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n ds, #N!

* $ " ' " 8%9 #/! " 89 ' #N! $

% " ( #N! ' H1() H(div) * H1() H(div) ( #N! =

*

Im(0) H1/2() , *

* #/! n() '

H1/2()

$ ' - "% H10 () H(div) $

H0(div) H(div)

( Cc () H(div) 5

B' " % !

*

H(div)

1 Wm,p()

Wm,p() m 0

1 p + 5 @( Lp() % #! #!! 5

% & A" #) % *' &

Lp() 4 % %

" . m 0 " Wm,p() & Wm,p() = {v Lp() || m, v Lp()} , #G!

% v

?

uWm,p() =

||m

up

1/p

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE


$ $

% " Wm,p() @( 5 ' " ' % FIH!

' ' + -* 4

2 1 % ' = RN = RN+

p < N W 1,p() Lq() q [1, p] % 1/p = 1/p 1/N p = N W 1,p() Lq() q [1,+[ p > N W 1,p() C(),

#)!

% ; & " W 1,p() E % " C " u W 1,p()

uE CuW1,p(). p = 1 m = N "'

1'%

! WN,1(RN) / 0! 1 RN Cb(R

N ) u WN,1(RN ) uL(RN ) uWN,1(RN ). #I!

1 u Cc (RN ) x = (x1, ..., xN )

u(x) =

x1

xN

Nu

x1 xN (y) dy1...dyN ,

:

uL(RN ) Nu

x1 xN L1(RN ) uWN,1(RN ).

$ Cc (RN) WN,1(RN )

D( ##

H1(RN)! . ' #I! u WN,1(RN) * Cc (R

N ) L(RN) Cb(RN) * ! . #I! " " * WN,1(RN )

'

*

A " V B' V ' * V D( ) AE L2() & L2() $ 1'%0 H10 () ;

&

$ H1() $ % % FGH!

' H1()

H1() =

{f = v0 +

Ni=1

vixi

v0, v1, ..., vN L2()}

.

2! ! H10 () L H1() H10 ()

L() =

(v0

Ni=1

vi

xi

)dx

v0, v1, ..., vN L2()OP & H1() % %

* L2() *' " % *' . #C! .% Q % " "

% % ' ;

( ' % %!

" v L2() . 1 i N & ! vxi

H1() !

vxi

, H1,H10() =

v

xidx H10 (), #C !

& vxi

H1()

vL2() .

" v H1() ! vxi

3 L2() v

$ % * " ' #C ! *

H10 () " Li H1() "

Li() =

v

xidx.

$ % * " v Li L2() H1() " % * v % *' v H1()! " %& * L2() Li =

vxi

# OP ( AE "

B' % 5 "

; H1() H10 () 0 + H10 ()

! L2() $ ;& L2() ! ; % %!

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE


- $

H1() H10 () % !

H10 () L2() (L2()

) H1(),

:

' % +

D ( % ( '

% FNH! 1 % ' ( '

1'% ' ("

' " ( (" 1'%

( % " % ! 5

( '

" " " (

" % R * % (

% & ' ' 5

* ' " % " % !

'*

( '

1 % RN $ Cc () D()! * C& $ Cc () 89 & " % Cc () $ " (n)n1 C

c ()

% % Cc () n K

n % * K %

D() 89 D() & * 89 D() D() % 4 D()

4 T D() ! D() &lim

n+T (n) = T ()

(n)n1 Cc () Cc () & 1$ T, = T () ' T D()

* D() K 89 Tdx 0 +

% " f * ' ! ' Tf

Tf , =

f dx.

" * ' Tf f $ D() % K " Tn D()

*+ % T D() D()lim

n+Tn, = T, .

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE


" $

5 % ' % *'

! & % 8 9

. * K T D()

Txi

D() Txi

, = T, xi

D().

$ % "+% %

Txi

' & " '

%' S 5 & ' $

% " f * %' " % ' T % %

@ - = % *' . #C

% ' . Lp()

1'% Hm() ' D() 0 % " % " %

' " *! 0 *

% " %

" ! "

'

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE


$

u L2() .0 80 A iA;. 80)0+'+) CC + !

$ : + $%: xN = 0 B f f(x, xN ) = f(x,xN ) " ! B : " ! + + = {xN > 0} > {xN = 0}

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE


- "

)9.0;5) CC N, . 0 f ;"< 6 ? L2()A .0 f H1() ; L8M

"

R . ; 0< , RN A f L2() g L2() , ;"$< . R . I

, F ) ,. ; 0< : 6

5 "- , . ;"$<

0 ( ? 6 6

', ? .

,

1.3) 7 0 , 8 .

* . 8 . , ;"$<

8 v u : ) ( 8

4 ;< ;. ;&

"

1 .) 8 . ;"- 0 ""

|A(x)| || RN . /CG!$

div(A) = N

i,j=1

xi

(aij(x)

xj)

, /C)!

' {div(Au) = f ,u = 0 .

/CI!

% (" & /CI! .

( " %

! " %

A(x) & ! A(x) " & + % 0 + A =

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE


"

As+Aa "As = (A+At)/2 "Aa = (AAt)/2

"

div(Au) = div(Asu) + V u % Vj(x) =Ni=1

1

2

(aji aij)xi

(x),

: V

% %

/0

u H10 () f L2()

$ .( /CI!

4

" /C)!

u

nA=(A(x)u

) n =

Ni,j=1

aij(x)u

xjni = 0 ,

:

unA

% u & /C)! 5 4

% (" " L

= Au " L ' n = 0

/0 f L2() g L2() ! ! { div(Au) + u = f ,

unA

= g .

! $

1 6 . 6

. 1 1 6

A . 0 RN f L2() 5 " 8

u H10 () { u = f

u = 0 . ;"'959

/ O . . ' ;Q

? 8 N = 1 Y . + < '

4%,0?9) CC $0'*+'3) &5 9.>'9596 f 0 ) )"" $ ' u 0 ) )"" $ ,

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE


"

)9.0;5) CC 8 6

8 . # J ,6

A ( ;f 0

"

G(vn) % % G(v) L2() . 1 i N G(vn)xi G(v) vxi

G(vn)G(v) vxi

+(suptR

|G(t)|)vn

xi v

xi

. /#C!5

|G(vn)G(v)|

vxi

% " % !

; 2 (sup |G(t)|) vxi

" & L2() (

% ' % ( CG FNH! *

% % L2() ; /#C! % % L2() G(vn) 5( H

10 () " B

' K % % w H10 () % "w = G(v) L2() " w

xi= G(v) v

xi L2() :

4 ( * t max(t, 0) *Gn(t) " v

+ & H10 () 1 G(t) *

C1(R) "

G(t) = 0 t 12, 0 G(t) 1 1

2 t 1, G(t) = 1 1 t.

$ Gn(t) = G(nt)/n n 1

" Gn(v) H10 ()

Gn(v)xi

= Gn(v)vxi

. % "

|Gn(v) v+| suptR

|Gn(t) t+| 1n,

Gn(v) % % v+

L2() $ 1 i N Gn(v)xi 1v>0 vxi = |Gn(v) 1v>0|

vxi

100v

>)0+'+) CC " $ v v+ L2() B( H1() B u = 0 $ u1(0)

%(5-.0'1%

/ , 0

.

4%,0?9) CC $&) 0%(5-.0'1%6 " $ $" m 0, " $ -" /$ RN Cm+2, " f Hm(), ' $ "$ u H10 () =,9))"$" Hm+2(), 8 )' ))"$ f u " $ $"$ Hm()$ Hm+2()' " &" $ $"$" C > 0 "

uHm+2() CfHm().

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE


"

* 5 "$ 5 "

; 8 Hm() N/2' "$ -"$$ u H10 () =,9 " $ "$" ))"$" C2(),

$ )"' " $ -" /$ RN C' " f C()' "$ u H10 () =,9 " $ C(),

)9.0;5) CC 0

9 uA 0 . uA ? 8A 1,51): . u ? H V E GA . N = 1 [ . . , 8

.

