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Texte intégral

(1)

J

C1([a, b],R) C2([a, b],R)

C1 C2 [a, b] R a

b

H = h2C2([a, b],R), h(a) =h(b) = 0 S= y2C2([a, b],R), y(a) =y0, y(b) =y1 .

S

8u2S,kuk=kuk1+ku0k1

S H

S

L2C0([a, b]R2,R)

J :u2S7 ! Z b

a

L(x, u(x), u0(x)) dx.

@L

@y

@L

@y x,u(x),u0(x)

(2)

f Rn R n k|fk|

f

k|fk|=supn |f(x1, ..., xn)|

kx1kRn...kxnkRn,(x1, ..., xn)2Rn {0}o k.kRn Rn

J

J J

S dJ

J dJ

J x2[a, b] (y, z)2R27 !L(x, y, z) C2 @L

@y

@L

@z [a, b]R2

J u 2 S

h2H

dJ(u)(h) = Z b

a

@L

@yh(x) +@L

@zh0(x) dx u2S h2H

J(u+h) J(u) = Z b

a

L(x, u(x) +h(x), u0(x) +h0(x)) L(x, u(x), u0(x)) dx.

x2[a, b]

K(x) =L(x, u(x) +h(x), u0(x) +h0(x)) L(x, u(x), u0(x)).

x2[a, b] (y, z)2R27 !L(x, y, z) x2[a, b]

K(x) = @L

@yh(x) +@L

@zh0(x) +p

h2(x) +h02(x)"x(p

h2(x) +h02(x))

(3)

"x lim

y!0"x(y) = 0

dJ(u) : H ! R

h 7!

Z b a

@L

@yh(x) +@L

@zh0(x) dx

h2H h h0 @L

@y

@L

@z [a, b]R2 x2[a, b]7 ! @L

@yh(x) +@L

@zh0(x) [a, b]

dJ(u) H

J k·k

Z b a

ph2(x) +h02(x)"(p

h2(x) +h02(x)) dx=khk (khk)

kulimk!0 (kuk) = 0

L C2

x2[a, b]

H(x) =

h(x) h0(x)

U(x) =

u(x) u0(x)

x2[a, b]

|p

h2(x) +h02(x)"x(p

h2(x) +h02(x))|h2(x) +h02(x) 2

supn

k|d2L(U(x)+tH(x))k|, t2[0,1]o x2[a, b]

h2(x) +h02(x)2khk2.

d2L(U+.H) : (t, x)2[0,1][a, b]7 !d2L(U(x) +tH(x))

[0,1]

[a, b] M > 0

(t, x)2[0,1][a, b]

k|d2L(U(x) +tH(x))k|M

(4)

Z b a

ph2(x) +h02(x)"x(p

h2(x) +h02(x)) dx (b a)khk2M

J

L u2 S J

Z b a

@L

@yh(x) +@L

@zh0(x) dx= 0 h2H

dJ

u v u < v ]u, v[ a b

x2]u, v[ (x)>0 x2R ]u, v[ (x) = 0

h2 H g [a, b]

R Z b a

g(x)h(x) dx= 0 g [a, b]

u v

u < v (x) =

((x u)2(v x)2 x2]u, v[

0 x2R ]u, v[

H

u v R (a) = (b) = 0

x2[a, b] (x)>0

x2]u, v[

(x) (u)

x u = (x u)(v x)2

0(u) = 0 0(v) = 0

0(x) =

(2(x u)(v x)2+ 2(x u)2(v x) x2]u, v[

0 x2R ]u, v[

(5)

g

[a, b] R h2H

Z b a

g(x)h(x) dx= 0 g

c2[a, b] g(c) g(c)>0 g

]u, v[ c g(x)>0 x2]u, v[

(x) > 0 x2]u, v[ (x) = 0 x2R ]u, v[

Z b a

(x)g(x) dx= Z v

u

(x)g(x) dx >0 x2[a, b] g(x) = 0.

J

x2[a, b] (y, z)2R2 7 !L(x, y, z) C2 @L

@y [a, b]R2

@L

@z 2C1([a, b],R)

h2H u2S

Z b a

@L

@yh(x) +@L

@zh0(x) dx= 0

u2S x2[a, b]

@L

@y d dx

@L

@z = 0

dJ

h2H u2S

@L

@z 2C1([a, b],R) Z b

a

@L

@zh0(x) dx=h@L

@zh(x)ib a

Z b a

h(x) d dx

@L

@z dx

(6)

h(a) =h(b) = 0 dJ

h2H u2S

dJ(u)(h) = Z b

a

(@L

@y d dx

@L

@z)h(x) dx

h2H u2 S dJ(u)(h) = 0 x2[a, b] @L

@y d dx

@L

@z = 0.

x2[a, b] g(x) =@L

@y d dx

@L

@z.

