Asymptotic analysis of some partial differential equations in domains becoming unbounded

Cette thèse a pour objet l'étude de quelques problèmes elliptiques dans des domaines qui deviennent infinis dans une ou plusieurs directions. Dans la première partie de la thèse, nous étudions le problème⌠-div(A ∇uℓ) = f dans Ωℓ⌡uℓ = g sur ∂Ωℓ.où Ωℓ est le cylindre lω1 x ω2 avec 1 et ω2 deux domaines bornés de Rk et Rn-k respectivement (avec 1 ≤ k ≤ n - 1). On note Ω ∞ le cylindre infini Rk x ω2, et l'on prend f ∈ H-1loc (Ω ∞) et g ∈ H1loc (Ω ∞), de sorte que f ∈ H-1loc (Ω ∞) et g ∈ H1loc (Ω ∞), tout ℓ > 0. Ce travail se fonde sur les méthodes développées dans [23] et [19]. On démontre qu'il est possible de passer indifféremment à la limite dans la suite de cylindres puis de résoudre le problème, ou de d'abord résoudre le problème sur le cylindre Ωℓ puis de passer à la limite. Ici, la limite doit être comprise dans le sens d'un principe de Saint-Venant, c'est-à-dire que la convergence a lieu pour des restrictions de uℓ à des cylindres Ωℓ plus petits (avec 0 0. This work is based on the methods developed in [23] and [19]. We show that it is possible to indifferently pass to the limit in the sequence of cylinders and then to solve the problem on the infinite cylinder, or to first solve the problem on the cylinder Ωℓ and then to pass to the limit. The limit here is to be understood in the sense of a Saint-Venant type principle, which is to say that the convergence takes place for the restrictions of uℓ to smaller domains Ωℓ (with 0 < ℓ’ < ℓ) contained in Ωℓ. After that, we give some optimality results concerning the domain in which the sequence of solutions uℓ converges to u ∞. In the second chapter, we construct some correctors that enable us to extend the convergence on the whole cylinder. The construction of these correctors is inspired from the ones made in [17] and [18]. In the third chapter of the thesis we prove that, under some decreasing conditions at in_nity for the data f, it is possible to recover the same convergence on the whole cylinder, without the adjunction of correctors. In the last part of the thesis, we study the Stokes problem∫ - µ∆uℓ + ∇pℓ = f dans Ωℓ∫ div u = 0 dans Ωℓ⌡u = 0 sur ∂Ωℓon a domain Ωℓ = Bℓ x ω, where Bℓ ⊂ Rk (1 ≤ k ≤ n-1) is the ball of radius ℓ centered at the origin. Here, an important hypothesis is that f has some radial invariant properties with respect to the first k coordinates. One of the major tools in the proof of the result of this chapter (concerning especially the case k ≥ 2) is a result of the divergence-problem type. More precisely, based on a construction inspired by [9], we prove the following result (here below Dℓ = Ωℓ +1 \ Ωℓ) :If g ∈ W1;p(Dℓ) is a radial function along X1 such that g = 0 on (Bℓ+1 \ Bℓ) x ∂ω and ∫Dℓ gdx = 0. then there exists u ∈ (W1;p 0 (Dℓ))n such that∫div u = g in Dℓ⌡ ||∇u || Lp (Dℓ) ≤ C(|| g || Lp (Dℓ) + ||∇x2g || Lp (Dℓ))the constant C being independent of ℓ for ℓ ≥ 1, (C depends on k, n, p and !). Thanks to this result, used in the case p = 2, we finally prove that in this case, we also have an exponential rate of convergence of uℓ Ωℓ2 to u ∞