Nonlocal transmission conditions arising in homogenization of p ε (x )-Laplacian in perforated domains V. Prytula 1 L. Pankratov 2,3 1 Universidad de Castilla-La Mancha Departamento de Matem´ aticas 2 B. Verkin Institute for Low Temperature Physics Ukraine Mathematical Division 3 Universit´ e de Pau, Laboratoire de Math´ ematiques Appliqu´ ees MP2 Workshop, 2009 Prytula, Pankratov pε(x )-Laplacian, Nonlocal effects
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Nonlocal transmission conditions arising in homogenization of … · 2009-11-13 · Nonlocal transmission conditions arising in homogenization of p "(x)-Laplacian in perforated domains
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Nonlocal transmission conditions arising inhomogenization of pε(x)-Laplacian in perforated
domains
V. Prytula1 L. Pankratov2,3
1Universidad de Castilla-La ManchaDepartamento de Matematicas
2B. Verkin Institute for Low Temperature Physics UkraineMathematical Division
3Universite de Pau, Laboratoire de Mathematiques Appliquees
Let uε be a solution of BVP extended by uε(x) = Aε in Fε. Letgrowth assumptions and (C.1) hold. Then there is a subsequenceuε, ε = εk → 0 that converges weakly in W 1,p0(·)(Ω) to afunction u(x) such that the pair u(x),A is a solution of
Constant A remains unknown. Supposing that the function c(x , b)is differentiable with respect to b we obtain:−div
(|∇u|p0(x)−2∇u
)+ |u|σ(x)−2 u = g(x) in Ω \ Γ;
u = 0 on ∂Ω; [u]±Γ = 0,[|∇u|p0(x)−2 ∂u
∂ν
]±Γ
= c ′u(x , u − A);∫Γ
c ′u(x , u − A) dS = 0,
where ν is a normal vector to Γ, [ · ]±Γ is the jump on Γ, c ′u is thepartial derivative of c with respect to u. This means that problemcontains a non–local transmission condition.
Let Ω be a bounded Lipschitz domain in R3. Fε consists of thinintersecting cylinders of radius
r (ε) = e−1/ε.
The axes of the cylinders belong to a plane Γ b Ω and form anε–periodic lattice in R2. Ωε = Ω \ Fε.Let pε(ε>0) be a class of smooth functions in Ω given by:
pε(x) =
2 + ε `(x) in N (Fε, ε2);2 + `ε(x) elsewhere,
N (Fε, ε2) denotes the cylindrical ε2–neighborhood of the set Fεand where `, `ε are smooth strictly positive functions in Ω,maxx∈Ω `ε(x) = o(1) , ε→ 0. It is clear that pε converges
Let Ω be a bounded Lipschitz domain in R3. Fε consists of thinintersecting cylinders of radius
r (ε) = e−1/ε.
The axes of the cylinders belong to a plane Γ b Ω and form anε–periodic lattice in R2. Ωε = Ω \ Fε.Let pε(ε>0) be a class of smooth functions in Ω given by:
pε(x) =
2 + ε `(x) in N (Fε, ε2);2 + `ε(x) elsewhere,
N (Fε, ε2) denotes the cylindrical ε2–neighborhood of the set Fεand where `, `ε are smooth strictly positive functions in Ω,maxx∈Ω `ε(x) = o(1) , ε→ 0. It is clear that pε converges
Let uε be the solution extended by the equality uε(x) = Aε in Fε.Then uε converges weakly in H1(Ω) to u the solution of−∆u + |u|σ(x)−2 u = g(x) in Ω \ Γ;
In the case of a surface distribution of Fε, with a constant growthpε(x) = 2 + α, α > 0 is a parameter independent of ε, there is no3D lattice for the corresponding problem which leads tohomogenization because the capacity of the lattice goes to infinityas ε→ 0. However our Theorem gives an example of the growthpε ∼ 2 + ε (in a small neighborhood of the lattice) which leads toa non trivial homogenization result.
In the case of a surface distribution of Fε, with a constant growthpε(x) = 2 + α, α > 0 is a parameter independent of ε, there is no3D lattice for the corresponding problem which leads tohomogenization because the capacity of the lattice goes to infinityas ε→ 0. However our Theorem gives an example of the growthpε ∼ 2 + ε (in a small neighborhood of the lattice) which leads toa non trivial homogenization result.