Espaces de Sobolev

H1(Ω)
ΩRd
H1(Ω) = u∈L2(Ω) ∀i∈J1, dKdu
dxi
∈L2(Ω)
H(x)H1(] −1,1[)
f(x) = |x|
H1(Ω)
(f, g)H1=ZΩ f(x)g(x)dx +
d
X
i=1 ZΩ
∂f(x)
∂xi
∂g(x)
∂xi
dx
k·kH1
kfkL2≤ kfkH1k∂ifkL2≤ kfkH1
ΩRd
D(Ω) ⊂H1(Ω) ⊂L2(Ω)

Hk(Ω)
Hk(Ω) = u∈L2(Ω) ∀α∈Nd,|α| ≤ k, ∂αu∈L2(Ω)
Ω =] −1,1[ x7→ sin x H1(Ω)
H2(Ω) x7→ |x|H1(Ω) H2(Ω)
Hk(Ω)
(f, g)Hk=ZΩ f(x)g(x)dx +X
1≤|α|≤kZΩ
∂αf(x)∂αg(x)dx
k·kHk
H2(Ω)
(f, g)H2=ZΩ f(x)g(x)dx+
d
X
i=1 ZΩ
∂f(x)
∂xi
∂g(x)
∂xi
dx+X
i,j ZΩ
∂2f(x)
∂xi∂xj
∂2g(x)
∂xi∂xj
dx
H1
0(Ω)
D(Rd)H1(Rd)H1f∈
H1(Rd)φn∈ D(Rd) limnkf−φnkH1(Rd)= 0
Ω⊂RdΩ6=RdD(Ω) H1(Ω)
H1
0(Ω) D(Ω) H1(Ω)
H1(Ω)
H1
0(Ω) = f∈H1(Ω) ∃φn∈ D(Ω) tq φn→f H1(Ω)
H1
0(Rd) = H1(Rd)
Ω6=RdH1
0(Ω) ⊂H1(Ω)
D(Ω) H1
0(Ω) H1(Ω)
H1
0(Ω) H1(Ω)

H1
0(Ω)
H1
ΩRd
CΩ
∀u∈H1
0(Ω),kukL2(Ω) ≤CΩk∇ukL2(Ω) .
u∈C0(¯ Ω) u ∂Ω
γ(u) : ∂Ω−→ R
x7→ u(x).
γ(u) = u|∂Ω
γ:C0(¯ Ω) −→ C0(∂Ω)
u7→ γ(u)
Ω
L2(Ω)
u:x7→ sin(1/x) Ω =]0,1[ ∂Ω
0 1
u u 0
H1(Ω)
∂ΩH1(Ω)
γ:H1(Ω) −→ L2(∂Ω)
u7→ γ(u)
∀u∈H1(Ω) ∩C0(¯ Ω) γ(u) = u|∂Ω
u∈H1(Ω) γ(u)∂Ω
L2(∂Ω)

H1
0(Ω) = u∈H1(Ω), γ(u) = 0.
H1
0(Ω) H1(Ω)
Ω