Homogenization in porous media andassociated spectral problems:

Robin boundary conditions with largeadsorption parameters.

M. Eugenia Perez

Departamento de Matematica Aplicada y Ciencias de la Computacion

Universidad de Cantabria, Santander (SPAIN)

Conca60

60th Anniversary of Prof. Carlos Conca.

12 - 13 December 2014, BCAM, Bilbao, Spain

Homogenization problems in perforateddomains with nonlinear Robin type boundary

conditions on the boundary of the cavities

Joint work: U. de Cantabria– Lomonosov Moscow State U. (1996–2014)

- M. Lobo, O.A. Oleinik, T.A. Shaposhnikova (1997-1998) - Linear problems

- D. Gomez, M. Lobo, A.V. Podolskiy, T.A. Shaposhnikova, V.V. Sukharev, M.N. Zubova

(2011 →)

Asymptotic behavior of the solution uε of P ε, ε→ 0

P ε

−∆uε = f in Ωε

uε = 0 on ∂Ω

∂uε∂n

+ β(ε)σ(x, uε) = 0 on⋃∂T ε

Ω ⊂ Rn, n = 2, 3, ... & T ε perforations ⊂ ΩΩε = Ω \

⋃T ε

The functions:

• f ∈ L2(Ω)

• σ ≡ σ(x, u), σ ∈ C1(Ω× R) /

σ(x, 0) = 0, 0 < k1 ≤∂σ

∂u(x, u) ≤ k2, x ∈ Ω, u ∈ R,

...that can be weakened:

0 ≤ ∂σ

∂u(x, u) ≤ k2(1 + |u|δ), δ ∈

[0,

2

n− 2

],

... and include some adsorption lows: Langmuir function

σ(u) =k1u

1 + k2u, k1, k2 > 0, u ∈ R+,

and others strictly increasing functions σ : [0,∞)→ [0,∞)

The parameters:

• ε the period: ε→ 0

• aε the size of the perforations: aε = o(ε) or aε = O(ε)

• β(ε) the adsorption parameter: β(ε)ε→0−−−→∞

As ε→ 0, we find very different behaviours of the solutionuε of P ε, depending on the relations between ε, aε and β(ε)!

Critical relations for parameters: β(ε)|⋃∂T ε| = O(1)

⇒ a certain relation between ε, aε and β(ε)

I We deal with: critical sizes, strange terms or foreign terms,...

I Relations between parameters for which we have a different asymp-totic behavior from extreme cases/behaviors: it may depend...

V.A.Marchenko & E.Ya. Khruslov (1974), E.Sanchez-Palencia (1982), D.Cioranescu &

F.Murat (1982), G.Allaire (1989), ...

C.Conca (1985), D.Cioranescu & P.Donato & H.Ene (1996), M.Lobo & O.A.Oleinik &

M.E.Perez & T.A.Shaposhnikova (1998),...

Outline

• A general situation:

critical relations for parameters in P ε

• The case of perforations by tubes

• Application to the spectral convergence

• The case of perforations by balls along a plane

• Bounds for convergence rates of eigenvalues

Outline

• A general situation:

critical relations for parameters in P ε

• The case of perforations by tubes

• Application to the spectral convergence

• The case of perforations by balls along a plane

• Bounds for convergence rates of eigenvalues

The critical relations for parameters: β(ε)|⋃∂T ε| = O(1)

β(ε) very large or very small compared with |⋃∂T ε|−1

aε large or small compared with the classical critical size

• The most critical case: β(ε)|⋃∂T ε| = O(1) + critical size T ε

⇒ the strange term contains a non-linear term H(x, u) solutionof the functional equation

H = cσ(x, u−H)

where c > 0 depends on n and the geometry of T ε, and Hsatisfies the same properties of smoothness of σ

• β(ε)|⋃∂T ε| = O(1)+ large sizes of T ε

⇒ the strange term contains cσ(x, u)

• Other relations: β(ε) very large + critical size of T ε

⇒ a linear strange term not depending on σ

• Other extreme relations do not take into account perforationsor adsorption parameters

