Periodic Homogenization of Parabolic Nonstandard Monotone Operators
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Periodic Homogenization of Parabolic Nonstandard Monotone Operators Rodrigue Kenne Bogning · Hubert Nnang
Received: 25 April 2012 / Accepted: 30 October 2012 / Published online: 16 November 2012 © Springer Science+Business Media Dordrecht 2012
Abstract We study the periodic homogenization for a family of parabolic problems with nonstandard monotone operators leading to Orlicz spaces. After proving the existence theorem based on the classical Galerkin procedure combined with the Stone-Weierstrass theorem, the fundamental in this topic is the determination of the global homogenized problem via the two-scale convergence method adapted to this type of spaces. Keywords Global solution · Periodic homogenization · Two-scale convergence · Nonstandard monotone operators · Orlicz spaces Mathematics Subject Classification (2010) 35B27 · 35B40 · 46E30 · 74G25
1 Introduction be the Fenchel’s conjugate Let B : [0, ∞) → [0, ∞) be a differentiable N -function and let B of B such that ∈ Δ , t 2 ≤ B(ρ0 t) B
and
1 < ρ1 ≤
tb(t) ≤ ρ2 for any t > 0, B(t)
(1)
where ρ0 > 0, ρ1 , ρ2 are constants, and b is odd, increasing homeomorphism from R onto t R such that B(t) = 0 b(s)ds (t ≥ 0). Let Ω be a smooth bounded open set in Rdx (the space of variables x = (x1 , . . . , xd ) (integer d ≥ 1)). Let T be a positive real number. Let
R.K. Bogning Faculty of Sciences, Department of Mathematics, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon e-mail: [email protected] H. Nnang () École Normale Supérieure de Yaoundé, University of Yaounde I, P.O. Box 47, Yaounde, Cameroon e-mail: [email protected]
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R.K. Bogning, H. Nnang
f ∈ LB (0, T ; W −1 LB (Ω; R)) ≡ LB (0, T ; W01 LB (Ω; R)) , where Q = Ω × (0, T ). For each given ε > 0, we consider the initial-boundary value problem x t ∂uε − div a , , Duε = f in Q, ∂t ε ε (2) uε = 0 on ∂Ω × (0, T ), uε (x, 0) = 0 in Ω, where D denotes the usual gradient operator in Ω, i.e., D = ( ∂x∂ i )1≤i≤d , LB (Q) and W01 LB (Ω; R) are Orlicz and Orlicz-Sobolev spaces (respectively) which will be specified later, and where a is a function from Rd × R × Rd into Rd satisfying the following properties: for each λ ∈ Rd , the function (y, τ ) → a(y, τ, λ), denoted by a(·, ·, λ), is measurable from Rd × R into Rd a(y, τ, ω) = ω almost everywhere (a.e.) in (y, τ ) ∈ Rdy × R, where ω is the origin in Rd there exist three constants c0 , c1 , c2 > 0 such that, a.e. in Rdy × R:
−1 B c1 |λ − μ| (i) a(y, τ, λ) − a(y, τ, μ) ≤ c0 B
(ii) a(y, τ, λ) − a(y, τ, μ) · (λ − μ) ≥ c2 B |λ − μ|
(3)
(4)
(5)
for all λ, μ ∈ Rd , where the dot denotes the usual Euclidean inner product in Rd for each λ ∈ Rd , a(·, ·, λ) is periodic, that is (i.e.), a.e. in Rdy × R: a(y + k, τ + l, λ) = a(y, τ, λ) for all (k, l) ∈ Zd × Z.
(6)
Provided the differential operator u → div a( xε , εt , Du) is rigorously defined, and an existence and uniqueness result for (2) is sketched, we are interested in this paper to the homogenization of (2), i.e., the limiting behaviour, as ε → 0, of uε (the solutio
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