)9.0;5) CC 5 "$ ' "&

.0 /

? : .0 ; 6

"

"

" % /##! 4 %

' #$ " f L2(RN ) u H1(RN ) 5 H2(RN) ' !6! f Hm(RN ) m 0 u Hm+2(RN)

8(

9 h RN h = 0 " +

Dhv(x) =v(x + h) v(x)

|h|

" ' & H1(RN ) v H1(RN ) $ % " (Dhv) = Dh(v) " v, L2(RN) 8* 9

RN

(Dhv) dx =

RN

v(Dh) dx.

. " Dhv

/C . *% /#/! v = Dh(Dhu) " '

RN

(|(Dhu)|2 + |Dhu|2

)dx =

RN

fDh(Dhu) dx.

$ ;

Dhu2H1(RN ) fL2(RN )Dh(Dhu)L2(RN ).

$ /#)!

Dh(Dhu)L2(RN ) (Dhu)L2(RN ) DhuH1(RN ).

. DhuH1(RN ) fL2(RN ) 1 i N Dh uxi

L2(RN )

fL2(RN ),

" "

/C "

uxi

& H1(RN) &

" u H2(RN )1 " f H1(RN) ? " u

xi "

H1(RN )

ui + ui = fxi

RN . /#N!

1 % "

uxi

&H2(RN ) & " u H3(RN )

% U% *% /#/! % *

xi

Cc (RN )RN

(u

xi+ u

xi

)dx =

RN

f

xidx.

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE


"

5

u H2(RN) f H1(RN ) 'RN

( u

xi + u

xi)dx =

RN

f

xi dx /#G!

" %' H1(RN ) $ " /#G! *

% /#N! "

uxi

= ui ' "

H1(RN ) /#N! f Hm(RN) u Hm+2(RN) m

% * m = 1

# . v L2(RN ) h RN h = 0 & 7

Dhv(x) =v(x + h) v(x)

|h| L2(RN ).

" v H1(RN) !

DhvL2(RN ) vL2(RN ). /#)!

8! v L2(RN ) 9 C h = 0

DhvL2(RN ) C, /#I! v H1(RN) e vL2(RN ) C e RN

. ' /#)! v Cc (RN )

Dhv(x) =

10

h

|h| v(x + th) dt,

" ;

|Dhv(x)|2 1

0

|v(x + th)|2 dt.

6 x %

Dhv2L2(RN ) 1

0

RN

|v(x + th)|2 dx dt 1

0

v2L2(RN )dt = v2L2(RN ).

/#)! % * H1(RN )1 v L2(RN) " % /#I! $ "

Cc (RN) RN

Dhv dx

CL2(RN ).$

RN

(Dhv) dx =

RN

v(Dh) dx,

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE


" "

h = te % e RN e = 0 limt0

Dh(x) = e (x).

" " 1 i N Cc (RN) RN

v

xidx

CL2(RN ), " " v & H1(RN) % .#C!

# (, 0"

/C % = RN ! D( /N $

## 1% . C/ % % J

##! % % (i)0iI (i)0iI "

i Cc (i), 0 i(x) 1,I

i=0

i(x) = 1 .

% 0 & * 0 ! " % i i 1 % ' 1 u H10 () " /# ! u =

Ii=0

iu % ( iu 0u u 6

" 0u

" H1(RN )

(0u) + 0u = f0 RN ,% f0 = 0(f u) 20 u u0 " & L2(RN) /C " 0u H2(RN) 5 f0 % /C " f Hm() " " 0u Hm+2() iu i 1 * ' 89 ' = RN+ 6 * " /C = RN+ 5 ' (

" % & %'

1 /C! K % & FGH 0

* "

%

% '

>)93-) &) :'*(5-.0'1%

># :'*(5-'?0):A ,6?6 6

, , 0 . 56

"$ ' "& 8 9 A 0 ,

80 ;,6?6 ? , 0. H1()< ,# 8 ;,6?6 8 .0< 0 /

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE


$ "

0

12

1

2

0

" W , ;

" &

)99) CC &" $ $ "$ / =,= $ H1()' $$)

u(r, ) =( rR

) k

cos(k

). ;"" 0 (2+N) > 0 H 1A

, B ; X A

. " 6< C > 0

|e(v)|2dx C

|v|2dx

v H10 ()N 5A , * ; A . * < C > 0 A v H10 ()N A

|v|2dx C

|v|2dx.

3 A ? .

2|e(v)|2dx +

|divv|2dx Cv2H1().

5 67 , ,

8 . ;""&

"

8 8 ' )

/ ,.

n = g , ;"" !!/ "

$

f (Mx + b) dx +

g (Mx + b) ds = 0 b RN , M = M t RNN

M $ $ $

H1()N $ + $ $ @ *A ( ! 2

)9.0;5) CC : 6

# 1 A ,

. # ; ' 0 2 + > 0

>)0+'+) CC $ ! . !! " ! . !! *(B $ ! .

div(u)(( + )divu) = f .

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE


$ "

1 A 30,/-?9) &5 +':.'--)9)*1 .*1'3-.*A

# , ) 0 ,

? , 0 '

8 ,

,

00 ,0 ' 6

, .

>)0+'+) CC :

$ $

: %* * L > 0 ( 4 * RN1 M G ( = (0, L)( x ( x = (x, xN ) 0 < xN < L x

div (2e(u) + tr(e(u)) Id) = 0 n = g (0, L)u = 0 {0, L}(n) n = 0 {0, L}

;"$)0+'+) CC >

a(x) = 0 " : % P1 * " $ :

10

f(x) dx = ,

$ 0

)0+'+) CC G % E 0% 2

6% & $ % $ (

: + B Kh + M bh E "B >

>)0+'+) CC (n + 2) * xj = j/(n + 1) 0 j n + 1 B k > 0 + % * fj 6$%:% * $* : $

L %%

$ : $

'$ #

* . ) P1 6

, . 6 $ /

) ,0 ,3%0.1)50 &!'*1)03,-.1',* rh ; $

$ ' !

' ) 0 A . A 8

H1(0, 1) . 0 ) ,6 rhv , 8 v 8 : [. v xj ;. 2 $

' ! N = 1 $

)9.0;5) CC ;$< 5 $$ .

0 f ? H1(0, 1) , ;$"< . . ) P1 ' I6

? hA ) P1 .

+ 5 $$ . .0

Kh 0 bh . 'A 6 ) P1 . , 8

Kh bh ;. T"U

$ ' !

%9,*:10.1',*C v C([0, 1]) * )A , rhv 8: A x ]xj , xj+1[A

v(x) rhv(x) = v(x)(v(xj) +

v(xj+1) v(xj)xj+1 xj (x xj)

)

= xxj

v(t) dt x xjxj+1 xj

xj+1xj

v(t) dt

= (x xj)v(xj + x) (x xj)v(xj + j)

= (x xj) xj+xxj+j

v(t) dt,

;$-'959 &':+0)16 $ )) f 0) )"" $ ]0, 1[, ' "$ uh ))&"$ -"$$%, ) "# $" *$ P1 -* uh 0 $ [0, 1],%9,*:10.1',*C uh ;$

& ' !

, , *,09) &':+0?1) Vh

' ! N = 1 &

1 ' ,

{ u = 6x 2 + 1/2 0 < x < 1u(0) = u(1) = 0 ;$ : *

>)0+'+) CC G % P1 6%

=]0, 1[2 * * 7* 2" * Kh B $ % E + h2 ( bh E

$ W 7 8 ,

>)0+'+) CC $' ; n * B B %

@* *A * * "

* Kh P1 $ n2 * $ 2n n *

" B % B : * =]0, 1[3

+ $ n3 * $ 2n2 4 n * $ B

)9.0;5) CC ' , ,9 $

0 ; 8O . 8

0< D KhA ,6?6 , ' ;.A

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE


- ' !

