1. Introduction 2. Espaces vectoriels normés

Ω⊂R3
f: Ω →Ru(x, y, z) (x, y, z) Ω
−δ2
δ2xu(x, y, z)−δ2 δ2yu(x, y, z)−δ2 δ2zu(x, y, z) = f.
∆u=δ2
δ2xu+δ2 δ2yu+δ2 δ2zu,
u u 7→ ∆u
f'0
u'0
EF=R C
E
k·k:E→[0,∞[, x 7→ kxk,
x, y ∈E λ ∈F
x= 0 kxk= 0
kλxk=|λ| kxk
kx+yk ≤ kxk+kyk
E=R3k(x1, x2, x3)k=k(x1, x2, x3)k2=px2
1+x2 2+x2 3
E=Fnx= (x1, ..., xn),kxk=kxk∞= max1≤i≤n|xi|
E[0,1]
E→[0,∞[, f 7→ kfk=kfk∞,[0,1] = sup
x∈[0,1]
|f(x)|,
E
Ek · k k · k∗
c, C > 0x∈E
ckxk ≤ kxk∗≤Ckxk.
k·k E(E, k·k)

(E, k · k)E
(E, k·k)
E
d:E×E→R
(x, y)7→ kx−yk
(E, d)
(E, k·kE) (F, k·kF)
f:E→F a ∈E f
a∀ε > 0,∃δ(ε, a)>0∀x∈E
kx−akE≤δ(ε, a) =⇒ kf(x)−f(a)kF≤ε.
f A E f
A
f:E→F f
E M > 0x∈E
kf(x)kF≤MkxkE.
f:E→F
kfk= sup
x∈E
kxkE=1
kf(x)kF= sup
x∈E\{0}
kf(x)kF
kxkE
.
kfkf
E F
k·k
E=F=R3k(x1, x2, x3)k=k(x1, x2, x3)k2=px2
1+x2 2+x2 3
f((x1, x2, x3)) = (a1x1, a2x2, a3x3),
a1, a2, a3∈R
E N F=R C k · k
E
E
E E
(E, k·k)
F
Ff:E→F f E