQ i
u xi
UE Q I H N C T E Y P O L L E C O k1 RN
Ck
(i )0iI
0 ,
I i , i=0
I i , i=1
i
Ck
i {1, ..., I} i
Q = y = (y , yN ) RN 1 R, |y | < 1, |yN | < 1 , Ck
i (i ) = Q y = (y , yN ) RN 1 R, yN > 0 = Q+ , i (i ) = Q y = (y , yN ) RN 1 R, yN = 0 .
2
OLE C
LY PO
CH TE
N
UE IQ
div(x)(x) dx =
(x) (x) dx +
C 1 ()
C
LE O
PO
LY T
(x) n(x) (x) ds, C 1 ()
HN EC
UE IQ
N = 3
R3 = (1 , 2 , 3 )
PO LE CO
rot =
TEC LY
R3 .
HN
UE IQ
3 2 1 3 2 1 , , x2 x3 x3 x1 x1 x2 C 1 ()
rot dx
( n) ds.
rot dx =
f
u C 2 () X X = C 1 () = 0 .
u X
OLE C
LY PO
CH TE
N
UE IQ u
u(x) v(x) dx =
f (x)v(x) dx
v X.
u C 1 () u C 2 () v v = u f
C
LE O
LY T PO
HN EC
UE IQ
u v X u(x)v(x) dx =
v = 0 v X
C
LE O
PO
LY T
u(x) v(x) dx +
HN EC
UE IQu (x)v(x) ds. n
f (x)v(x) dx =
u(x) v(x) dx,
u X u(x) + f (x) v(x) dx = 0
v X.
N C H T E Y POL O L E C C ()
(u + f ) u(x) = f (x) x u X u = 0 u RN g(x)
UE IQ
g(x)(x) dx = 0,
g
g(x0 ) > 0 g x0 g(x) > 0 x g(x)(x) dx =
x0 g(x0 ) = 0
g(x)(x) dx = 0,
g
g(x) = 0 x
u X a(u, v) = L(v)
C
LE O
a(u, v) =
LY T POu(x) v(x) dx
v X,
HN EC
UE IQ
a(, ) X L() X
C
X X = {v C 1 (), v = 0 } 1 H0 () X
LE O
LY T PO
L(v) =
HN ECf (x)v(x) dx,
UE IQ
C 2 () u u u C 1 ()
C
OLE
LY POu = f u n = 0
CH TE .
N
UE IQ
u(x) v(x) dx =
f (x)v(x) dx
v C 1 (). C 2 ()
f (x)dx = 0
(u) = f u=0 u n = 0
v v C 2 () v n X u C 4 () u u X
u(x)v(x) dx =
C
LE O
f (x)v(x) dx
LY T PO
HN EC v X.
UE IQ
C
V ,
LE O
LY T PO
HN EC
UE IQ
u V a(u, v) = L(v)
v V.
a L L() V v L(v) V R C > 0 |L(v)| C v v V ;
a(, ) V w a(w, v) V R v V v a(w, v) V R w V a(, ) M > 0 |a(w, v)| M w
a(, ) > 0 a(v, v) v2
OLE C
LY PO
v w, v V ;
CH TE
N
UE IQ a(, )
v V.
V L() V a(, ) V L w V v a(w, v) V V A(w) a(w, v) = A(w), v v V.
a(w, v) w A(w) v = A(w)
C
LE OA(w)2
= a(w, A(w)) M w
LY T PO
a(w, v) A(w) ,
HN EC
UE IQ
A(w) M w w A(w) V f f V = L V
C
A V V u u L a(w, v) w2
LE O
u V A(u) = f.
LY T PO
L(v) = f, v v V.