A )0+'+) CC $ B = (bij)1i,jn " ( i(

bii > 0,n

k=1

bik > 0, bij 0 j = i.

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE


' ! N 2 -"

" " M

>)0+'+) CC N = 2 ) uh % 6% % P1

* * Ki Th * + /2 " uh(x) 0 f(x) 0 L I ( > 0(Kh + Id "( 4 Kh *

$ W 5 f , ;$$

-$ ' !

>)0+'+) CC " * Kh

% Pk E $' !22

:

>)0+'+) CC $

!&; % $ : O N = 2 - * * Th $

Vh ={v C1() v |Ki P5 Ki Th

}.

" :9 p P5 * K 2

v(aj),v(aj),v(aj), p(bj)n

j = 1, 2, 3, ;$- N/2 *$$

' !

% h

! " C " " ' ! %

dist(, h) Ch2.$ uh % Vh

Th & Pk 0 ( h Vh 1'% V " (( u 4

! " "

4 N = 2 % FC/H! " P1 u H2() ;

u uhH1(h) ChuH2(), N/#!

" P2 u Hk+1()

u uhH1(h) Ch3/2uHk+1(). N//! " ( * P1 " %

N/#! " N/! %

Pk % k 2 $ & 8 "9 K ' & '8'9 ' * N * * & A" NC !

N = 2 % * < " *

* 89 ' 5

( * h $ " N/! % F H FC/H!

4 & NCN "

0

(" 1 K N k k $ rK * v K

rKv = p Pk " p(x) = v(x) x k. N/N!.

NCC " < Pk "

% k K " N/N! ' rK

=!

#! ( 12, k + 1 > N/2 1 rK H

k+1(K) Hk+1(K) C(K) v Hk+1(K)

v rKvHk+1(K) C(K)|v|Hk+1(K), N/G!

% |v|Hk+1(K) !1! &

|v|2Hk+1(K) =

||=k+1

K

|v|2dx = v2Hk+1(K) v2Hk(K).

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE


' ! N 2

k+1 > N/2 D( #C/ " " Hk+1(K) C(K) % * Hk+1(K) '

* " rKv < - v Hk+1(K) " Hm(K) % m N$ " rK H

k+1(K) .

vHk+1(K) C(K)(|v|Hk+1(K) + rKvHk+1(K)

), N/)!

% ;& * M %

#/! ! 6

vn Hk+1(K) "1 = vnHk+1(K) > n

(|vn|Hk+1(K) + rKvnHk+1(K)

). N/I!

( N/I! " " vn ' Hk+1(K)

D( A( #C vn " % Hk(K)

N/I! " " % vn || = k + 1 % % E L2(K) " vn % Hk+1(K)% v " % & N/I!!

|v|Hk+1(K) = 0, rKvHk+1(K) = 0. NN ! NN ! " v Pk K #/! N/N! rK rKv = v v Pk NN ! " rKv = v = 0 " % ( N/I! ' N/G! " N/)! & (vrKv) " " rK(v rKv) = 0 " |v rKv|Hk+1(K) = |v|Hk+1(K) " % k + 1 < Pk

%

NCI @'B' "

N/G! K $

%

# $ k + 1 > N/2 diam(K) 1 : C K v Hk+1(K)

v rKvH1(K) C (diam(K))k+1

(K)|v|Hk+1(K). NN!

$ A" NC " " N K * N * K0 N#N! %' B % b K! " x K x0 K0 %

x = Bx0 + b. NN!

' NN! N/G! ' K0 " (

%' NN! 5 % % & K 4 "

' FC/H! 7'

( %' det(B) % K ' & %

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE


' !

K0 % B1

C K " * v(x) % v0(x0) = v(Bx0 + b)

|v0|Hl(K0) CBl | det(B)|1/2 |v|Hl(K)|v|Hl(K) CB1l |det(B)|1/2 |v0|Hl(K0).

$ N/G!

|v rKv|H1(K) CBk+1B1|v|Hk+1(K)v rKvL2(K) CBk+1|v|Hk+1(K).

% * "

B diam(K)(K0)

, B1 diam(K0)(K)

.

0 ' ' NN!

' # v Hk+1() rhv N K rKv "

v rhv2H1() =

KiTh

v rKiv2H1(Ki).

$ " ; NN! & (" Ki % C !

N#I! ; *

diam(Ki)/(Ki) $

v rhv2H1() Ch2k

KiTh

|v|2Hk+1(Ki) Ch2kv2Hk+1()

"

>)0+'+) CC " * *(

P1( $ $ rh N = 2 3

v rhvL2() Ch2vH2().

(

# ;,6?6 . 6# 8

' ! N 2 "

$ W 9 N = 2

%@*'1',* CC " $ -" $$& )! RN , :$ 9.'--.() 0)+1.*(5-.'0) " $ $/ Th N "$0 $$ 0$ (Ki)1in -*$"

, Ki " = ni=1Ki', $""$ Ki Kj & N "$0 "$" " $ m"$0' -

0 m N 1' $" " " $" " Ki " Kj, $ $$ N = 2' $""$ & "$0 " " -' " $" $' " $ $ $"+,

:,99)1: *,)5&: 0 Th $" " N "$0 Ki )$", $-$"$' )+" h 0$ & +" N "$0 Ki,

/ ) ,0 Qk #X ? : RN

R

8 ? k 3.0 0.33,01 E +4.;5) 2.0'./-)A ,6?6 p Qk , 8

p(x) =

0i1k,...,0iNki1,...,iNx

i11 xiNN . x = (x1, ..., xN ).

+ p H ? kA ( ,Qk Pk

* k 1 ) 10)'--': &!,0&0) k N 6 K ,0

k ={x K xj lj

Lj lj {0,1k, ...,

k 1k

, 1} 1 j N}

. ;$$ 0 ; (.

(

L0M 6

V A . , 0 ) 00 ; . . # $")0+'+) CC ) V (x) = Ax x A : " u(x) = exp(A1/2x x/2) &/ = tr(A1/2) H

/ &

1 0 6

=0 ;. T$U$"' " A = A,

%@*'1',* CC " A $ ))"$ $ $"$ V $ V , $ " A " *$ )"- Ax, x > 0 ) "" x V $$ $,

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( $

@ , ) 6I

0 0 / . , )

6I

+,93.+1): )

,

%@*'1',* CC :$ $/ K V " " )" ' "" " (un)n1$" K' $ )" &" $ " un $-0$" $ K,

:$ $/ K V " " "-$" )" ' "" " (un)n1$" K' $ )" &" $ " un $-0$" $ V ,

0 A V )A 60 V 8 0 7A , . ) 9 (A 60 I 8 0

, . .

)99) CC 8$ $ ) @/" V $$ $*$' / $" $" > )",

%9,*:10.1',*C ' , )A

46 ) (en)n1 ' 6 0 ? 0 8 * A n = p

en ep2 = en2 + ep2 2en, ep = 2,

. , 6 en , '#

%@*'1',* CC " V " W & ) @/" " A $ ))"$ $$"$ V $ W , $ " A " )" 0 ) A / $" V " "-$" )" $ W ,

1 .A A A 0 xn V A 6 Axn . W W V )A ' , . W V )A , .

>)0+'+) CC " $ Id O V $ 5 &2

>)0+'+) CC ) $ O 2 x = (xi)i1 i1 |xi|2 < +( x, y =

i1 xiyi ) (ai)i1

( |ai| C < + i 1 $ A Ax = (aixi)i1 A " A limi+ ai = 0

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COLE P

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( "

>)0+'+) CC ) U ( V W O ( A

V W ( B U V " $ AB A B ' $ $ 5 $

* #

6 .

