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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
MP2 Workshop, 2009
Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects
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Introduction
Applications and physical motivation:
Image restoration based on a variable exponent Laplacian 1
E (u) =
∫Ω
|∇u(x)|p(x) + |u(x)− I (x)|2 dx
Modeling of non–newtonian fluids, in particularelectrorheological fluids (ER).2
−div (K (x)|∇u|p(x)−2∇u) + R(x)|u|σ(x)−2u = g(x),
1Y. Chen, S. Levine, R. Rao, Functionals with p(x)–growth in image processing. Duquesne University, Dep. of
Math. And Comp. Sci. Tech. Rep.2
M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000
Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects
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Introduction
Applications and physical motivation:
Image restoration based on a variable exponent Laplacian 1
E (u) =
∫Ω
|∇u(x)|p(x) + |u(x)− I (x)|2 dx
Modeling of non–newtonian fluids, in particularelectrorheological fluids (ER).2
−div (K (x)|∇u|p(x)−2∇u) + R(x)|u|σ(x)−2u = g(x),
1Y. Chen, S. Levine, R. Rao, Functionals with p(x)–growth in image processing. Duquesne University, Dep. of
Math. And Comp. Sci. Tech. Rep.2
M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000
Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects
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Introduction
Applications and physical motivation:
Image restoration based on a variable exponent Laplacian 1
E (u) =
∫Ω
|∇u(x)|p(x) + |u(x)− I (x)|2 dx
Modeling of non–newtonian fluids, in particularelectrorheological fluids (ER).2
−div (K (x)|∇u|p(x)−2∇u) + R(x)|u|σ(x)−2u = g(x),
1Y. Chen, S. Levine, R. Rao, Functionals with p(x)–growth in image processing. Duquesne University, Dep. of
Math. And Comp. Sci. Tech. Rep.2
M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000
Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects
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Introduction: Mathematical Motivation
We are interested in the study of the following nonlinear elliptical
equations:
− div (F (|∇u|, x),∇u) + R (|u|, x) u = g(x).
Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects
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Introduction: Mathematical Motivation
In this work:
pε–Laplacian in perforated domain
−div(|∇uε|pε(x)−2∇uε
)+ |uε|σ(x)−2 uε = g(x), Ωε ⊂ Rn;
uε = Aε on ∂Fε;
uε = 0 on ∂Ω;
∫∂Fε|∇ε|pε(x)−2∂uε
∂~νds = 0.
Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects
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Introduction: History of the problem
Existence results:V. V. Zhikov, On some variational problems, J. Math. Phys. 5 (1997),
#1,
E. Acerbi, G Mingione, Regularity results for stationary
electro-rheological fluids, Arch. Ration. Mech. Anal, 164 (2002),
P. Hasto, On the variable exponent Dirichlet energy integral, Comm.
Pure and Appl. Anal, 5 (2006), #3.
Homogenization Dirichlet problem for p-Laplacian (p isconstant)
E. Hruslov, L.Pankratov, Asymptotical behaviour of p-Laplacian
equations in domains with complex boundary, J. Math. Phys., Anal.,
Geom., FTINT, (1987),
I. Skrypnik, Methods for analysis of nonlinear elliptic boundary value
problems, Providence, RI: American Mathematical Society (AMS),
1994,
A. Braides and A. Defranceschi, Homogenization of Multiple Integrals,
Oxford Lecture Ser. Math. Appl. vol. 12, Clarendon Press, Oxford
(1998).
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Introduction: History of the problem
Existence results:V. V. Zhikov, On some variational problems, J. Math. Phys. 5 (1997),
#1,
E. Acerbi, G Mingione, Regularity results for stationary
electro-rheological fluids, Arch. Ration. Mech. Anal, 164 (2002),
P. Hasto, On the variable exponent Dirichlet energy integral, Comm.
Pure and Appl. Anal, 5 (2006), #3.
Homogenization Dirichlet problem for p-Laplacian (p isconstant)
E. Hruslov, L.Pankratov, Asymptotical behaviour of p-Laplacian
equations in domains with complex boundary, J. Math. Phys., Anal.,
Geom., FTINT, (1987),
I. Skrypnik, Methods for analysis of nonlinear elliptic boundary value
problems, Providence, RI: American Mathematical Society (AMS),
1994,
A. Braides and A. Defranceschi, Homogenization of Multiple Integrals,
Oxford Lecture Ser. Math. Appl. vol. 12, Clarendon Press, Oxford
(1998).