HN EC
UE IQ
a(w, w) = A(w), w A(w)
w ,
w A(w) w V,
A A Im(A) = V V Im(A) V Im(A) = {0} V = {0} = ( Im(A) ) = Im(A) = Im(A) A(wn ) Im(A) b V
n p wn V w V A A(wn ) A(w) = b b Im(A) Im(A) v Im(A) a(w, v) v2
C
OLE
wn wp A(wn ) A(wp )
LY PO
CH TE
N
A
UE IQ
a(v, v) = A(v), v = 0,
v = 0 Im(A) = {0} A A1 1 1 w = A (v) A u f
V A a(w, v) A V = RN Au, v = f, v v RN Au = f
C
LE O
LY T PO
HN EC
UE IQ
u T V V
PO LE COT (v) T (w)2
TEC LY
HN=
UE IQ , M2
T (w) = w A(w) f
u V T (u) = u
= v w A(v w) 2 = v w 2 2 A(v w), v w + 2 A(v w) = v w 2 2a(v w, v w) + 2 A(v w) 2 (1 2 + 2 M 2 ) v w 2 (1 2 /M 2 ) v w 2 .
2
E O L C
LY T POJ(v) =
N CH E
UE IQ
v, w V J(v)
v V
a(w, v) = a(v, w)
1 a(v, v) L(v). 2
u
u V J(u) = min J(v).vV
u V
u a
1 1 J(u + v) = J(u) + a(v, v) + a(u, v) L(v) = J(u) + a(v, v) J(u). 2 2
C
LE O
LY T PO
HN EC
J(v)
u
UE IQ
u + v u V
PO LE CO
v V j(t) = J(u + tv) R R t t = 0 j j (0) = 0
TEC LYvV
V u
J(u) = min J(v).
HN
UE IQJ
V
a J(v)
a(, ) X
OLE C
LY POa(u, v) =
u(x) v(x) dx
CH TE
N
UE IQ
L(v) =
f (x)v(x) dx,
L()
V
V = v C 1 (), v = 0 .
V w, v =
w(x) v(x) dx,
1/2
v =
|v(x)| dx2
.
V v = 0 v = 0
C
LE O
LY T PO|v(x)|2 dx = 0
HN EC
UE IQ
v = 0 v = 0 v a(, ) a L
C
C f v L V V C 1 V V V 1 V V H0 () V V
LE O
f (x)v(x) dx
LY T PO
HN EC1/2
UE IQ1/2
|f (x)|2 dx
|v(x)|2 dx
C v ,
RN C > 0 v C 1 () |v(x)|2 dx C
OLE C
LY PO
CH TE
N
UE IQ
|v(x)|2 dx.
x x1 < a C 1 () x1 b < + v v C 1 N R x x1
v(x) =a
x1
|v(x)|2 (x1 a)
C
a
LE O
v (t, x2 , ..., xN ) dt (b a) x1
LY T PO 2
v (t, x2 , ..., xN ) dt, x1
HN ECb a
UE IQ2
v (t, x2 , ..., xN ) dt. x1
C
LE O
LY T PO|v(x)| dx (b a)2
HN ECb a 2
UE IQ2
v (t, x2 , ..., xN ) dt dx, x1 t x1 |v(x)|2 dx.
|v(x)|2 dx (b a)2
v (x) dx (b a)2 x1
V RN N = 1 1 < x < n1 , x 1 2 (n/2)x 1 + 1/(2n) n1 x n1 , un (x) = x1 n1 < x < 1.
N = 2 0 < < 1/2
un (x) = | log(|x|2 + n1 )|/2 | log(1 + n1 )|/2 .
N 3 0 < < (N 2)/2
un n
C
OLE
LY PO
CH TE
N
UE IQ
un (x) =
1 1 . (|x|2 + n1 )/2 (1 + n1 )/2
V
V
C
LE O
LY T PO
HN EC
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
OLE C
LY PO
CH TE
N
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
L2
OLE C
LY PO
CH TE
N
UE IQ
C
LE O
LY T PO
HN EC
UE IQ
RN L2 ()
C
LE O
O LY Pf
N C H TEf (x)g(x) dx,
UE IQ
f, g =
L2 ()
1/2 L2 ()
=
|f (x)|2 dx
f f f g f (x) = g(x) E E f (x) = g(x) x ( \ E) Cc () D() C Cc () Cc ()
f L ()2
OLE C
LY POlim f fn
CH TE
N
UE IQ
Cc () L2 () fn Cc () n+ L2 ()
= 0.