4%,0?9) CC " V $ ) @/" $$ $*$ " A $))"$ $ $"$' *$ )"-' ">$"' )" V $ V , - )) A $" $ " (k)k1 ""$" )" "$ - ' " &" $ / #/"$$ (uk)k1 V -")) A' -

Auk = kuk ) k 1.)9.0;5) CC ' 5 &-A . H 6

A 0 &%+,93,:'1',* :3)+10.-) v V

v =+k=1

v, ukuk . v2 =+k=1

|v, uk|2.

>)0+'+) CC %:%

$ ( $

$ % * " |Au, u| Au2 max(|m|, |M |) A.

A ; ' u, v V

4Au, v = A(u+ v), (u + v) A(u v), (u v) Mu + v2 mu v2 max(|m|, |M |)

(u + v2 + u v2

) 2max(|m|, |M |)

(u2 + v2

).

$ A = supu=v=1Au, v " Au = supv=1Au, v $ "A max(|m|, |M |) :

m M % K A = M 0 A = m % m 0 5 A = M 0 A = m " 2 A A! 1 (un)n1 %

V " M & "

limn+

Aun, un = M un = 1.

5

A " Aun % V % v .

Aun, un Aun A = M,: " limn+ Aun = M & " v = M J

Aun Mun2 = Aun2 + M2 2MAun, un, ' " limn+ Aun Mun = 0 n " " un% % v/M M = 0 M M = 0 % " " "A = 0! A " Aun % % Av/M % v! " Av/M = v & " v % v = M = 0! & % M

" V * A ! V V . > 0 ! & ] ,+[ 1 ! &

? " % V

" % A % "|| > 0 $ 1 " (uk)k1 V % (k)k1 "

Auk = kuk |k| k 1.$ Ek % * (u1, u2, ..., uk) 5

Ek1 Ek % vk Ek " ( & Ek15

|k| vk/k ' V " A " Avk/k % V 5 j < k

Avkk

Avjj

= vk + (A k Id) vkk

Avjj

. G !

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COLE P

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COLE P

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( &

$ % * " AEk Ek " (Ak Id)Ek Ek1 & G ! & Ek1 5

vk ( & Ek1 G ! Avkk Avjj

vk = 1, " % % Avk/k

%& "

G " ' %

A % "

G " ' '' %

A % % % 4 (k) % A Vk = Ker(A k Id)

G " (" Vk ! $ " " Vk ( & K + vk Vk vj Vj % k = j

A ;

Avk, vj = kvk, vj = vk, Avj = jvk, vj,

: " vk, vj = 0 " k = j 1 W ( V Vk

W =

{u V, K 1 " u =

Ki=1

uk, uk Vk}

.

$ * ' (' W ' ( (" Vk ( ( ! ? " * W = V " % " (k) " V ! $ ( W

W = {u V " u, v = 0 v W} .

5

W ' A AW W ! % " W ' A Au, v = u,Av = 0 u W v W $ A &W " ;

G W = {0} % % u W " % % A 5 ' - % * " W ;& % A " W W = {0} " W = {0}

W * " W = {0} = V

D( G) %' A % % K %

% % % ' KerA & % !

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COLE P

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COLE P

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- ( $

/ 0 " +

!& $

/ . . ' ,H

F , , ? 0 ( 1

=0 V 8 0 a(, )A :B9%10';5)A .A ,6?6 a(w, v) = a(v, w)A M > 0 > 0

|a(w, v)| MwV vV w, v V

a(v, v) v2V v V.* . A

. A ? . =0 H / 8 ,#8 . {

V H . I V H.

;&)0+'+) CC =]0, L1[]0, L2[ ]0, LN [( 4 (Li > 0)1iN 0 6%

&&

5 &" ,

/ .

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COLE P

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COLE P

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( $

,0,--.'0) CC " $ -" /$ 0 RN $" $"+ ) $ & )" >$" 0+ N " D - )0+'+) CC + $' & 0

>

. '62 6

. 8 A , .

>)0+'+) CC %:%

( $ # "

4%,0?9) CC $&) L0)'*519.*6 $ )$ $""$ " #!)"#+

A#+ F,3,=, $ )) -" " $$&, )+ -)) 1 " ) ,, ) )) )$$" " $$ " ) -" )) )" E" # )" ) )"" $ ,

)9.0;5) CC 5 & B6+ )

LM ;,6?6 , u ? . R $

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COLE P

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$ ( $

)9.0;5) CC ,0 6

: A ,6?6 0

. .{ div(Au) = u u = 0

R A(x) # . ;. 6 " 0 . e(u) =(u+(u)t

)/2 +

: .) > 0 2+N > 0 9 ,0 8 f ; ,. < ;&-< ,.

, 6 & ' ?

(, u) 0 . .{ div (2e(u) + tr(e(u)) Id) = u u = 0 , ;& 0'

$ $$ $" u0 L2()' " $ " f L2(]0, T [;L2()), "$ #

u

tu = f ),), $ ]0, T [

u = 0 ),), ]0, T [u(x, 0) = u0(x) ),), $ .

;- 0,$ $+ $ $$ $" (u0, u1) H10 () L2() " $ " f L2(]0, T [;L2()), "$ $

2u

t2u = f ),), $ ]0, T [

u = 0 ),), ]0, T [u(x, 0) = u0(x) ),), $ u

t(x, 0) = u1(x) ),), $ .

;- 1, 01 =

u0u1 dx, ;-)0+'+) CC ) * RN u1 6%( 1 $ % u1 > 0 0 ) f = 0( u0 L2() u $ * .2

) > 0 " $ K

Ku1(x) u(x, ) Ku1(x) x , ;-< $ C

maxx

|u(x, t)| Ce1t t > . ;-)0+'+) CC ) * RN ) u0 L()( f L(R+)( u C([0, T ];L2())L2(]0, T [;H10 ()) $ .2 "

uL(R+) u0L() + D2

2NfL(R+), ;- 0 x . ;-" 0' u " C $ x " t $ [, T ],

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COLE P

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$ ) #$

&%) &) &%9,*:10.1',*C *X , ; F 6

A . ,9 -"< 8

0 , ;

8 = RN A . ,9 -

) $ $

v2W2m() =|()mv|2dx

&%

' H2m() ( ) #* (wk)

+! C

([, T ],W 2m())

: #

" '%

"

( RN A " ( %

( % ' *

'

' "

'"! 5 " ( ( RN , {

utu = 0 ]0,+[RN

u(x, 0) = u0(x) RN .

)#I!

" % " ' )#I!

% u0 % O =

t

%& "" u0 L2(RN ) 2 ;!

$ ) #$

"

0

|k|2 eikx

% * RN A" " J |k| u(k, t) K k ! & + " (% 0 )#I !

"$ )/ ! * O 0

G(x, t) = 1(2t)N/2

e|x|24t

)/ ! * u(t) = G(t) u0 &

u(x, t) =

RN

G(x y, t)u0(y) dy.

* O " ( RN R+ " " % "{

GtG = 0 ]0,+[RN

G(x, 0) = 0(x) RN .

' : 0 . & 5 % % ' & ( "

'" + ( 89 %

% " ( (

/0 " u0 L2(RN ) t > 0 S(t)u0 ! ! #, -. S(t) ! L2(RN ) L2(RN) /

S(0) = Id L2(RN)) &. (S(t))t0 %" !0 t !%'% &. S(t+ t) = S(t)S(t) t, t 01 f C1(R+;L2(RN )) 2 {

utu = f ]0,+[RN

u(x, 0) = u0(x) RN .

u C(R+;L2(RN)) C1(R+ ;L2(RN))

u(t) = S(t)u0 +

t0

S(t s)f(s) ds,

!%'%

u(x, t) =

RN

u0(y)e |xy|

2

4tdy

(2t)N/2+

t0

RN

f(y, s)e |xy|

2

4(ts) dy ds

(2(t s))N/2 .

* )/ ! % ' )#I!

