· = ·, λP getdoceca0. 357KB Jun 04 2011 12:05:09 AM

prove the second statement, choose ϕ = w λ in 19 and plug the relation f , w λ 2 ≤ | f | 2 |w λ | 2 ≤ 12λ| f | 2 2 + λ2|w λ | 2 2 into the right-hand side to obtain λ|w λ | 2 2 + a i j D i w λ , D j w λ 2 ≤ | f | 2 2 λ. From 5, we deduce Λ|Dw λ | 2 2 ≤ | f | 2 2 λ and the result follows. Proof of Lemma 2.2. The proof is quite similar to that of Proposition 2.6 below. So we let the reader check the details. Generator on the random medium associated to the reflection term Proof of Proposition 2.4. The resolvent properties of the family R λ λ are readily derived from those of the family G λ λ . So we first prove 1. Consider ϕ, ψ ∈ L 2 Ω. Then, by using 26 and 27, we obtain R λ ϕ, ψ 2 =P G λ P ∗ ϕ, ψ 2 = G λ P ∗ ϕ, P ∗ ψ = B λ G λ P ∗ ψ, G λ P ∗ ϕ =B λ G λ P ∗ ϕ, G λ P ∗ ψ = G λ P ∗ ψ, P ∗ ϕ = ϕ, R λ ψ 2 so that R λ is self-adjoint in L 2 Ω. We now prove 2. Consider ϕ ∈ L 2 Ω satisfying λR λ ϕ = ϕ for some λ 0. We plug g = G λ P ∗ ϕ ∈ W 1 into 28: λ|R λ ϕ | 2 2 + 1 2 Z Ω + a + i j ∂ i G λ P ∗ ϕ ∂ j G λ P ∗ ϕ dµ + = P G λ P ∗ ϕ, ϕ = R λ ϕ, ϕ 2 . 49 Since λR λ ϕ = ϕ, the right-hand side matches R λ ϕ, ϕ 2 = λ|R λ ϕ | 2 2 so that the integral term in 49 must vanish, that is R Ω + a + i j ∂ i G λ P ∗ ϕ ∂ j G λ P ∗ ϕ dµ + = 0. From 5, we deduce ∂ G λ P ∗ ϕ = 0. Thus, G λ P ∗ ϕ0, · is G ∗ -measurable. Moreover, we have λG λ P ∗ ϕ0, · = λPG λ P ∗ ϕ = λR λ ϕ = ϕ so that ϕ is G ∗ -measurable. Hence ϕ = M 1 [ϕ]. Conversely, we assume ϕ = M 1 [ϕ], which equivalently means that ϕ is G ∗ measurable. We define the function u : Ω + → R by ux 1 , ω = ϕω. It is obvious to check that u belongs to W 1 and satisfies ∂ u = 0. So B λ

u, · = ·, λP

∗ ϕ for any λ 0. This means u = λG λ P ∗ ϕ in such a way that λR λ ϕ = λPG λ P ∗ ϕ = PλG λ P ∗ ϕ = Pu = ϕ. We prove 3. Consider ϕ ∈ L 2 Ω. Since the relation 49 is valid in great generality, 49 remains valid for such a function ϕ. Since the integral term in 49 is nonnegative, we deduce λ |R λ ϕ | 2 2 ≤ R λ ϕ, ϕ 2 ≤ |R λ ϕ | 2 |ϕ| 2 . Hence |λR λ ϕ | 2 ≤ |ϕ| 2 for any λ 0. So the family λR λ ϕ λ is bounded in L 2 Ω and we can extract a subsequence, still indexed by λ 0, such that λR λ ϕ λ weakly converges in L 2 Ω towards a function ˆ ϕ. Our purpose is now to establish that there is a unique possible weak limit ˆ ϕ = M 1 [ϕ] for the family λR λ ϕ λ . By multiplying the resolvent relation λ − µR λ R µ ϕ = R µ ϕ − R λ ϕ by µ and passing to the limit as µ → 0, we get λR λ ˆ ϕ = ˆ ϕ. This latter relation implies see above that ˆ ϕ is G ∗ -measurable. To prove ˆ ϕ = M 1 [ϕ], it just remains to establish the relation ϕ, ψ 2 = ˆ ϕ, ψ 2 for every G ∗ - measurable function ψ ∈ L 2 Ω. So we consider such a function ψ. Obviously, it satisfies the relations M 1 [ψ] = ψ and λR λ ψ = ψ see the above item 2. We deduce ϕ, ψ 2 = ϕ, λR λ ψ 2 = lim λ→0 λR λ ϕ, ψ 2 = ˆ ϕ, ψ 2 . 1009 As a consequence, we have ˆ ϕ = M 1 [ϕ] and there is a unique possible limit for each weakly con- verging subsequence of the family λR λ ϕ λ . The whole family is therefore weakly converging in L 2 Ω. To establish the strong convergence, it suffices to prove the convergence of the norms. As a weak limit, ˆ ϕ satisfies the property | ˆ ϕ | 2 ≤ lim inf λ→0 |λR λ ϕ | 2 . Conversely, 49 yields lim sup λ→0 |λR λ ϕ | 2 2 ≤ lim sup λ→0 λR λ ϕ, ϕ 2 = ˆ ϕ, ϕ 2 = | ˆ ϕ | 2 2 and the strong convergence follows. The remaining part of this section is concerned with the regularity properties of the operator G λ P ∗ Propositions 2.5 and 2.6 and may be omitted upon the first reading. Indeed, though they may appear a bit tedious, they are a direct adaptation of existing results for the corresponding operators defined on ¯ D not on Ω + . However, since we cannot quote proper references, we give the details. Given u ∈ L 2 Ω + , we shall say that u is a weakly differentiable if, for i = 1, . . . , d, we can find some function ∂ i u ∈ L 2 Ω + such that, for any g ∈ C c Ω + : Z Ω + u ∂ i g d µ + = − Z Ω + ∂ i u g d µ + . It is straightforward to check that a function u ∈ W 1 is weakly differentiable. For k ≥ 2, the space W k is recursively defined as the set of functions u ∈ W 1 such that ∂ i u is k − 1 times weakly differentiable for i = 1, . . . , d. Proposition C.1. If ϕ belongs to C , then G λ P ∗ ϕ ∈ T ∞ k=1 W k . Proof of Proposition C.1. The strategy is based on the well-known method of difference quotients. Our proof, adapted to the context of random media, is based on [7, Sect. 7.11 Th. 8.8]. The properties of difference quotients in random media are summarized below see e.g. [19, Sect. 5]: i for j = 2, . . . , d, r ∈ R \ {0} and g ∈ C c Ω + , we define ∆ j r g x 1 , ω = 1 r g x 1 , τ r e j ω − g x 1 , ω. ii for each r ∈ R \ {0} and g ∈ C c Ω + , we define ∆ 1 r g = 1 r g x 1 + r, ω − g x 1 , ω. iii for any j = 1, . . . , d, r ∈ R \ {0} and g , h ∈ C c Ω + , the discrete integration by parts holds Z Ω + ∆ j r g h d µ + = − Z Ω + g ∆ j −r h d µ + provided that r is small enough to ensure that ∆ j r g and ∆ j r h belong to C c Ω + . iv for any j = 1, . . . , d, r ∈ R \ {0} and g ∈ C c Ω + such that ∆ j r g ∈ C c Ω + , we have Z Ω + |∆ j r g | 2 d µ + ≤ Z Ω + |∂ j g | 2 d µ + . 1010 Up to the end of the proof, the function G λ P ∗ ϕ is denoted by u. The strategy consists in differen- tiating the resolvent equation B λ

u, · = ·, P