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doi: 10.3934/dcds.2020385

Homogenization for nonlocal problems with smooth kernels

Monia Capanna 1, , Jean C. Nakasato 2, , Marcone C. Pereira 3, and  Julio D. Rossi 1,,

1. 

CONICET and Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Ciudad Universitaria, Pabellon I, (1428), Buenos Aires, Argentina
2. 

Dpto. de Matemática, ICMC, Universidade de São Paulo, Avenida Trabalhador São-Carlense, 400, São Carlos - SP, Brazil
3. 

Dpto. de Matemática Aplicada, IME, Universidade de São Paulo, Rua do Matão 1010, São Paulo - SP, Brazil

* Corresponding author: Julio D. Rossi

Received  May 2020 Revised  October 2020 Published  November 2020

Fund Project: The first and last authors (MC and JDR) are partially supported by CONICET grant PIP GI No 11220150100036CO (Argentina), UBACyT grant 20020160100155BA (Argentina), Project MTM2015-70227-P (Spain).
The third author (MCP) has been partially supported by CNPq 303253/2017-7 and FAPESP 2020/04813-0 (Brazil).
The second author (JCN) supported by CAPES - INCTmat grant 465591/2014-0 (Brazil)

In this paper we consider the homogenization problem for a nonlocal equation that involve different smooth kernels. We assume that the spacial domain is divided into a sequence of two subdomains $ A_n \cup B_n $ and we have three different smooth kernels, one that controls the jumps from $ A_n $ to $ A_n $, a second one that controls the jumps from $ B_n $ to $ B_n $ and the third one that governs the interactions between $ A_n $ and $ B_n $. Assuming that $ \chi_{A_n} (x) \to X(x) $ weakly-* in $ L^\infty $ (and then $ \chi_{B_n} (x) \to (1-X)(x) $ weakly-* in $ L^\infty $) as $ n \to \infty $ we show that there is an homogenized limit system in which the three kernels and the limit function $ X $ appear. We deal with both Neumann and Dirichlet boundary conditions. Moreover, we also provide a probabilistic interpretation of our results.

Keywords: Heterogeneous media, homogenization, nonlocal equations, Neumann problem, Dirichlet problem.
Mathematics Subject Classification: Primary: 45A05, 45C05; Secondary: 45M05.
Citation: Monia Capanna, Jean C. Nakasato, Marcone C. Pereira, Julio D. Rossi. Homogenization for nonlocal problems with smooth kernels. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020385
References:
[1]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, vol. 165. AMS, 2010. doi: 10.1090/surv/165.  Google Scholar

[2]

P. W. Bates and A. Chmaj, An integrodifferential model for phase transitions: Stationary solutions in higher dimensions, J. Statist. Phys., 95 (1999), 1119-1139.  doi: 10.1023/A:1004514803625.  Google Scholar

[3]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland Publishing Company, 1978.  Google Scholar

[4]

L. Caffarelli and A. Mellet, Random homogenization of fractional obstacle problems, Netw. Heterog. Media, 3 (2008), 523-554.  doi: 10.3934/nhm.2008.3.523.  Google Scholar

[5]

M. Capanna and J. D. Rossi, Mixing local and nonlocal evolution equations, Preprint, arXiv: 2003.03407v1. Google Scholar

[6]

C. Carrillo and P. Fife, Spatial effects in discrete generation population models., J. Math. Biol., 50 (2005), 161-188.  doi: 10.1007/s00285-004-0284-4.  Google Scholar

[7]

P. Cazeaux and C. Grandmont, Homogenization of a multiscale viscoelastic model with nonlocal damping, application to the human lungs, Math. Models Methods Appl. Sci., 25 (2015), 1125-1177.  doi: 10.1142/S0218202515500293.  Google Scholar

[8]

E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl., 86 (2006), 271-291.  doi: 10.1016/j.matpur.2006.04.005.  Google Scholar

[9] D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford University Press, New York, 1999.   Google Scholar
[10]

C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, How to approximate the heat equation with neumann boundary conditions by nonlocal diffusion problems, Arch. Rational Mech. Anal., 187 (2008), 137-156.  doi: 10.1007/s00205-007-0062-8.  Google Scholar

[11]

M. D'Elia, Q. Du, M. Gunzburger and R. Lehoucq, Nonlocal convection-diffusion problems on bounded domains and finite-range jump processes, Comput. Methods Appl. Math., 17 (2017), 707-722.  doi: 10.1515/cmam-2017-0029.  Google Scholar

[12]

M. D'Elia, M. Perego, P. Bochev and D. Littlewood, A coupling strategy for nonlocal and local diffusion models with mixed volume constraints and boundary conditions, Comput. Math. Appl., 71 (2016), 2218-2230.  doi: 10.1016/j.camwa.2015.12.006.  Google Scholar