f L2 () Cc () f (x)(x) dx = 0,
f (x) = 0 fn Cc ()
f
L2 () 0 = lim
C
LE On+
f (x)fn (x) dx =
LY T PO
|f (x)|2 dx,
HN EC
UE IQ
f (x) = 0
Lp () 1 p + 1 p < p + Lp () 1/p f L () = |f (x)|p dx ,
C
Lp () L ()
p = + f C > 0 |f (x)| C f L () = inf C R+ |f (x)| C ,
LE O
LY T POp
HN EC
UE IQ
L () Lq () 1 q p +
Lp ()
L2 ()
v L2 () v L () wi L2 () i {1, ..., N } Cc ()
OLE C2
LY POv(x)
CH TE
N
UE IQ
(x) dx = xi
wi (x)(x) dx.
wi
i
v
v xi
v wi = xi wi v L2 () v
v L2 () C Cc () i {1, ..., N }
v
LY T P O LE COv(x)
(x) dx C xi
L2 () ,
HN EC
UE IQ> 0
L
C
L() Cc () L L2 () Cc () L2 () L L2 () (wi ) L2 ()
LE O
LY T POL() =
L() = v(x) (x) dx. xi
HN EC
UE IQ
wi (x)(x) dx,
v
L2 ()
L () > 1/22
= (0, 1) x
C 1
L2 () v x 1 i N C v(x) = C vi
O LE C
C 1 L2 ()
EC LY T PO L2 ()
NIQ H
UE
Cc ()
> 0 (t)
v(x)
(x) dx = 0. xi
Q =] , + [N Cc ( , + )
+
(t) dt = 1.
Cc (Q) xi
(x , xi ) =
C
LE O
(t)
LY T PO+
HN EC
UE IQ
(x , s) ds (x , t) dt,
N 1
x = (x , xi ) x R Cc (Q)
PO LE CO Q
(x , xi ) = (xi ) xi
TEC LY+
HN+
xi R
UE IQ
(x , s) ds (x , xi ).
=Q
(x , s) ds dx dxi v(x , xi )(xi ) dxi dx ds
v(x)(x) dx
v(x)(xi ) (x , s)Q
+
=
+
v(x) =
v(x , s)(s) ds,
OLE C 2
v xi Q xi v(x) Q v(x)
LY PO
CH TE
N
UE IQ
RN L () L2 ()N L2() w L2() Cc () (x) (x) dx =
w(x)(x) dx.
w
w = div
C
LE O
LY T PO
HN EC
div
UE IQ
E U I Q N ECH LY T O P LE CO Cc ()
N
C >0
L ()
2
(x) (x) dx C
L2 () ,
1 2 = 1 2 C 1 1 2 L2 ()
Lp () 1 p + p = 2 Lp () v(x)
(x) dx C xi
Lp
()
Lp () Lp ()
OLE H C () 1
RN
LY PO v
N C H TE
1 1 =1 + p p
1 < p +,
UE IQ
H 1 () v L2 () , xi
H 1 () =
v L2 ()
i {1, ..., N }
v xi
u H 1 ()
C
LE Ou, v =
u(x)v(x) + u(x) v(x) dx
LY T PO
HN EC
UE IQ
u
H 1 ()
PO LE CO
H 1 () 1
H 1 () H () (un )n1 H 1 () H 1 () (un )n1 ( un )n1 i {1, ..., N } L2 () L2 () xi u wi un u un xi wi L2 () un Cc () un (x) (x) dx = xi
T E C LY =
|u(x)|2 + |u(x)|2 dx
HN
UE IQ1/2
un (x)(x) dx. xi
n +
u(x) (x) dx = xi
u u u xi u 1 H ()
C
C 1 H 1 ()
N 2 H 1 ()
OLE
LY PO
CH TE
wi (x)(x) dx,
wi i H 1 () (un )n1 u
N
UE IQ
RN N = 2 1 u(x) = | log(|x|)| H (B) 0 < < 1/2 N 3 u(x) = |x| H 1 (B) 0 < < (N 2)/2 N = 1 H 1 () = (0, 1)
B
v H 1 (0, 1) x, y [0, 1] y
v(y) = v(x) +
x [0, 1] v v(x) H 1 (0, 1) 1 R H (0, 1) v H 1 (0, 1) [0, 1]
C
LE O
LY T POx
v (s) ds.