J J

E f

C1 E R f x2E

x f df(x) = 0.

(E,k.kE) f

C1 E R x2E f f

C1 f h2E

limt!0

f(x+th) f(x)

t =df(x)(h) h2E

f(x+th) f(x) =tdf(x)(h) +t"(t)

tlim!0"(t) = 0

x f

>0 y2E f(x) f(y)<0 kx ykE<

h 2 E |t|khk < f(a+th) f(a) < 0 tdf(x)(h) +t"(t)<0

t 0 t >0 df(x)(h)0

df(x)(h) 0 t 0 t <0

h2E df(x)(h) = 0 J

J

(7)

x2[a, b] (y, z)2R2 7 !L(x, y, z) C1 @L

@y [a, b]R2

@L

@z 2C1([a, b],R)

u2S J

u

@L

@y d dx

@L

@z = 0.

u2 S J u

J dJ(u) = 0

J

J

(a, y0) (b, y1)

y=u(x) x

[a, b] u S

u : [a, b] ! R2 x 7!

u(x) x

u2S

u2S L( u) =

Z b

a k u0(x)kR2dx

k.kR2 R2

(8)

J:u2S 7 ! Z b

a

p1 +u02(x) dx.

(y, z) 2 R2 L(y, z) =p

1 +z2 L

d dx

u0 p1 +u02

= 0

u00= 0.

S x 2 [a, b] 7 ! u(x) = y0 y1

a b x+ay1 by0

a b u J

J

J

J

J

J

x 2 [a, b] (y, z) 2 R2 7 !

L(x, y, z) C3 @2L

@y2,@2L

@z2

@2L

@y@z [a, b]R2

u2S (h, k)2H2,

d2J(u)(h, k) = Z b

a

@2L

@y2h(x)k(x)+ @2L

@y@z(h0(x)k(x)+k0(x)h(x))+@2L

@z2h0(x)k0(x) dx

J(u+h) J(u) = Z b

a

L(x, u(x) +h(x), u0(x) +h0(x)) L(x, u(x), u0(x)) dx.

(9)

x2[a, b]

K(x) =L(x, u(x) +h(x), u0(x) +h0(x)) L(x, u(x), u0(x)).

x2[a, b] (y, z)2R2 7 !L(x, y, z)

C2 K x2[a, b]

K(x) =@L

@yh(x) +@L

@zh0(x) +@2L

@y2h2(x) + 2@2L

@y@zh0(x)h(x) +@L

@zh02(x) + (h2(x) +h02(x))"(h2(x) +h02(x))

" lim

y!0"(y) = 0

d2J(u) : HH ! R

(h, k) 7!

Z b a

@2L

@y2h(x)k(x) + @2L

@y@z(h0(x)k(x) +k0(x)h(x)) +@L

@zh0(x)k0(x) dx H

J k·k

J C2 u J

d2J(u)

u2S J

h2 H dJ(u)(h) = 0 u J

J

t2R h2H

J(u+th) J(u) =d2J(u)(th, th) +kthk2"(kthk)

zlim!0"(z) = 0

u J

u > 0 k.k B(u,⌘) =

{y2S,ku yk<} y2B(u,⌘) J(u) J(y)0

t |t|khk<

J(u+th) J(u) =t2d2J(u)(h, h) +t2khk2"(|t|khk)>0

t >0 |t|khk< d2J(u)(h, h) +khk2"(|t|khk)>0

t 0 d2J(u)(h, h) 0 d2J

(10)

J

J J u 2 S

J C2 k >0

h2H d2(u)(h, h)> kkhk2

u J

J t2R h2H

J(u+th) J(u) =t2d2J(u)(h, h) +kthk2"(kthk)

zlim!0"(z) = 0

J

t2R h2H

J(u+th) J(u)> t2khk2

k+"(kthk)

zlim!0"(z) = 0 > 0 kthk< "(kthk) k2

kthk<

J(u+th) J(u) k

2t2khk2 0

x 2[a, b] 7 ! u(x) = y0 y1

a b x+ ay1 by0

a b u

h2H d2J(u)(h, h) =

Z b a

h02(x) (1 +u(x)02)(p

1 +u(x)02)dx

(11)

J

J

u2S h2H

d2J(u)(h, h) = Z b

a

@2L

@z2

2(x) dx

C1 [a, b] R u h

@2L

@z2 [a, b]

x

L C2

L x2[a, b]