[13]

M. D'Elia, D. Ridzal, K. J. Peterson, P. Bochev and M. Shashkov, Optimization-based mesh correction with volume and convexity constraints, J. Comput. Phys., 313 (2016), 455-477.  doi: 10.1016/j.jcp.2016.02.050.  Google Scholar

[14]

Q. Du, X. H. Li, J. Lu and X. Tian, A quasi-nonlocal coupling method for nonlocal and local diffusion models, SIAM J. Numer. Anal., 56 (2018), 1386-1404.  doi: 10.1137/17M1124012.  Google Scholar

[15]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, In Trends in Nonlinear Analysis, 153-191, Springer, Berlin, 2003.  Google Scholar

[16]

C. G. Gal and M. Warma, Nonlocal transmission problems with fractional diffusion and boundary conditions on non-smooth interfaces, Comm. Partial Differential Equations, 42 (2017), 579-625.  doi: 10.1080/03605302.2017.1295060.  Google Scholar

[17]

A. Gárriz, F. Quirós and J. D. Rossi, Coupling local and nonlocal evolution equations, Calc. Var. Par. Diff. Equations., 59 (2020), Paper No. 112, 24 pp. arXiv: 1903.07108. doi: 10.1007/s00526-020-01771-z.  Google Scholar

[18]

C. Kipnis and C. Landim, Scaling Limits of Interacting Particle Systems, Grundlehren der mathematischen Wissenschaften, Springer, Berlin, New York, 1999. doi: 10.1007/978-3-662-03752-2.  Google Scholar

[19]

D. Kriventsov, Regularity for a local-nonlocal transmission problem, Arch. Ration. Mech. Anal., 217 (2015), 1103-1195.  doi: 10.1007/s00205-015-0851-4.  Google Scholar

[20]

T. M. Liggett, Interacting Particle Systems, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, 1985. doi: 10.1007/978-1-4613-8542-4.  Google Scholar

[21]

M. C. Pereira, Nonlocal evolution equations in perforated domains, Math. Methods Appl. Sciences, 41 (2018), 6368-6377.  doi: 10.1002/mma.5144.  Google Scholar

[22]

M. C. Pereira and J. D. Rossi, An obstacle problem for nonlocal equations in perforated domains, Potential Analysis, 48 (2018), 361-373.  doi: 10.1007/s11118-017-9639-5.  Google Scholar

[23]

M. C. Pereira and J. D. Rossi, Nonlocal problems in perforated domains, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 305-340.  doi: 10.1017/prm.2018.130.  Google Scholar

[24]

R. W. Schwab, Periodic homogenization for nonlinear integro-differential equations, SIAM J. Math. Anal., 42 (2010), 2652-2680.  doi: 10.1137/080737897.  Google Scholar

[25]

L. Tartar, The General Theory of Homogenization. A Personalized Introduction, Lecture Notes of the Unione Matematica Italiana, Springer-Verlag, 2009. doi: 10.1007/978-3-642-05195-1.  Google Scholar

[26]

V. S. Varadarajan, Weak convergence of measures on separable metric spaces, The Indian Journal of Statistics., 19 (1958), 15-22.   Google Scholar

[27]

M. Waurick, Homogenization in fractional elasticity, SIAM J. Math. Anal., 46 (2014), 1551-1576.  doi: 10.1137/130941596.  Google Scholar

[28] D. Williams, Probability with Martingales, Cambridge University Press, 1991.  doi: 10.1017/CBO9780511813658.  Google Scholar

show all references

References:
[1]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, vol. 165. AMS, 2010. doi: 10.1090/surv/165.  Google Scholar

[2]

P. W. Bates and A. Chmaj, An integrodifferential model for phase transitions: Stationary solutions in higher dimensions, J. Statist. Phys., 95 (1999), 1119-1139.  doi: 10.1023/A:1004514803625.  Google Scholar

[3]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland Publishing Company, 1978.  Google Scholar

[4]

L. Caffarelli and A. Mellet, Random homogenization of fractional obstacle problems, Netw. Heterog. Media, 3 (2008), 523-554.  doi: 10.3934/nhm.2008.3.523.  Google Scholar

[5]

M. Capanna and J. D. Rossi, Mixing local and nonlocal evolution equations, Preprint, arXiv: 2003.03407v1. Google Scholar

[6]

C. Carrillo and P. Fife, Spatial effects in discrete generation population models., J. Math. Biol., 50 (2005), 161-188.  doi: 10.1007/s00285-004-0284-4.  Google Scholar

[7]