HN EC
UE IQ
v H (0, 1) 1
PO LE CO x 0
LY T x
w(x) =
HN EC x
w(x) [0, 1]
UE IQ
v (s) ds.
0
1 0
v (s) ds
x0
|v (s)|2 ds
|v (s)|2 ds < +.
x
w [0, 1]x y
|w(x) w(y)| =y
v (s) ds
|x y|
|v (s)|2 ds
|x y|0
1
|v (s)|2 ds.
w w = v Cc (0, 1) 2 T T = {(x, s) R , 0 s x 1} 1 1 x
w(x) (x) dx =0 0 0
v (s) ds (x) dx =T
v (s) (x) ds dx.
T
1
1 s
v (s) (x) ds dx =
w
C
OLE1 0
w(x) (x) dx v
LY PO0
(x) dx v (s) ds =
CH TEL2 (0,1)
N1
UE IQ
(s)v (s) ds,
0
L2 (0,1) .
1 0
1 0
w (s)v (s) ds,
1
w (x)(x) dx =0 Cc (0, 1)
w(x) (x) dx =
w = v w v (0, 1) |v(x)| |v(y)| + |y x| y1 0 1 0 y x
|v (s)|2 ds |v(y)| +0
1
|v (s)|2 ds,
|v(x)| 0 1
|v(y)| dy + |v(y)|2 dy +
|v (s)|2 ds1 0
v v(x)
PO LE CO
LY T|v (s)|2 ds
HN EC 2 vH 1 (0,1) ,
UE IQ
H 1 (0, 1)
PO LE CO
H 1 (0, 1) v H 1 (0, 1) v H 1 (0, 1) v H 1 (0, 1)
TEC LY H 1 ()
HN1
H (0, 1)
E U I Q
u H 1 ()
=
RN +
{x RN + N R xN > 0}
Cc () H 1 () C C () Cc () = C () Cc () Cc () Cc ()
OLE C
LY PO
=R
N
Cc () H 1 ()
CH TE
N
UE IQC 1
C 1
C
LE O
LY T PO
HN EC
UE IQ
C
H 1 ()
LE O
LY T H () PO1 0
HN EC
UE IQ
H 1 ()
C Cc () 1 H0 () Cc ()
1 H0 () 1 H () 1 Cc () H0 () Cc () H 1 () Cc () = RN N = R = Cc (RN ) H 1 (RN ) 1 H0 (RN ) = H 1 (RN ) RN
1 H0 ()
E O L C
LY PO
CH TE
N
UE IQ
H 1 ()
1 H0 ()
H 1 ()
1 v H0 ()
RN C > 0 |v(x)|2 dx C Cc ()
|v(x)|2 dx.
v 1 1 Cc () H0 () v H0 () vn Cc () n+
lim
v vn
C
LE O2 H 1 ()
= lim
n+
LY T PO
HN EC
UE IQ
|v vn |2 + |(v vn )|2 dx = 0.
2 2
lim
n+
C
LE O
LY T PO|vn | dx =
|v| dx
HN ECn+
UE IQ|v|2 dx.
lim
|vn |2 dx =
|vn (x)|2 dx C
|vn (x)|2 dx.
n +
H 1 () 1 H0 ()
O LE C 1 |v|H0 () =
LY PO
1 H0 ()RN
CH TE1/2
N
UE IQ
|v(x)|2 dx
1 H0 ()
1/2
H 1 ()
1 v H0 ()
1 |v|H0 () v
H 1 ()
=
|v|2 + |v|2 dx
v2 H 1 ()
(C + 1)
1 |v|H0 ()
C
LE O
LY T PO
|v|2 dx = (C + 1)|v|2 1 () , H0
HN EC vH 1 ()
UE IQ
C
N 2 H 1 () v H 1 () v v| H 1 ()
LE O RN +
LY T PO
HN EC
UE IQ
=
0H 1 () C() v L2 () C() 0 (v) = v| .
C 1
0 H 1 () L2 () 0 C > 0 v H 1 ()
R , xNN
OLE C
H 1 () L2 ()
LY POvL2 () +
C v
CH TEH 1 () .