@2L

@z2 >0

@2L

@y@z C1 [a, b]

d2J u2S h2H

d2J(u)(h, h) = Z b

a

@2L

@z2h02(x) +B(x)h2(x) dx

B =@2L

@y2 d dx

@2L

@y@z

(12)

u2S h2H d2J(u)(h, k) =

Z b a

@2L

@y2h2(x) + 2@2L

@y@z(h0(x)h(x)) +@2L

@z2h02(x) dx

@2L

@y@z C1 [a, b]

x2 [a, b] 2 @2L

@y@z(h0(x)h(x)) = @2L

@y@z d dxh2(x)

Z b a

2h(x)h0(x)@2L

@y@zdx=h

h2(x)@2L

@y@z ib

a

Z b a

h2(x) d dx

@2L

@y@zdx h(a) =h(b) = 0

Z b a

2h(x)h0(x)@2L

@y@zdx= Z b

a

h2(x) d dx

@2L

@y@zdx

L

y2= @2L

@z2(B+y0) B= @2L

@y2 d dx

@2L

@y@z

w

u2S h2H

d2J(u)(h, h) = Z b

a

@2L

@z2

h0(x) +w(x)

@2L

@z2

h(x)2

dx u 2 S h 2 H

Z b J

a

d dx

w(x)h2(x)

dx w C1

[a, b]

Z b a

d dx

w(x)h2(x)

dx=w(a)h2(a) w(b)h2(b) = 0

(13)

h(a) =h(b) = 0

d2J(u)(h, h) =d2J(u)(h, h) + Z b

a

d dx

w(x)h2(x) dx

d2J(u)(h, h) = Z b

a

@2L

@z2h02(x) +B(x)h2(x) + d dx

w(x)h2(x)⌘⌘

dx

B =@2L

@y2 d dx

@2L

@y@z

w

y2= @2L

@z2(B+y0)

f : R[a, b] ! R

(t, x) 7! t2

@2L

@z2

B(x)

x C1 t

w x2[a, b]

@2L

@z2h02(x) +B(x)h2(x) + d dx

w(x)h2(x)

=@2L

@z2

h0(x) +w(x)

@2L

@z2

h(x)2

w

@2L

@z2

(14)

n2N

U Rn I R f :IU !Rn

(t, y)2IU f(t, y) =A(t)y A2C0(I, Mn(R))

y0=f(t, y)

L C2 @2L

@z2

@2L

@y@z C1

d dx

@2L

@z2v0 @2L

@y2 d dx

@2L

@y@z

v= 0

L C2 @2L

@z2

@2L

@y@z C1

v [a, b]

[a, b]

w= v0 v

@2L

@z2 C1

(15)

v

c2]a, b] a v(a) =v(c) = 0

[a, b]

U Rn f : [a, b]U ! Rn

=U [a, b]

y0=f(t, y)

: ! U

(y0, t) 7! y(t) (y0, t)2 (y0, t) =y(t)

(y0=f(t, y) y(a) =y0

f

L C2

@2L

@z2

@2L

@y@z C1

a ]a, b]

u [a, b]

a [a, b]

x=u y=u0 8>

>>

><

>>

>>

: x0=y y0 =

@2L

@y2 d dx

@2L

@y@z

x y d dx

@2L

@z2

@2L

@z2

(16)

=R2[a, b]

R2 (v(a), v0(a))

t2[a, b] ((1,0), t)

p : R2 ! R

(x, y) 7! x t2[a, b] v(t) =p( ((0,1), t)) v

a ]a, b] t 2]a, b] v(t) 6= 0

p p

> 0 (x0, y0) 2 R2 k(x0, y0) (0,1)k1 < t 2 [a, b] p( ((x0, y0), t)) 6= 0

R2 (x, y)2R2 k(x, y)k1=max{|x|,|y|}

(x0, y0)2RR

|x0|< |y0|< ˜v=p( ((x0, y0), t)) [a, b]

u2S d2J(u)

u2S L C2

@2L

@z2

@2L

@y@z C1

a u2S d2J(u)

a

[a, b]

J

u2S J h2H

J(u+h) J(u) = 1

2d2J(u)(h, h) + Z b

a

⌘(h)h02(x) + (h)h2(x) dx

0 khk 0

u2S J

h2H 2]0,1[

(17)

J(u+h) J(u) =dJ(u)(h) +1

2d2J(u+✓h)(h, h) = 1

2d2J(u+✓h)(h, h) u

1

2d2J(u+✓h)(h, h)

J

L C2

@2L

@z2

@2L

@y@z C1 u2S

a ]a, b] u

J

>0 u2S

h2H

K( , h) =d2J(u)(h, h) Z b

a

h02(x) dx= Z b

a

(@2L

@z2 )h02(x) +B(x)h2(x) dx

K K( , h) =

Z b a

l(x, h(x), h0(x)) dx

l : [a, b]R2 ! R

(x, t, w) 7! (@@z2L2 )w2+B(x)t2 l

x 2 [a, b] l

@2l

@w2 = 2@2L

@z2

@2l

@t@w = 0 C1 [a, b]