P. Cazeaux and C. Grandmont, Homogenization of a multiscale viscoelastic model with nonlocal damping, application to the human lungs, Math. Models Methods Appl. Sci., 25 (2015), 1125-1177.  doi: 10.1142/S0218202515500293.  Google Scholar

[8]

E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl., 86 (2006), 271-291.  doi: 10.1016/j.matpur.2006.04.005.  Google Scholar

[9] D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford University Press, New York, 1999.   Google Scholar
[10]

C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, How to approximate the heat equation with neumann boundary conditions by nonlocal diffusion problems, Arch. Rational Mech. Anal., 187 (2008), 137-156.  doi: 10.1007/s00205-007-0062-8.  Google Scholar

[11]

M. D'Elia, Q. Du, M. Gunzburger and R. Lehoucq, Nonlocal convection-diffusion problems on bounded domains and finite-range jump processes, Comput. Methods Appl. Math., 17 (2017), 707-722.  doi: 10.1515/cmam-2017-0029.  Google Scholar

[12]

M. D'Elia, M. Perego, P. Bochev and D. Littlewood, A coupling strategy for nonlocal and local diffusion models with mixed volume constraints and boundary conditions, Comput. Math. Appl., 71 (2016), 2218-2230.  doi: 10.1016/j.camwa.2015.12.006.  Google Scholar

[13]

M. D'Elia, D. Ridzal, K. J. Peterson, P. Bochev and M. Shashkov, Optimization-based mesh correction with volume and convexity constraints, J. Comput. Phys., 313 (2016), 455-477.  doi: 10.1016/j.jcp.2016.02.050.  Google Scholar

[14]

Q. Du, X. H. Li, J. Lu and X. Tian, A quasi-nonlocal coupling method for nonlocal and local diffusion models, SIAM J. Numer. Anal., 56 (2018), 1386-1404.  doi: 10.1137/17M1124012.  Google Scholar

[15]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, In Trends in Nonlinear Analysis, 153-191, Springer, Berlin, 2003.  Google Scholar

[16]

C. G. Gal and M. Warma, Nonlocal transmission problems with fractional diffusion and boundary conditions on non-smooth interfaces, Comm. Partial Differential Equations, 42 (2017), 579-625.  doi: 10.1080/03605302.2017.1295060.  Google Scholar

[17]

A. Gárriz, F. Quirós and J. D. Rossi, Coupling local and nonlocal evolution equations, Calc. Var. Par. Diff. Equations., 59 (2020), Paper No. 112, 24 pp. arXiv: 1903.07108. doi: 10.1007/s00526-020-01771-z.  Google Scholar

[18]

C. Kipnis and C. Landim, Scaling Limits of Interacting Particle Systems, Grundlehren der mathematischen Wissenschaften, Springer, Berlin, New York, 1999. doi: 10.1007/978-3-662-03752-2.  Google Scholar

[19]

D. Kriventsov, Regularity for a local-nonlocal transmission problem, Arch. Ration. Mech. Anal., 217 (2015), 1103-1195.  doi: 10.1007/s00205-015-0851-4.  Google Scholar

[20]

T. M. Liggett, Interacting Particle Systems, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, 1985. doi: 10.1007/978-1-4613-8542-4.  Google Scholar

[21]

M. C. Pereira, Nonlocal evolution equations in perforated domains, Math. Methods Appl. Sciences, 41 (2018), 6368-6377.  doi: 10.1002/mma.5144.  Google Scholar

[22]

M. C. Pereira and J. D. Rossi, An obstacle problem for nonlocal equations in perforated domains, Potential Analysis, 48 (2018), 361-373.  doi: 10.1007/s11118-017-9639-5.  Google Scholar

[23]

M. C. Pereira and J. D. Rossi, Nonlocal problems in perforated domains, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 305-340.  doi: 10.1017/prm.2018.130.  Google Scholar

[24]

R. W. Schwab, Periodic homogenization for nonlinear integro-differential equations, SIAM J. Math. Anal., 42 (2010), 2652-2680.  doi: 10.1137/080737897.  Google Scholar

[25]

L. Tartar, The General Theory of Homogenization. A Personalized Introduction, Lecture Notes of the Unione Matematica Italiana, Springer-Verlag, 2009. doi: 10.1007/978-3-642-05195-1.  Google Scholar

[26]

V. S. Varadarajan, Weak convergence of measures on separable metric spaces, The Indian Journal of Statistics., 19 (1958), 15-22.   Google Scholar

[27]

M. Waurick, Homogenization in fractional elasticity, SIAM J. Math. Anal., 46 (2014), 1551-1576.  doi: 10.1137/130941596.  Google Scholar

[28] D. Williams, Probability with Martingales, Cambridge University Press, 1991.  doi: 10.1017/CBO9780511813658.  Google Scholar
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