Bath Institute For Complex Systems - ENS de Cachan - Antenne de ...

Proof. Similarly to the proof of the previous Lemma, u i ∈ H 1 0 (D ∩ D i) is the weak solution of anew problem −div(a∇u i ) = g i in D ′ (D ∩ D i ) withg i := fχ i + a∇u · ∇χ i + div(au∇χ i ).As in Lemma A.4, since again div is linear and continuous from H s to H s−1 , we can establish thatg i ∈ H s−1 (D ∩ D i ) and‖g i ‖ H s−1 (D∩D i ) ‖a‖ C t (D)a min‖f‖ H s−1 (D).Now let Q = {(y ′ , y d ) ∈ R d−1 ×R : |y ′ | < 1 and |y d | < 1}, Q 0 = {(y ′ , y d ) ∈ R d−1 ×{0}| ‖y ′ ‖ < 1}and Q + = Q ∩ R d +. For 1 ≤ i ≤ p, let α i be a bijection from D i to Q such that α i ∈ C 2 (D i ),αi −1 ∈ C 2 (Q), α i (D i ∩ D) = Q + and α i (D i ∩ ∂D) = Q 0 .For all y ∈ Q + , we define w i (y) := u i (αi−1 (y)) ∈ H0 1(Q +) with ∇w i (y) = Ji−t (y)∇u i (αi−1 (y)),where J i (y) := Dα i (αi −1 (y))) is the Jacobian of α i . Furthermore, for x ∈ D i ∩ D and ϕ ∈ H0 1(Q +),we define v(x) := ϕ(α i (x)). Then v ∈ H0 1(D i ∩ D) and ∇v(αi −1 (y)) = Ji t(y)∇ϕ(y), for all y ∈ Q +,so that∫∫whereD i ∩Da(x)∇u i (x) · ∇v(x) dx =We define F i ∈ H s−1 (Q + ) by=∫a(αi−1Q +(y)) ∇u i (α −1iQ +A i (y) ∇w i (y) · ∇ϕ(y) dy ,A i (y) := a(αi−1 (y)) |detJ i (y))| −1 (J i Ji t )(y) ∈ S d (R).(y)) · ∇v(α −1 (y)) |detJ i (y))| −1 dy〈F i , ϕ〉 H s−1 (Q + ),H 1−s0 (Q + ):= 〈g i , ϕ ◦ α i 〉 H s−1 (D i ∩D),H 1−s0 (D i ∩D), for all ϕ ∈ H1−s 0 .Indeed, since we assumed that α i and αi−1 are in C 2 , we have ϕ ◦ α i ∈ H0 1−s (D i ∩ D) and moreover‖ϕ ◦ α i ‖ H 1−s (D i ∩D) ‖ϕ‖ H 1−s (Q + ) (cf. [18, Theorems 6.2.17 and 6.2.25(g)]), which implies thatF i ∈ H s−1 (Q + ) and‖F i ‖ H s−1 (Q + ) ‖g i ‖ H s−1 (D∩D i ).We finally get that v i ∈ H 1 0 (Q +) solves∫Q +A i ∇v i · ∇ϕ dy = 〈F i , ϕ〉 H s−1 (Q + ),H 1−s0 (Q + )for all ϕ ∈ H 1 0 (Q + ).In order to apply Lemma 3.3 we check first that A i ∈ C t (Q + , S d (R)) and that it is coercive, andthen define an extension of A i to R d +. Recalling that α i is a C 2 –diffeomorphism from D i ∩ D toQ + , with α −1i∈ C 2 (Q + ), we have for any y ∈ Q + and ξ ∈ R d :• Coercivity:• Boundedness:A i (y)ξ · ξ = a(α −1i(y))|detJ i (y)| −1 |J t i (y)ξ|2 a min |ξ| 2 . Hence A min a min .A max := ‖A i ‖ C 0 (Q + ,S d (R)) = max a(x) ‖|detJ i | −1 J i Ji t ‖ Cx∈D i ∩D0 (Q + ,S d (R)) a max .i• Regularity:A i ∈ C t (Q + , S d (R)) and‖A i ‖ C t (Q + ,S d (R))≤ a max ‖|detJ i | −1 J i J t i ‖ C t (Q + ,S d (R))+ |a| C t (D) ‖|detJ i| −1 J i J t i ‖ C 0 (Q + ,S d (R)) ‖a‖ C t (D) .27