K( )

d dx

@2l

@w2v0 @2l

@t2 d dx

@2l

@t@w

v= 0

(18)

d dx

@2L

@z2

u0 Bu= 0

x2[a, b]

@2L

@z2 >0

>0 x2[a, b] @2L

@z2 > x2[a, b] @2L

@z2 >0 <

a ]a, b]

= 0

K v : (x, ) 2 [a, b]]0;⌘[7 ! v(x, )

x2]a, b] x

a ]a, b]

(x,0) v ">0 >0

2]0,⌘[ < |v(x, ) v(x,0)|<" v(x,0)6= 0 v(x,0)>0 "= v(x,0)2

<

v(x, )>v(x,0) 2 >0 a ]a, b]

K a ]a, b]

h2H d2K(h)

>0 h2H

1

2d2J(u)(h, h)>

2 Z b

a

h02(x) dx

R= Z b

a

⌘h02(x) + h2(x) dx

0 khk 0

>0 >0 & >0

(19)

(khk< ||<

khk<& ||<

max=max( ,&) khk< max

khk< max

|R|<

Z b a

h02(x) +h2(x) dx x 2 [a, b]

h2(x) =⇣ Z x

a

h0(t) dt2

(x a) Z x

a

h0(t)2dt(x a) Z b

a

h02(x) dx.

Z b a

h2(x) dx (b a)2 2

Z b a

h02(x) dx

khk< max

|R|

1 + (b a)2 2

⌘ Z b

a

h02(x) dx 2 Z b

a

h02(x) dx

= 2(1 +(b a)2 2)

khk< max

J(u+h) J(u) =1

2d2J(u)(h, h) + Z b

a

⌘h02(x) + h2(x) dx

>2 Z b

a

h02(x) dx 2

Z b a

h02(x) dx= 0

u J

a

(20)

J a [a, b]

[a, b]

u (c1, c2)2R2

u1 = @u

@c1

u2 = @u

@c2

C2

[a, b] L C2

(u1, u2)

u1 u2

u

d dx

@L

@z

@L

@y = 0

@

@c1

d dx

@L

@z

@L

@y

= 0

L C2

@

@y

@L

@y

@u

@c1

+@L

@z

@u0

@c1

= @

@y

@L

@c1

= @2L

@c1@y d

dx

@

@z

@L

@y

@u

@c1 +@L

@z

@u0

@c1

⌘⌘= d dx

@

@z

@L

@c1

= @

@c1

d dx

@L

@z

@

@y

@L

@y

@u

@c1

+@L

@z

@u0

@c1

d dx

@

@z

@L

@y

@u

@c1

+@L

@z

@u0

@c1

⌘⌘= @

@c1

d dx

@L

@z

@L

@y

= 0

(21)

u01= d dx

@u

@c1

= @u0

@c1

@2L

@y2u1+ @2L

@y@zu01 d dx

@2L

@z@y +@2L

@z2u01

= 0

d dx

@2L

@z2u01 @2L

@y2 d dx

@2L

@y@z

u1= 0

u2

u1 u2

(u1, u2)

↵u1+ u2 (↵, )2R2 c

a ↵u1(a) + u2(a) =↵u1(c) + u2(c) = 0 u1(a)u2(c) =u2(a)u1(c)

L C2 @2L

@z2

@2L

@y@z C1

u1= @u

@c1

u2= @u

@c2 C2 [a, b]

c a [a, b]

u1(a)u2(c) =u2(a)u1(c)

x2[a, b]7 !u(x) =y0 y1

a b x+ay1 by0

a b

c a [a, b] c

u1(a)u2(c) =u2(a)u1(c) A= y0a by1 B =aya b1 by0

(22)

x2[a, b]

u1(x) = @u

@A(x) =x u2(x) = @u

@B(x) = 1

c=a a

]a, b] u

z2R7 !L(z) =p 1 +z2

m

~ m

F =Fx~x+Fy~y+Fz~z

F~ w=Fxdx+Fydy+Fzdz

m F~

F~

w

Ep C1 R3 R

w= dEp

Ep

R3 5.2

F~ = gradE~ p

(23)

~ x

w=Fxdx Fx= d

dx(Ep)x

Ep x

M m

t1 t2

x:t 2[t1, t2]7 !x(t) S m

A= Z t2

t1

1

2m(x0(t))2 Ep(t, x(t), x0(t)) dt A

1 2mx02 M

Ep

1

2m(x0(t))2 Ep(t, x(t), x0(t)) dt

t1 t2

M

~x M

g

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