Outline

• A general situation:

critical relations for parameters in P ε

• The case of perforations by tubes

• Application to the spectral convergence

• The case of perforations by balls along a plane

• Bounds for convergence rates of eigenvalues

a basis of the domain the periodicity cell

the perforated domain

Domain of R3 perforated by tubes.Homogenized problems P 0: aε << ε, ε→ 0

D.Gomez, M.Lobo, E.Perez, T.A.Shaposhnikova, M.N.Zubova (DM 2013, M2AS 2014, →)

P 0

−∆u + ς(x, u) = f in Ω,u = 0 on ∂Ω,

• If β(ε)aεε−2 → C2 > 0 and ε2 ln(aε)→ −α2 < 0

ς(x, u) =2π

α2H(x, u), 2πH = |∂G|α2C2σ(x, u−H)

• If β(ε)aεε−2 → C2 > 0 and ε2 ln(aε)→ 0

ς(x, u) = |∂G|C2σ(x, u)

• If β(ε)aεε−2 →∞ and ε2 ln(aε)→ −α2 < 0

ς(x, u) =2π

α2u

Domain of R3 perforated by tubes:the spectral problem P ε

Asymptotics for the eigenelements (λε, uε) as ε→ 0depending on the relations between ε, aε and β(ε)

a(x) ∈ C1(Ω), a(x) > 0

Fixed ε : 0 < λε1 ≤ λε2 ≤ · · ·λεi ≤ · · ·i→∞−−−→∞

The eigenfunctions uεi∞i=1 basis in L2(Ωε) and H1(Ωε)

P ε

−∆uε = λεuε in Ωε

uε = 0 on ∂Ω \(∂Ω ∩

⋃Gε)

∂uε

∂n+ β(ε)a(x)uε = 0 on

⋃∂Gε

The homogenized spectral problems

• when β(ε)aεε−2 → 0 or ε2 ln(aε)→ −∞

P 0 is the spectral Dirichlet problem in Ω,

• when β(ε)aεε−2 →∞ and ε2 ln(aε)→ −α2 < 0

P 0

−∆u +2π

α2u = λu in Ω,

u = 0 on Ω,

• when β(ε)aεε−2 → C2 > 0 and ε2 ln(aε)→ 0

P 0

−∆u + |∂G|C2a(x)u = λu in Ω,u = 0 on Ω,

• when β(ε)aεε−2 → C2 > 0 and ε2 ln(aε)→ −α2 < 0

P 0

−∆u +2π

α2

|∂G|α2C2a(x)

2π + |∂G|α2C2a(x)u = λu in Ω,

u = 0 on Ω,

Convergence of (λε, uε) towards the eigenelements of P 0 with con-servation of the multiplicity!

Outline

• A general situation:

critical relations for parameters in P ε

• The case of perforations by tubes

• Application to the spectral convergence

• The case of perforations by balls along a plane

• Bounds for convergence rates of eigenvalues

Critical sizes/relations: ε−κ | ∪ ∂T ε| = O(1)

P 0

−∆u = λu in Ω+ ∪ Ω−

[u] = 0 on Σ[ ∂u∂x3

]= A(x)u on Σ

u0 = 0 on ∂Ω

A(x) = C20 |∂T |a(x) when κ = 2(α− 1)

A(x) = C20 |∂T |

a(x)

a(x)C0 + 1when κ = α = 2

A(x) = C0 |∂T | for κ > 2 , α = 2A(x) = 0 for κ < 2 , α = 2

2

21

Diri

chle

tpro

blem

inU

Dirichlet problem in

n = 3

Convergence rates for eigenelements:the most critical case α = κ = 2

TheoremLet λεk∞k=1 and λk∞k=1 be the eigenvalues of problem P ε andP 0, respectively. Then, for each fixed k there exists a constantCk independent of ε such that

|λεk − λk| ≤ Ckε1/16,

holds for sufficiently small ε. In addition, for any eigenvalueλk of P 0 with multiplicity s (λk = λk+1 = · · · = λk+s−1), and forany u eigenfunction corresponding to λk, with ‖u‖L2(Ω) = 1, thereexists uε, uε a linear combination of eigenfunctions uεrr=k+s−1

r=k

of P ε corresponding to λεrr=k+s−1r=k , such that

‖uε − u‖L2(Ωε) ≤ Ckε1/16.