"%

5 ';

% : u )/ ! ' )#I! % u0 L2(RN )

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COLE P

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COLE P

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)" $ * $"

/0 " (+ +, 2 T > 0

1

2

RN

u(x, T )2dx +

T0

RN

|u(x, t)|2dx dt = 12

RN

u0(x)2dx.

/0 "" ( 0, 2 u0 L(RN) u(t) L(RN )

u(t)L(RN ) u0L(RN ) t > 0.2 u0 0 RN u 0 RN R+

/0 "! (4 +, 2 u C(RN R+ )

/0 "$ ( , 2

lim|x|+

u(x, t) = 0 t > 0, limt+

u(x, t) = 0 x RN .

/0 " (* + , 2 u0 0 u0 0, u(x, t) > 0 RN R+

1 4 % 3+

: "$&

/ .

, A ( ,

8A I? . ' A

0%2)0:'/'-'1% )* 1)93:

0,3,:'1',* CC " $ -" /$ 0 RN ' " $ ") *$ T >0, " (v0, v1) H10 () L2()' " $ " f L2(]0, T [;L2()), "$ $ 0%10,(0.&) )* 1)93: $"0 $ $"$" ") )"

T

2vt2 v = f ),), $ ]0, T [v = 0 ),), ]0, T [v(x, T ) = v0(x) ),), $ vt (x, T ) = v1(x) ),), $

;-" 0 " $)$$" t " $$ !"+ ndl$ ) 0 h' )" )$ $$ $" U0' $/ b' " T ,

)9.0;5) CC

MU U 1) -$ , 8 MU U = |u|2dx . u V0h 8 U 0 V0h ;M 0 ) .

)' &

.

i =1 (1 )ti

1 + ti.

1 0 0 UnM = Un C * A

: 0 |i| 1 iA , , ;-$< 0 < 1/2A I 8 1/2

)9.0;5) CC , ;-$< A : . bnki 9 8A ? 0

* A H . 1/2 < 1 6 0A 0 = 1 ; ,)0+'+) CC % .!/ 1/2 1 UM =MU U 6 $ $* $* .&

Un02M + tn0n=0

KUn Un C(U02M +

T0

f(t)2L2()dt +O(1))

.

- ( .!/ Un = Un+1 + (1 )Un

>)0+'+) CC " % ? .

>)0+'+) CC P1 % $ % .2 N = 1 M * C & $' &

K 2 $ $' " 07 . 2 : t Ch2

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COLE P

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& ) #$

J % % (

" '

)NI ! "

% % % & FC/H

' "! " u +>!! ;,

)( * &"

: =

@ )* ):3.+) :)5-)9)*1 8 . ;-"< ,6

;-

&$ ) #$

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@ &'A%0)*+): @*'): )* 1)93: #6

, ( ;-$$3-'+'1) ; , 8 . M

)( * &&

+ = 1/2 /C% 6 9 6A A 8 0 ; , .)0+'+) CC " % >Q= $

= 1/2( $ 2 = 1/2 = 1/12( $ = 1/2 = 1/12 + $

@ 0 1) -$ * .

A , 0

> / ;. +

&- ) #$

A ;-&< .A A ?

Un+1i 2Uni + Un1i(t)2

+i

(Un+1i + (

12

+ 2)Uni + (12 + )Un1i

)= bni , ;-&+*< . n . . . ? @ tij I . i . j ;.6 ( tji

+ $ -"

(tij) ;. 2

-$ + , #

>)93-) CC $+,*:,99.1',* &): 9%*.():6 @

n # 8 . p Rn+ . ? b > 0A H 8 , u(x) Rn+ R ; .

+ $ -&

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, , 0 0 , , 6

/ .# ' "

. 0 RN f L2() 1, * "&A 0 1 A 8

, J(v) ) v H10 ()

J(v) =12

|v|2dx

fv dx.

3 A .

infvH10 ()

J(v).

@ H 0

% ,9 "A

% . ?

infvH10 ()N " divv=0

{J(v) =

2

|v|2dx

f v dx}

.

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. ? \ .

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A ) A 8 , , 8 f ' ,. . ? 6 A 0 { u = f + v

u = 0 ,

R u . 0 v 8 X ? ' X # F6

. @ ) ,0 X 0

K = {v(x) vmin(x) v(x) vmax(x) v = 0 \ } ,

R vmin vmax 8 @ X u 0 , u0A ,G @ )

J(v) =12

(|u u0|2 + c|v|2) dx,. c > 0 0 X ,

infvK

J(v).

0 G ? 8 v 0 ;. 6 *(

, .0 J

) / 0

; , : , 0 0

+ $ -

0 9)A +0'1?0)A =,*+1',* +,M1A =,*+1',*

,/

+ , #

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(un)n0 " $ K , limn+ u

n = + = limn+J(u

n) = + . ;)0+'+) CC " $ ' /( / /&

>)0+'+) CC ) a b 0 < a < b( n N( Pn $ :9 P * * + n P (0) = 1 - P Pn( P = maxx[a,b] |P (x)|

"

infPPn

P ;)0+'+) CC ) (Li)iI M V " supiI Li V C( J V " J * supLiJ Li 4 Li

M

* 8 . ,# (

0 .

0,3,:'1',* CC J " $ $"$ $-& $ $/ $-& K' "")$" $ J K " $ $ 0/ " $/ )$" $ " $ $/ $-& -$"$" -,

) J " ""$" $-&' &" ) $ )$" $,

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> 0 , w K , w u < = J(w) J(u) . ; " #)

/ . ,

R J 8 . ;6.':1)*+) &!5* 9'*'959H +.: =,01)9)*1 +,*2)>)6 " K

$ $-& $$ - $ @/" V " J $ $"$ $-& $"$ K, ' &" $ $ $ u J K " $

v u2 4[J(v) J(u)] v K . ;$ J(x0)A J (x0) 0 .6 h 8 h . 0 1 H 06 J (x0) 0 x0 ]a, b] x = x0 h + ;, , < x0 ]a, b[ J .0 x0A J (x0) 0 ; (A J(x0) +

h2

2J (x0) + o(h2) J(x0) h F 0 x0 [a, b[, x0 h . h > 0 x0 ]a, b]< J &'0)+1',*: .&9'::'/-): @ ) . ;

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)9.0;5) CC 1) 8 .0 V E ; , ;)0+'+) CC $)::)*1')- P6 ) a : V V ) L V J(u) = 12a(u, u) L(u) " J V J (u), w = a(u,w) L(w) u,w V

>)0+'+) CC ) A : N N b RN - x RN ( J(x) = 12Ax x b x " J J (x) = Ax b x RN

>)0+'+) CC $' ;2 V = L2() RN ( a(u, v) =

uv dx( L(u) =

fu dx f L2() ' V V (

J (u) = u f

>)0+'+) CC $' ;2 V = H10 () RN $

u, v =

(u v + uv) dx.

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COLE P

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COLE P

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- # %

a(u, v) =u v dx( L(u) =

fu dx f L2() "

J (u) = u f V = H1() " ( V V ( J (u) = u0 4 u0 $ H10 () { u0 + u0 = u f

u0 = 0

>)0+'+) CC ) RN 4N = 1 =]0, 1[ ) L = L(p, t, x) RNR( + p t ( Lp

Lt %#

V = H10 () J(v) =

L(v(x), v(x), x)dx " J H10 ()

J (u), w =

(L

p(u(x), u(x), x) w(x) + L

t(u(x), u(x), x)w(x)

)dx .

2 ) N = 1 =]0, 1[( ( u H10 (0, 1) J (u) = 0( u

d

dx

(L

p

(u(x), u(x), x

)) Lt

(u(x), u(x), x

)= 0 , ;)0+'+) CC ) a : V V ) L

V J(u) = 12a(u, u)L(u) " J

V J (u)(v, w) = a(v, w) u, v, w V G ' ;( ;( ;!

J 8 .0 . . J . .A 8 .

