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

714 D. Bucur / Journal of Functional Analysis 236 (2006) 712–725
lim inf
n→∞
Ω
a(x,∇un)∇vn dx
Ω
a(x,∇u)∇vdx. (2)
For the simplicity of the exposition, our results are presented for the p-Laplace operator. We
point out the fact that the convergence of (vn) into the sense of obstacles is independent on the
choice of the operator −div(a(x, ·)).
Section 2 contains a review of the main tools used in the paper, Section 3 contains the proof of
the characterization result and the last section is devoted to some examples. A particular attention
is given to uniformly oscillating obstacles.
2. Obstacles, capacity and γ -convergence
2.1. Capacity and relaxation measures
is
Let Ω ⊆ R N be a bounded open set and let 1 0 there exists a continuous
(respectively lower semi-continuous) function fɛ : Ω → R such that capp ({f = fɛ},Ω)