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Introduction: History of the problem
Opposite case: logarithmic growth
E. Khruslov, L. Pankratov, Homogenization of the Dirichlet variational
problems in Orlicz–Sobolev spaces, Fields Institute Communications 25
(2000) 345–66,
M. Goncharenko, V. Prytula, Homogenization of the electrostatic
problems in nonlinear medium with thin perfectly conducting grids, J.
Math. Phys. Anal. Geom. 2, 424–448 (2006),
D. Lukkassen, N. Svanstedt, On Γ–convergence in Anisotropic
Orlicz–Sobolev Spaces, Rend. Instit. Mat. Univ. Trieste, XXXIII,
281-287 (2001).
pε(x) case:
B. Amaziane, S. Antontsev, L. Pankratov, A. Piatnitski,
Homogenization of pε(x)–Laplacian in perforated domains, Ann. Institut
H. Poincare (C). Analyse non Lineaire (2009).
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Introduction: History of the problem
Opposite case: logarithmic growth
E. Khruslov, L. Pankratov, Homogenization of the Dirichlet variational
problems in Orlicz–Sobolev spaces, Fields Institute Communications 25
(2000) 345–66,
M. Goncharenko, V. Prytula, Homogenization of the electrostatic
problems in nonlinear medium with thin perfectly conducting grids, J.
Math. Phys. Anal. Geom. 2, 424–448 (2006),
D. Lukkassen, N. Svanstedt, On Γ–convergence in Anisotropic
Orlicz–Sobolev Spaces, Rend. Instit. Mat. Univ. Trieste, XXXIII,
281-287 (2001).
pε(x) case:
B. Amaziane, S. Antontsev, L. Pankratov, A. Piatnitski,
Homogenization of pε(x)–Laplacian in perforated domains, Ann. Institut
H. Poincare (C). Analyse non Lineaire (2009).
Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects
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Statement of the problem
−div(|∇uε|pε(x)−2∇uε
)+ |uε|σ(x)−2 uε = g(x), Ωε ⊂ Rn, d ≥ 2;
uε = Aε on ∂Fε;
uε = 0 on ∂Ω;
∫∂Fε|∇ε|pε(x)−2∂uε
∂~νds = 0,
Aε is an unknown constant
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Statement of the problem: Growth Conditions
pε ∈ C (Ω) such that ∀ε > 0:
(i) pε is bounded in Ω, i.e., 1 < p− ≤ p−ε ≡ minx∈Ω pε(x) ≤pε(x) ≤ maxx∈Ω pε(x) ≡ p+
ε ≤ p+ ≤ n;
(ii) pε is log–continuous, i.e., for any x , y ∈ Ω,|pε(x)− pε(y)| ≤ ωε(|x − y |), where limτ→0 ωε(τ) ln
(1τ
)≤ C ;
(iii) pε converges uniformly in Ω to a function p0, where p0 islog–continuous;
(iv) pε satisfies the inequality: pε(x) ≥ p0(x) in Ω.
σ be a log–continuous function in Ω such that ε > 0:
(v) 1 < σ− ≡ minx∈Ω σ(x) ≤ σ(x) ≤ maxx∈Ω σ(x) ≡ σ+ ≤n p0(x)/(n − p0(x)) in Ω.
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Statement of the problem: Growth Conditions
pε ∈ C (Ω) such that ∀ε > 0:
(i) pε is bounded in Ω, i.e., 1 < p− ≤ p−ε ≡ minx∈Ω pε(x) ≤pε(x) ≤ maxx∈Ω pε(x) ≡ p+
ε ≤ p+ ≤ n;
(ii) pε is log–continuous, i.e., for any x , y ∈ Ω,|pε(x)− pε(y)| ≤ ωε(|x − y |), where limτ→0 ωε(τ) ln
(1τ
)≤ C ;
(iii) pε converges uniformly in Ω to a function p0, where p0 islog–continuous;
(iv) pε satisfies the inequality: pε(x) ≥ p0(x) in Ω.
σ be a log–continuous function in Ω such that ε > 0:
(v) 1 < σ− ≡ minx∈Ω σ(x) ≤ σ(x) ≤ maxx∈Ω σ(x) ≡ σ+ ≤n p0(x)/(n − p0(x)) in Ω.