>)0+'+) CC " J V - ; ;! +

J (u)(w,w) 0 J (u)(w,w) w2 u,w V . ;")0+'+) CC ' % *# *#

$ f(v) v RN ( *# (

u( T ( *F f(v)

=RN

f(v) dv , u =RN

v f(v) dv ,12u2 +

N

2T =

12

RN

|v|2f(v) dv .;-

F (u) 0 , p 0 , p F (u) = 0 , J (u) + p F (u) = 0 . NC!

5 C % 1 "

' * & " '

! u ' p ' . " (u, p) & " u '

minvV

L(v, p) .

"% (( *

' " '

u 5 C ; (( * % &

J

$# 5 C

' "" RN %

minvRN , F (v)=Bvc0

{J(v) =

1

2Av v b v

}, N#!

: A N N " % b RN B M N c RM

L(v, q) = 12Av v b v + q (Bv c) (v, q) RN (R+)M . N/!

4 % ;& * N! J " ' N#! 0 ' q (R+)M '

minvRN

L(v, q)

" " v L(v, q) * * % 5 %

Lv

(v, q) = Av b + Bq = 0 v = A1(bBq) $ '

G(q) = L(A1(bBq), q

),

'

supq0

(1

2q BA1Bq + (BA1b c) q 1

2A1b b

). NN!

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COLE P

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COLE P

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- - # %

5 * & NN!

(" 6 ' % * ""

5 5 C " $ % "

* " * B M BA1B %! ? % ' NN!% * " (q 0) * ' " ' M % & 1 /C "

% (

' D * " '

( ;

/0 $# 1 ! !"

A =

1 0 4 2 3 53 2 1 2 5 24 2 2 0 1 22 4 1 6 2 21 2 6 3 1 1

.

(

5 ! " i ! j ! ! 6 & ! " "

5 !7! aij ! A 5 !&

aij 2 " ! ' % 3 5 % &! ! ! A 8

6 #

. ""

1 "

&

( ( ((

@

5( / % % " ' %{u = f u = 0 ,

NG!

: % ' RN f L2() "% &

minvH10 ()

{J(v) =

1

2

|v|2dx

fv dx

} N)!

% /G! 4 % % & 0 I) " N)! "

& 0 # " " 0 * %

NG! (" N)! % NG!

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COLE P

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COLE P

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-

* ' " u * f! "

" * * $ "

( & NG! 0&

; " !

(" " N)!

4 & N)! ' "

*

%' e L2()N e = v N)! "%&

minvH1

0() , eL2()N

e=v

{J(v, e) =

1

2

|e|2dx

fv dx

}.

$ '

M(e, v, ) = J(v, e) +

(v e) dx ,

% L2()N $ e ' ((

L(v, ) = mineL2()N

M(e, v, ).

5

e M(e, v, ) * % "

* "

L(v, ) = 12

| |2dx

fv dx +

v dx. NI!

$ % " ' NI! ' N)!(max

L2()NL(v, )

)= J(v) ,

" '

(min

vH10()L(v, )

)= G( ) =

1

2

| |2dx div = f

G !

$

%& $ : (u, ) L(v, ) H10 ()L2()N

L(u, ) = maxL2()N

minvH10()

L(v, ) = minvH10()

maxL2()N

L(v, ).

2! u !!! J(v) H10 () !!! G( ) L2()N

J(u) = minvH10()

J(v) = maxL2()N

G( ) = G(),

= u

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE


- # %

$ ' G ! (" 5

maxG( ) = min(G( )) ! "

12

| |2dx ' (

& % "' * div = f . * v =

"' div = f , " D( # " ; G( ) J(v) " ' ' A" "

J(u) = G() =1

2

fu dx

" " % *

% "

5 C

" "

! ( ! & ' $

' G ! " div = f " x 4 % * %" 0 +

G( ) L(v, ) J(v),

" ' "

u % $

u NG!

"

J(u) =1

2

|u|2dx

fu dx = 12

|u|2dx = 12

fu dx .

1 = u NG! " div = f '

G( ) J(u) = G(),

& "

G (u, ) L(v, )

@ '

$ 0 II '

"" $ +

! y(t) & % RN{dy

dt= Ay + Bv + f 0 t T

y(0) = y0

G!

: y0 RN f(t) RN v(t) RM

" A B % N N N M

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE


-

$ % (

v & ""

J(v) =1

2

T0

Rv(t) v(t)dt+ 12

T0

Q(y z)(t) (y z)(t)dt+ 12D (y(T ) zT ) (y(T ) zT ) ,

: z(t) ; 8'9 zT 8'9 R,Q,D " % R % A" " * y(t) %' v & % G!

% " ( (

( v B' L2(]0, T [;RM ) * ]0, T [ RM ' 8 9 * (

B'! %

% * % K RM " '

' '

infv(t)L2(]0,T [;K)

J(v). G!

5

2 % " G! '

$# f(t) L2(]0, T [;RN) v(t) L2(]0, T [;K) 2 C! y(t) H1(]0, T [;RN) [0, T ]

5 ' f v 6 L2 $ *

y(t) = exp(tA)y0 +

t0

exp((t s)A

)(Bv + f)(s) ds

" % y H1(]0, T [;RN )

#CC " y [0, T ]

$

' $ : u L2(]0, T [;K) !! C# - !1! ! u T

0

Q(yu z) (yv yu)dt + T

0

Ru (v u)dt+D(yu(T ) zT ) (yv(T ) yu(T )) 0 ,

GC!

v L2(]0, T [;K) % yv C !! v

$

" " v y * 0 + G! yv = yv + y : yv {

dyvdt

= Ayv + Bv 0 t Tyv(0) = 0

G#!

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE


- # %

y {dy

dt= Ay + f 0 t T

y(0) = y0

6 " y v v yv L2(]0, T [;K) H1(]0, T [;RN) " v J(v) * "" % v

* "" * ! J % * % R % 5

L2(]0, T [;K)

% * % D( IN & &

u G! . D( J (u), v u 0 ( -

lim0

J(u + w) J(u)

= J (u), w.

5

J(v) "" " yu+w = yu + yw $ ' GC! " " yu yv = yu yv

$ 0 G! *

J (w) w L2]0, T [

u! "

(

" % 1 /! $ ' T

0

J (w)v dt =

T0

Rw v dt + T

0

Q(yw z) yvdt+D(yw(T ) zT ) yv(T ) ,

G/!

: v * "" L2]0, T [

GC! * ' S 0

+ u (" * v yv , *2 % ' '

J (u) & G/! & 6 " * (

-

D ! > ;! C# T0

(Bp + Ru) (v u) dt 0 v L2(]0, T [;K). G)!

$ * GG! w L2]0, T [ J (w) =Bp+Rw " & p GN! y &

w D( #N ; GN! 5 + * % *

G/! " (" * v G! %

v

1 p GN! yv G#! GN! yv G#! p " % + D '

yv(0) = 0 p(T ) = D(y(T ) zT ) T

0

(dp

dt yv + p dyv

dt

)dt = D(y(T ) zT ) yv(T ). GI!

. " ' T0

(dp

dt yv + p dyv

dt

)dt =

T0

Q(y z) yv dt + T

0

Bv p dt. ) !

$ GI! ) ! G/!

% T0

J (u)v dt =

T0

Ru vdt + T

0

Bv pdt,

" GG! G)!

$" 5

' * ' GN! "

; J (v) X 0 * ' G! $ "

G!

%' v y

J(v) " p &

L(v, y, p) = T

0

Rv(t) v(t)dt + T

0

Q(y z)(t) (y z)(t)dt

+D (y(T ) zT ) (y(T ) zT ) + T

0

p (dy

dt+ Ay + Bv + f

)dt

p(0) (y(0) y0) ,

: p G! v y J

G! ' "

& "

Lv

=Ly

=Lp

= 0.

