Estimation optimale du gradient du semi-groupe de la chaleur sur le ...

lim
n→∞
Ω
lim sup
n→∞
+
D. Bucur / Journal of Functional Analysis 236 (2006) 712–725 721
|∇un| p−2 ∇un −|∇θn| p−2
∇θn ∇φn dx −
E
|∇un| p−2 ∇un −|∇θn| p−2
∇θn ∇φn
dx
E
|∇u| p−2 ∇u −|∇θ| p−2 ∇θ ∇φ dx.
For every ε>0, there exists M such that
Ω
p−2 p−2
|∇u| ∇u −|∇θ| ∇θ ∇φdx
|∇ρ| M ⇒ ∇(ρ + wΩ) p−2 ∇(ρ + wΩ) −|∇ρ| p−2 ∇ρ ε|∇ρ| p−1 .
Thus, for a given ε>0wehave
|∇un| p−2 ∇un −|∇θn| p−2
∇θn ∇φn
dx
E
ε
E∩{|∇θn|M}
|∇θn| p−1 |∇φn| dx + 3M p−1
C ε + 3M p−1 E ∩ |∇θn| t} ,
H
that {vn >t} γ -converges to μ and gn −→ g.
Applying inequality (12) to the sequence (un) constructed above, we get
lim inf
n→∞
|∇un|
Ω
p−2
∇un∇vn dx |∇u|
Ω
p−2 ∇u∇vdx,
or, decomposing the integrals by using vn − t = (vn − t) + − (vn − t) − ,
lim inf
n→∞
Ω
Ω
|∇un| p−2 ∇un∇(vn − t) +
dx −
Ω
|∇un| p−2 ∇un∇(vn − t) − dx
|∇u| p−2 ∇u∇(v − t) +
dx − |∇u| p−2 ∇u∇(v − t) − dx.
Ω