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Main steps in the analysis
Consider the associated variational problem
Prove convergence of the minimizers to the minimizer of thehomogenized functional
Obtain homogenized PDE
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Variational Formulation
Jε[u] ≡∫
ΩεFε (x , u,∇u) dx −→ inf, uε ∈W 1,pε(·)(Ωε);
uε = Aε on ∂Fε and uε = 0 on ∂Ω,
where
Fε (x , u,∇u) =1
pε(x)|∇u|pε(x) +
1
σ(x)|u|σ(x) − g(x) u,
Aε unknown; g ∈ C (Ω).
Existence
For each ε > 0, ∃!uε ∈W 1,pε(·)(Ωε).
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Variational Formulation
Jε[u] ≡∫
ΩεFε (x , u,∇u) dx −→ inf, uε ∈W 1,pε(·)(Ωε);
uε = Aε on ∂Fε and uε = 0 on ∂Ω,
where
Fε (x , u,∇u) =1
pε(x)|∇u|pε(x) +
1
σ(x)|u|σ(x) − g(x) u,
Aε unknown; g ∈ C (Ω).
Existence
For each ε > 0, ∃!uε ∈W 1,pε(·)(Ωε).
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Energy Characteristics of the set F ε
Local energy characteristics of the set Fε
For any piece S of Γ, introduce a layer Th(S) generated by thesurfaces Γ−h (S), Γ+
h (S):
C ε,h(S , b) = infvε
∫Th(S)
1
pε(x)|∇v ε|pε(x) + h−p+−γ |v ε − b|pε(x)
dxa
b ∈ R, γ > 0,
vε ∈W 1,pε(·) (Th(S)) , vε = 0, x ∈ Fε.
aV. A. Marchenko, E. Ya. Khruslov, Homogenization of Partial DifferentialEquations, Birkhauser, Berlin, 2006.
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Energy Characteristics of the set F ε
For any piece S of Γ, introduce a layer Th(S) generated by the surfaces Γ−h
(S), Γ+h
(S):
C ε,h(S , b) = infvε
∫Th(S)
1
pε(x)|∇vε|pε(x) + h−p+−γ |vε − b|pε(x)
dx , a
b ∈ R, γ > 0, vε ∈ W 1,pε(·) (Th(S)), vε = 0 on Fε.
aV. A. Marchenko, E. Ya. Khruslov, Homogenization of Partial Differential Equations, Birkhauser, Berlin,
2006.
Condition
(C.1) for any arbitrary piece S ⊂ Γ and any b ∈ R:
limh→0
limε→0
C ε,h(S , b) = limh→0
limε→0
C ε,h(S , b) =
∫S
c(x , b)dΓ,
where c(x , b) is a nonnegative continuous function on Γ such thatc(x , b) ≤ C
(1 + |b|p0(x)−1
)|b|.
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Main Result
Theorem
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
Jhom[u,A] ≡∫
ΩF0(x , u,∇u) dx +
∫Γ
c(x , u − A) dx −→ inf,
u ∈W1,p0(·)0 (Ω) with A = lim
ε→0Aε,
where
F0(x , u,∇u) =1
p0(x)|∇u|p0(x) +
1
σ(x)|u|σ(x) − g(x) u,
and constant A is finite.
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Main Result
Remark
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.
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Main Result
Lemma
Properties of the homogenized functional.Let the conditions of Theorem hold. Then we have the followingproperties:
The functional Jhom[u,A] is convex in A and strictly convex inu;
Jhom[u,A] is continuous in the space W1,p0(·)0 (Ω) with respect
to the variable u.
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Periodic Example
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
uniformly in Ω to p0 ≡ 2.
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Periodic Example
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
uniformly in Ω to p0 ≡ 2.
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Periodic Example
Theorem
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;
u = 0 on ∂Ω; [u]±Γ = 0 and[∂u∂ν
]±Γ
= 4π (u(z)− Al )µ(z) on Γ,
where
Al =
(∫Γµ(s) ds
)−1 ∫Γµ(s)u(s) ds and µ(z) =
e l(z) − 1
l(z).
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Periodic Example: Remark
Remark
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.
Polynomial growth faster then 2 : no 3D lattice,
Logarithmic growth : nontrivial homogenization results,
Our case, pε ∼ 2 + ε : nontrivial homogenization results.
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Periodic Example: Remark
Remark
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.
Polynomial growth faster then 2 : no 3D lattice,
Logarithmic growth : nontrivial homogenization results,
Our case, pε ∼ 2 + ε : nontrivial homogenization results.
Prytula, Pankratov pε(x)-Laplacian, Nonlocal effects