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE


- # %

% GG! "

% ; p " % y 6 *" * " 8'9 "

; D( #N ' "%

y

u % 0 #! (

" G! ( 0 '

& K = RM % 8

9 " ; p

u = R1Bp % G)!!

' $! K = RM f = 0 z = 0 zT = 0 " P (t) [0, T ] ! N {

dP

dt= AP PA + PBR1BP Q 0 t T

P (T ) = D )!

2 P (t) ! t [0, T ] p(t) = P (t)y(t) t [0, T ] " & y0 RN * (y, p, u)(t) : u = R1Bp

! 0 ;% % 0 # ';% RN RN "

y(t) p(t) RN RN P (t) N " p(t) = P (t)y(t) 0 % " G! GN! '

dP

dty = APy PAy + PBR1BPy Qy

y(t) " "" " y0 ! $ " )! $' P (T ) = D p(T ) = Dy(T ) y(T ) "" RN $ )!

/0 $ (

K = RM f = 0 z = 0 zT = 0 2 t [0, T ]

p(t) y(t) = Dy(t) y(t) + Tt

Qy(s) y(s) ds + Tt

R1Bp(s) Bp(s) ds .

/ t0 [0, T ] y(t0) = 0 y(t) = p(t) = 0 t [0, T ]9 !

/0 $ ( & 4,$$ ) 4,$

y

ty = v + f ]0, T [

y = 0 ]0, T [y(0) = y0

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE


- "

: y0 L2() f L2(]0, T [) v L2(]0, T [) !

infvL2(]0,T [)

J(v) =

T0

v2dt dx +

T0

|y z|2dt dx +

|y(T ) zT |2dx,

: z L2(]0, T [) zT L2()

/0 $# ; !! ! '

@ ? &

$ 0 I &

$ - # %

' $ 0 J(v) K

J (v) = p + cv ,

% p v ! E ! v

u = f + v , u H10 (), )G!p = u u0 , p H10 (), ))!

v = 1IP[vmin(x),vmax(x)]

(p

c

), )I!

% 1I 1 1 \ P[vmin(x),vmax(x)] / ! [vmin(x), vmax(x)]

& P[vmin(x),vmax(x)]w = min(vmax(x),max

(vmin(x), w(x)

))

5

## ( -

lim0

J(v + w) J(v)

=

J (v)w dx .

5

J(v) "" '

J (v)wdx =

((u u0)uw + cvw) dx,

: uw {uw = w uw = 0 .

I !

; K

I ! p )N! uw

p uw dx =

(u u0)uw dx

uw p dx =

wp dx

"

J (v)w dx =

(p + cv)w dx,

:

D(

(p + cv) (w v) dx 0 w K . I!

0 w & v * ' * ' % E 89 I! "! x (

p(x) + cv(x)) (

w(x) v(x)) 0 w(x) [vmin(x), vmax(x)] .

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE


-" % &

5 " v(x)

; ( p(x)/c [vmin(x), vmax(x)] % D( ! J ' )I! " " * K &

$ 5

A" #) "

%

* )N! " ; $ '

)/! " " )! "

%' v u! *

L(v, u, p) = 12

(|u u0|2 + c|v|2

)dx +

p(u + f + v) dx,

: p )! v u J ' "

& "

Lv

=Lu

=Lp

= 0.

% )I! "

% ; p " % u

6 #

@ ,

,0I # 6

A ,.330,+4)0 0 ,6

5 (.

0 ,6

' . J ? , 6

u0A (un)nN ,.A #A . u 0 , 3 . +,*2)0()*+) &) +): .-(,0'149): ;,6?6A

. (un) . u u0

- - # %

)9.0;5) CC /

# ; A 6

A 0 wn V A , . wn =J (un)J (un) , . . J(u

n+1) ; ,0

, 8 .

-" %

4%,0?9) CC $ )) J " $-& 5$"/ " J " )#"7$ "" /$ V ' "

M > 0, CM > 0, v+ w M J (v) J (w) CMv w . ;$< 0"# 0$" ) )" $-0 O " u0' " (un)*$ ) ,9 " ,= $-0 - "$ u ,,

%9,*:10.1',*C 8 f() = J(un J (un)) 8 .

.0 R ; J (un) = 0 Y A I? .A un = u V

- # %

4%,0?9) CC $ )) J " $-& 5$"/ " J " )#"7$ V ' " &" $ $"$" C > 0 "

J (v) J (w) Cv w v, w V . ; 0( $*% * + * a = b x0y0 = 0( * * 8

* $*% * + I * ( K

@ . 6 $ 7

@ 0 , .

infvK

J(v) , ;)0+'+) CC ' %:% - ;!;(

J F1, . . . , FM , E I(u)$ u( u 6 ;2 '(

(F i (u)

)iI(u)

( $ **

1, . . . , M J(u) +

Mi=1 iF

i (u) = 0( i = 0 i / I(u) "

( i {1, . . . ,M}

lim0

[2max (Fi(u), 0)

]= i .

)9.0;5) CC / . ? 6

8 L0M

@ 5 %B

@ ) V = RN 9 /C F 8 C2 RN RN u F F ,6?6

F (u) = 0 F (u) .0

8 5# . v

F (u) = F (v) + F (v)(u v) +O (u v2) ,,6?6

u = v (F (v))1 F (v) +O (v u2) . /C ? 8O . 6

* u0 RN A un+1 = un (F (un))1 F (un) n 0. ;" 0 " (x + y) Xad (x y) Xad * " x = (x+ y)/2+ (x y)/2 x x '"

8 . , :

# Xad

0,3,:'1',* CC &" $ "$ )" )0 $ "$

,' &" $ "$ )" /,

%9,*:10.1',*C ? "

x Xad ;

% "&

,0 Xad I,? . 0 ? z 0 .3 A z0 Xad (k1) @ , . x = z0 8 (k 1) (ai) 3 8 8A 0) 8 0 0

)9.0;5) CC 9 * $ c = 0 ; 0

"-

G , 0 x . 0 x @ . 8 c x c x

cB B1b cB B1(bNxN ) + cN xN . ;$ &5 +4.*()9)*1 &) /.:)

, # . G ckN .A 8 8 , * 0A

.

*'1'.-':.1',*

' . 0 0 , K ;+6

,0 xB = B1b 0 , . < P ? 0 * A

0 ;< m .0 ,A Idm 0 L0M , ;

% $

A

Bk+1 = BkEk . Ek =

1 l1

0

1

lj

1

0

ln 1

,

Ek 8 ? .

(Ek)1 =1lj

1 l1

01 lj1

1lj+1 1

0

ln 1

.

@ 8A 8 8A

(Bk)1 = (Ek1)1(Ek2)1 (E0)1(B0)1.

>)0+'+) CC C $*% *

minx10, x20, x30, x40, x50

x1 + 2x2

3x1 + 2x2 + x3 = 2x1 + 2x2 + x4 = 4x1 + x2 + x5 = 5

>)0+'+) CC C $*% *

minx10, x20

2x1 x2

x1 + x2 1 x2 x1 1/2 $ $ $

>)0+'+) CC C $*% *

minx10, x20, x30, x40

3x3 x4

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE


$

{x1 3x3 + 3x4 = 6x2 8x3 + 4x4 = 4

.

1 . B B% 0 -A

. ,A A

, . ,

;A A 0 #

Xad< . ? , Xad I 0 ,? . / , 6

.-(,0'149) &) 10.

% $

; )0+'+) CC C $E( $' /2

5 n *R n ( ( aij % (i, j)( %% Sn(

maxSn

1in

ai(i) . ;&)0+'+) CC - **

+ !2( $ + $' !

>)0+'+) CC @A :*

+ *% G = (N ,A)( $ t : A R+( +%% % [

**( % **( D() + $E '

O N( :*

>)0+'+) CC > &;( 4 Ji Rn R $ + $ $( $ 0% ;(

+ > C &; " C J(x) 2 0 J(x) ' *

min t y Rn, t R, Ji(y) t, i I ,

0 C 0 *( C = J(x)

@ " '; ""

" 0 , ( !

