1. Montrer que t → |t| mais pas 2. Soit le problème aux limites

t→ |t|H1(] −1,1[)
t→1x < 0,
−1x > 0.
−(a(x)u0(x))0=f(x),
u(0) = u(1) = 0,
a(x) = α1>0x < 1/2α2>0
H1
0(]0,1[). u0(1/2−)u0(1/2+)
h > 0f∈L2(]0,1[)
J(u) = inf
u∈H1
0(]0,1[),u≤hˆ1
0
u02
2−fu.
∃λ∈L2(]0,1[) λ≤0u
−u00 =f+λ,
(u−h)λ= 0,
u≤h.
Πu u P2
h u ∈H3(]0,1[),
|u−Πu| ≤ Ch2|u|H3.
C > 0H1(I)
|u(x)−u(y)| ≤ C(I)|x−y|1/2.
I H
C(I). H
∀ε > 0,∃δ > 0,∀x, y ∈I, ∀f∈H, |x−y|< δ ⇒ |f(x)−f(y)|< ε.
H C(I).
I H1C(I)
I
I u H1(I)
u0L2(I)u
u u

∃α > 0,∀x, β(x)≥α
−u00(x) + β(x)u(x) = f(x)
u(0) = u(1)