... and similar bounds for the rest of α and κ!

Some references

• M. Lobo, O.A.Oleinik, E. Perez, T.A. Shaposhnikova. On homogenization ofsolutions of boundary value problems in domains, perforated along manifolds. Annalidella Scuola Normale Superiore Pisa, Classe di Scienza, (4) V. 25, N. 3-4, p. 611–629,1998,

• M. Lobo, E. Perez, T.A. Shaposhnikova, V.V. Sukharev. Averaging ofboundary-value problem in domain perforated along (n-1) - dimensional manifold withnonlinear third type boundary conditions on the boundary of cavities. Doklady Math-ematics, V. 83, N. 1, p. 34–38, 2011,

• M. Lobo, D. Gomez, E. Perez, T.A. Shaposhnikova. Averaging in varia-tional inequalities with nonlinear restrictions along manifolds. Comptes Rendues deMecanique, 339, p. 406–410, 2011

• M. Lobo, D. Gomez, E. Perez, T.A. Shaposhnikova. Averaging of a variationalinequality for the Laplacian with nonlinear restrictions along manifolds. ApplicableAnalysis, V. 92, N. 2, p. 218–237, 2013,

• E. Perez, T.A. Shaposhnikova. Boundary homogenization of a variational in-equality with nonlinear restrictions for the flux on small regions lying on a part of theboundary. Doklady Mathematics, , V. 85, N. 2, p. 198–203, 2012

• D. Gomez, E. Perez, T.A. Shaposhnikova. On homogenization of nonlinear Robintype boundary conditions for cavities along manifolds and associated spectral problems.Asymptotic Analysis, N. 80, p. 289–322, 2012

References (cont.)

• D. Gomez, E. Perez, T.A. Shaposhnikova. On correctors for spectral problemsin the homogenization of Robin boundary conditions with very large parameters. In-ternational Journal of Applied Mathematics , V. 26, N. 3, p. 309–320, 2013

• D. Gomez, E. Perez, T.A. Shaposhnikova. Spectral boundary homogenizationproblems in perforated domains with Robin boundary conditions and large parameters.In, Integral Methods in Science and Engineering. Progress in Numerical and AnalyticTechniques, Birkhauser Boston, Springer p. 155- 174, 2013

• D. Gomez, M. Lobo, E. Perez, T.A. Shaposhnikova, M.N. Zubova. Ho-mogenization problem in domain perforated by thin tubes with nonlinear Robin typeboundary condition. Doklady Mathematics, V. 87, N. 1, p. 5–11, 2013

• D. Gomez, M. Lobo, E. Perez, A.V. Podolskiy, T.A. Shaposhnikova. Ho-mogenization for the p-Laplace operator and nonlinear Robin boundary conditions inperforated media along manifolds. Doklady Mathematics. V. 89, N.1, p.11-15, 2014

• E. Perez, T.A. Shaposhnikova, M.N. Zubova. A homogenization problem ina domain perforated by tiny isoperimetric holes with nonlinear Robin type boundaryconditions. Doklady Mathematics, V. 90, N. 1, p. 489-494, 2014.

• D. Gomez, M. Lobo, E. Perez, T.A. Shaposhnikova, M.N. Zubova. On criticalparameters in homogenization for nonlinear fluxes in perforated domains by thin tubesand related spectral problems. Mathematical Methods in Applied Sciences, p. 1-24,2014

• D. Gomez, M. Lobo, E. Perez, T.A. Shaposhnikova, M.N. Zubova. Correctingterms for perforated media by thin tubes with nonlinear flux and large adsorptionparameters. Integral Methods in Science and Engineering. Springer, to appear, 2015

Thank you very much…