N
UE IQ
= RN = {x + N ) > 0} v Cc (R+ x = (x , xN ) |v(x , 0)|2 = 20
v(x , xN )2
v (x , xN ) dxN , xN
2ab a + b2 + 0
|v(x , 0)|2
|v(x , xN )|2 +
v (x , xN ) xN
2
dxN .
x
RN 1
|v(x , 0)|2 dx
v L2 (RN ) v +
C
LE O
H 1 (RN ) +
LY T PORN +
|v(x)|2 +
v (x) xN
HN EC2
UE IQ
dx,
Cc (RN ) +
H 1 (RN ) +
C = RN + 1
C C1
LE O u u(x)
LY T POv(x)
HN EC
UE IQ
H 1 () 1 C
v v (x) dx = xi
H 1 ()
u(x)v(x)ni (x) ds,
u (x) dx + xi
n = (ni )1iN
C 1 Cc () H 1 () (vn )n1 Cc () H 1 () u un
n + u un un vn vn u xi v xi xi v xi L2 () 0 0 (un ) 0 (vn ) 0 (u) 0 (v) L2 () u v H 1 () 1 H0 ()
OLE C
vn dx = xi
LY POvn
un dx + xi
CH TE
(un )n1 v
N
UE IQ
un vn ni ds.
1 C 1 H0 () 1 H ()
1 H0 () Cc () 1 0 H0 () 1 H ()
C
LE O
LY T PO
HN EC
UE IQ
0 1 H0 () 0 Im(0 ) L2 () L2 () H 1/2 ()
C
LE O
LY T PO
HN EC
UE IQ
1 C > 0 v H0 ()|v(x)|2 dx C
1 vn H0 ()
1 vn H0 () vn L2 () 2 vn L () 1 vn H0 () 1 H0 () v
OLE C
1=
LY PO|vn (x)|2 dx > n
CH TE|vn (x)|2 dx.
|v(x)|2 dx,
N
UE IQ
|v(x)|2 dx = lim
n+
|vn (x)|2 dx lim
n+
1 = 0, n
v v v |v(x)|2 dx = lim n+
|vn (x)|2 dx = 1,
C 1
C
LE O
LY T PO
v = 0
HN EC
UE IQ
N D V
PO LE CO
V = {v H 1 () v = 0 D }.
V H 1 () 1 H 1 () H0 () C > 0 v V H 1 ()
TEC LY
HN
UE IQ
H 1 ()
C 1 (i)1iI i C 1 i j = i = j = I i v i=1 i vi = v|i H 1 (i ) v v H 1 ()
C
v(x)
OLE (x) dx xj
I
LY PO i vi (x)i=1 I i i=1 I i
CH TEI
N
UE IQ
v Cc ()
= = =i=1
(x) dx xj vi (x)(x)ni (x) ds j
vi (x)(x) dx + xj i=1 vi (x)(x) dx, xj
i
i
= i k i k ni (x) = nk (x) j j v vi (x)(x)ni (x) ds + j vk (x)(x)nk (x) ds = 0. j
v
C
LE O
v
LY T POv xj =i
vi . xj
HN EC
UE IQ
H 1 ()
D
C
LE O
LY T PO
HN EC
UE IQ
N
(i )1iI i y
C
OLE
LY PO
CH TEx
N
UE IQ
y=xr
R2 0 1 v L2 () 2 > 1
L2 () C > 0
C
LE O
LY T PO
HN EC
UE IQ
2
v L ()
C
L E O
LY T POv|
L2 ()
L2 ()
HN ECC vL2 () .
UE IQ
C 1 1 H () L2() 1 2 H () L ()
= RN H 1 (RN ) L2 (RN ) un (x) = u(x + ne) e u H 1 (RN ) u e un L2 (RN )1 H 1 () H0 ()
OLE C
LY PO
CH TE
N
UE IQ
N = 1N = 1 = (0, 1) (un )n1 H 1 (0, 1) K > 0 unH 1 (0,1)
x, y [0, 1]
|un (x)| CK |un (x) un (y)| CK
C
LE O
LY T PO
K n 1.