! ' * < !

$ " '

( ' < *

' " 8'9 " 0 #C

" ' & % ( - Y &

( , '

(

( "

1 C ! , ' '

" ' ' 6 4 6%

' 4 % "

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE


' 5 B " &

% ("

( *

' 4 % "" % %E %

B ' % ("

! , ' 4 ' 1 " &

" @ , ' 9'3

" ' 4 " "

% ' '

4 ' 9' ' &

* 4 4 K ' ( - & 53

% 1 5 B ' 4 $

" ' 4 " % 4

% B -

&

%

-

' 4 M '

& 0 #/ ' ( %

0 #N 4 $ " ' 4

%' " ' < K "

& " = 4 ' % ' " -! ? A O . 1 7(

J IGI " -! ! 5 B Z

II/

( & A$

( " ' "

; ' ' "

% " ( % & *

+ K ( (" " %

FCH! ' % F/H , ( %

(" " & %

' 1 " "

' (" " % %

. ( " (( '

" % % % &

' , (" + " %

' !

5 "

& "

' "

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE


#442': : '2: 2:4#2: !2 ;'

)9.0;5) CC . 8 K 6 .6 8 W 1 A ;< xW .

xW W, xW x, z = 0 z W.

9 (A ;< : y = xK z . z W 1 yn & " yn K %

dn = x yn d = infyK

x y " n +.

? " yn 5( 0 %

x 12(yn + yp)2 + 1

2(yn yp)2 = 1

2(d2n + d

2p).

$ % K (yn + yp)/2 K x 12(yn + yp)2 d2 "

yn yp2 2(d2n + d2p) 4d2,

" " yn 5( 5

V B' yn % % xK

K * xK & K " d = x xK 5

% * " xK

minyK x y

1 xK K

y K [0, 1] % KxK + (y xK) & K

x xK2 x (xK + (y xK))2.

0 % %

x xK2 x xK2 + 2y xK2 2x xK , y xK,

" > 0

0 2x xK , y xK+ z2.0 * % ' ! A" xK "% y K

x y2 = x xK2 + xK y2 + 2x xK , xK y x xK2,

" % " xK ' ; ( x K

%@*'1',* CC " V $ ) @/" ) )" , , $ )) / #/"$$ $// V $ $// (en)n1 $" V " "#$ ) )" " " ) -"$0$ ) "" " $ $ V ,

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE


"

0,3,:'1',* CC " V $ ) @/" ) )" , , "(en)n1 $ / #/"$$ V , "" $" x V ' &" $ $" (xn)n1 " )"

pn=1 xnen $-0 - x $

p "$ - $*$' " "" " " *$ ) xn = x, en, 8 )' $

x2 = x, x =n1

|x, en|2. ;)6 " K $ )" $-& $$ - " $ ) @/" V ' " x0 / K, &"

$ #!))$ V ) ""$" x0 " K' " &" $ $ L V " R "

L(x0) < < L(x) x K . ;")0+'+) CC " * Kh( 2 % P1 (

cond2(Kh) 42h2

. ;" 1 ?

6 9 A opt # , M1N A . .

>)0+'+) CC ) A % " ]0, 2[( % *

>)0+'+) CC " ( % ( 5

(M1N) |1 | , = 0,

$ * 0 < < 2

%@*'1',* CC $9%14,&) &5 (0.&')*16 " $ )+" = 0, $)) "# 0$" "# ""- )"$

M =1

Id " N =(

1

IdA)

.

0 . 6

A

8 f(x) = 12Ax x b x 0

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OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE

COLE P

OLYTEC

HNIQUE


*

)99) CC " A $ " 0$/ - )) 1 2 ... n, 1 0 n' "# 0$" $ $-0 ) $- , 0 < 1 ... n' "# 0$" $-0 "$" 0 < < 2/n' " )+" )"' $ (M1N)' "

opt =2

1 + n" min

(M1N) =

n 1n + 1

.

)9.0;5) CC 1 ... n < 0A #

) 8 * optA # , 8 n/1 A IA , cond2(A) A * A A 0 A .

%9,*:10.1',*C 1, $A

. (M1N) < 1 @ M1N = ( Id A)A

(M1N) < 1 |1 i| < 1 1 < 1 i < 1 , i.

' i > 0 1 i n * . A . H H . 1 0 nA A A 0 < 1 ... nA , 8 0 < < 2/n * optA 8 |1 | ], 1/] [1/,+[A

(M1N) = max{|1 1|, |1 n|}.

* A 8 (M1N) ,6 opt = 21+n

5 I

I

.

# # ) . ,

, ; 0

. A . 1)

*

B#. (Kk)k0 8 6 .6 Kk Kk+1 k 0 ' Kk RnA . 8 6 ? , * A .)

, k0A . 0 k0 n 1A {dimKk = k + 1 0 k k0,dimKk = k0 + 1 k0 k.

9 r0 = bAx0A 8 . A A ,xk ? , : [x0 + Kk1] ;) ,0 . x (x x0) Kk1

*

k0 B#.A ,6?6 A k k0A dimKk = k0 + 1 9 A AKk0 Kk0+1 = Kk0 A rk0+1 = b Axk0+1 = r0Ayk0+1 ? Kk0 * Ark0+1 = 0 xk0+1 #

)9.0;5) CC I

, ? , B#. / .

A .A , 0

* A xk [x0 + Kk1] [x0 + Kk1]

f(x) =12Ax x b x,

0 rk = bAxk Kk 8

g(r) =12A1r r

. r = bAx

>)0+'+) CC ) A : ) (xk)0kn

% % * 5*

rk = bAxk dk = xk+1 xk " $ N: Kk * +

Kk = [r0, ..., rk] = [d0, ..., dk],

(rk)0kn1 %*

rk rl = 0 0 l < k n 1,

(dk)0kn1 5* + A

Adk dl = 0 0 l < k n 1.

) , . I

9 (A , , rk ?Kk1A xk . 8 .

0,3,:'1',* CC " A $ " !" *$ )"-' " x0 Rn," (xk, rk, pk) " " *$ ) "$ $

p0 = r0 = bAx0, " ) 0 k

xk+1 = xk + kpkrk+1 = rk kApkpk+1 = rk+1 + kpk

; . ? m " %' i . * (r0, Ar0, . . . , A

m1r0) ' " (P1r0, , Pmr0) 5 " " dimKm1 = m m 1 k0 .: "m = k0 + 1 Em = Kk0

OP & * C)! A(Pir0) = i(Pir0) 5

Pir0 ' % A & % i 5

Em = Kk0 " Vk0+1 * ' Kk0 " % yi Rm " Pir0 = Vk0+1yi $ CN! yi '

Tk0+1yi = Vk0+1AVk0+1yi = V

k0+1APir0 = iV

k0+1Pir0 = iV

k0+1Vk0+1yi = iyi,

yi % Tk0+1 % i 5 "

, 8 6

F I,? , k0 +1A . Tk0+1 ) , . . A ' 6 F 0 ? 4.6= ; k0 , nA ,

" *

:

A(Vky) (Vky) = (ek y)vk+1. CI!$ Vky ' % A

Vky =

mi=1

Pi(Vky).

$ CI! % Vky & C)!

mi=1

(i ) |Pi(Vky)|2 = (ek y) (vk+1 Vky). C !

0 ; ( 5(

1(VE %

min1im

|i | Vky2 y vk+1 Vky.

5

Vk ( Vky = y '

min1im

|i | vk+1.5 " 5(1(VE < ek, y > C ! . %

min1im

|i | vk+1 |ek y|y ,

"

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COLE P

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COLE P

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$%&'

TU 3=N3 +BA 734/3/5 5A @+/ NEA "NJ RNA * =A 6

+.A /C N# ;

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COLE P

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