HN EC|x y|.
UE IQ
C
(xp )p1 [0, 1] Q [0, 1] p un (xp ) R R un (x1 ) un1 (x1 ) R n1 n2 un2 (x2 ) R un2 (x1 ) np unp (xp ) unp (xk ) 1 k p unp p un1 un2 p p unp um p 1 um (xp ) R x [0, 1] > 0 (xp )p1 [0, 1] xp |x xp | um (xp ) R m m0 m, m m0 |um (xp ) um (xp )|
LE O
LY T PO
HN EC
UE IQ
|um (x) um (x)|
um (x) R x [0, 1] um L2 (0, 1)
OLE C
|um (xp ) um (x)| + |um (xp ) um (xp )| + |um (xp ) um (x)| + 2CK
LY PO
CH TE
N
UE IQ
= (0, 1) un (x) = sin(2nx) un L2 () Cc () 1 n+
lim
un (x)(x) dx = 0,0
un L2 () 1 H () un
H m ()
H 1 () m 0 = (1 , ..., N )
C
LE O
LY T PO
HN EC
UE IQ
PO LE CO
N i 0 || = v || v v(x) = (x). x1 xN 1 N
m m v L2 () m m 1 2 v 2v v xi xj = xj xi
TEC LY
HN
UE IQ
N i=1
i
m 0 v
H m ()
H m () = v L2 ()
|| m, v L2 () ,
C
OLEu
u, v = ||m
LY POu, u
CH TE
N
UE IQ
u(x) v(x) dx
H m ()
H m ()
=
H m () m N m H m () N = 1 H 1 ()
H m()
C 1 m > N/2 C()
k 0 m N/2 > k H m () C k () k
C
LE O
LY T PO
HN EC
UE IQ
C
m H m () v H m () v H m|| () H 1 () H m () H m ()
LE O Cc ()
LY T PO
HN EC
UE IQ
H m ()
C m
= RN +
H m () m = 2
1
C 1
v n = u n 1 H 2 () L2 () C > 0 v H 2 ()
H 2 () C 1 () L2 () C() v v 1 (v) = , n
OLE C
LY POv nL2 ()
CH TEH 2 () .
N
UE IQ
C v
1 H 1 () v H 2 () v H 1 ()N v L2 ()N v n L2 ()
H 2 () C() v H 1 () C() 0 (v) = v| ,
C 2 0
H 2 () H 1 () 0 (v) = RN +
C
LE O
LY T PO
HN EC
UE IQ
U E I Q N ECH LY T PO L E C O
H 2 () C 2 C2 u H 2 ()
v H 1 ()
u(x)v(x) dx =
u(x) v(x) dx +
u (x)v(x) ds. n
C 2 2 1 H () H () 1
= RN
RN
OLE C H1 N
LY PO
CH TE C
N
UE IQ
(R )
Cc (RN )
nN
n
n
1
1 n
1 n
2n n
Cc (B)
B
PO LE CO
n
0 B (x) dx = 1
LY T
n
HN ECn 2n
UE IQ
U E I Q N ECH LY T O P E L O C n (x) = n (nx)N
1/n
v H 1 (RN )
vn (x) = v
n (x) =
n (x y)v(y) dy,
RN
C
vn v
v
vn = (v) n v2
L2 (RN )
L2 (RN )
vn v
vn
vn vn n v L2 (RN ) RN
|x| 1 (x) = 0 |x| 2 vn (x) = vn (x)n (x) v v
L (R ) Cc (RN ) x n (x) = n
N
01
(x) = 1
C 1 = RN + P H 1 () H 1 (RN ) v H 1 () P v| = v P v L (R ) C v L () P v H (R ) C v H () C > 0 1 = RN x = (x , xN ) + x = (x1 , ..., xN1 ) v H (RN ) + v(x , xN ) xN > 0 P v(x) = v(x , xN ) xN < 0. 1 i N 1 v (x , xN ) xN > 0 P v x (x) = v xi (x , xN ) xN < 0, x v (x , xN ) xN > 0 x P v (x) = v xN x (x , xN ) xN < 0. v v P v P C = 2 = RN + C 1
