Homogenization for nonlocal problems with smooth kernels

Monia Capanna; Jean C. Nakasato; Marcone C. Pereira; Julio D. Rossi

doi:10.3934/dcds.2020385

Previous Article
Properties of multicorrelation sequences and large returns under some ergodicity assumptions
DCDS Home
This Issue
Next Article
Forward untangling and applications to the uniqueness problem for the continuity equation

June 2021, 41(6): 2777-2808. 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 June 2021 Early access 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 and Continuous Dynamical Systems, 2021, 41 (6) : 2777-2808. 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.
[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.
[3]	A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland Publishing Company, 1978.
[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.
[5]	M. Capanna and J. D. Rossi, Mixing local and nonlocal evolution equations, Preprint, arXiv: 2003.03407v1.
[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.
[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.
[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.
[9]	D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford University Press, New York, 1999.
[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.
[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.
[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.
[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.
[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.
[15]	P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, In Trends in Nonlinear Analysis, 153-191, Springer, Berlin, 2003.
[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.
[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.
[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.
[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.
[20]	T. M. Liggett, Interacting Particle Systems, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, 1985. doi: 10.1007/978-1-4613-8542-4.
[21]	M. C. Pereira, Nonlocal evolution equations in perforated domains, Math. Methods Appl. Sciences, 41 (2018), 6368-6377. doi: 10.1002/mma.5144.
[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.
[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.
[24]	R. W. Schwab, Periodic homogenization for nonlinear integro-differential equations, SIAM J. Math. Anal., 42 (2010), 2652-2680. doi: 10.1137/080737897.
[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.
[26]	V. S. Varadarajan, Weak convergence of measures on separable metric spaces, The Indian Journal of Statistics., 19 (1958), 15-22.
[27]	M. Waurick, Homogenization in fractional elasticity, SIAM J. Math. Anal., 46 (2014), 1551-1576. doi: 10.1137/130941596.
[28]	D. Williams, Probability with Martingales, Cambridge University Press, 1991. doi: 10.1017/CBO9780511813658.

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.
[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.
[3]	A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland Publishing Company, 1978.
[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.
[5]	M. Capanna and J. D. Rossi, Mixing local and nonlocal evolution equations, Preprint, arXiv: 2003.03407v1.
[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.
[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.
[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.
[9]	D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford University Press, New York, 1999.
[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.
[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.
[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.
[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.
[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.
[15]	P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, In Trends in Nonlinear Analysis, 153-191, Springer, Berlin, 2003.
[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.
[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.
[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.
[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.
[20]	T. M. Liggett, Interacting Particle Systems, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, 1985. doi: 10.1007/978-1-4613-8542-4.
[21]	M. C. Pereira, Nonlocal evolution equations in perforated domains, Math. Methods Appl. Sciences, 41 (2018), 6368-6377. doi: 10.1002/mma.5144.
[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.
[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.
[24]	R. W. Schwab, Periodic homogenization for nonlinear integro-differential equations, SIAM J. Math. Anal., 42 (2010), 2652-2680. doi: 10.1137/080737897.
[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.
[26]	V. S. Varadarajan, Weak convergence of measures on separable metric spaces, The Indian Journal of Statistics., 19 (1958), 15-22.
[27]	M. Waurick, Homogenization in fractional elasticity, SIAM J. Math. Anal., 46 (2014), 1551-1576. doi: 10.1137/130941596.
[28]	D. Williams, Probability with Martingales, Cambridge University Press, 1991. doi: 10.1017/CBO9780511813658.

[1]	Sunghan Kim, Ki-Ahm Lee, Henrik Shahgholian. Homogenization of the boundary value for the Dirichlet problem. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 6843-6864. doi: 10.3934/dcds.2019234
[2]	Fioralba Cakoni, Houssem Haddar, Isaac Harris. Homogenization of the transmission eigenvalue problem for periodic media and application to the inverse problem. Inverse Problems and Imaging, 2015, 9 (4) : 1025-1049. doi: 10.3934/ipi.2015.9.1025
[3]	Wenxian Shen, Zhongwei Shen. Transition fronts in nonlocal Fisher-KPP equations in time heterogeneous media. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1193-1213. doi: 10.3934/cpaa.2016.15.1193
[4]	Ihsane Bikri, Ronald B. Guenther, Enrique A. Thomann. The Dirichlet to Neumann map - An application to the Stokes problem in half space. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 221-230. doi: 10.3934/dcdss.2010.3.221
[5]	Francis J. Chung. Partial data for the Neumann-Dirichlet magnetic Schrödinger inverse problem. Inverse Problems and Imaging, 2014, 8 (4) : 959-989. doi: 10.3934/ipi.2014.8.959
[6]	Mamadou Sango. Homogenization of the Neumann problem for a quasilinear elliptic equation in a perforated domain. Networks and Heterogeneous Media, 2010, 5 (2) : 361-384. doi: 10.3934/nhm.2010.5.361
[7]	Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199
[8]	Chuanqiang Chen, Li Chen, Xinqun Mei, Ni Xiang. The Neumann problem for a class of mixed complex Hessian equations. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022049
[9]	Antonin Chambolle, Gilles Thouroude. Homogenization of interfacial energies and construction of plane-like minimizers in periodic media through a cell problem. Networks and Heterogeneous Media, 2009, 4 (1) : 127-152. doi: 10.3934/nhm.2009.4.127
[10]	Yu-Hao Liang, Shin-Hwa Wang. Classification and evolution of bifurcation curves for a one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 1075-1105. doi: 10.3934/dcds.2020071
[11]	Grégoire Allaire, Yves Capdeboscq, Marjolaine Puel. Homogenization of a one-dimensional spectral problem for a singularly perturbed elliptic operator with Neumann boundary conditions. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 1-31. doi: 10.3934/dcdsb.2012.17.1
[12]	Fanghua Lin, Xiaodong Yan. A type of homogenization problem. Discrete and Continuous Dynamical Systems, 2003, 9 (1) : 1-30. doi: 10.3934/dcds.2003.9.1
[13]	Bo Guan, Heming Jiao. The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 701-714. doi: 10.3934/dcds.2016.36.701
[14]	Salvatore A. Marano, Sunra J. N. Mosconi. Some recent results on the Dirichlet problem for $(p, q)$-Laplace equations. Discrete and Continuous Dynamical Systems - S, 2018, 11 (2) : 279-291. doi: 10.3934/dcdss.2018015
[15]	Chunhui Qiu, Rirong Yuan. On the Dirichlet problem for fully nonlinear elliptic equations on annuli of metric cones. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5707-5730. doi: 10.3934/dcds.2017247
[16]	Martino Bardi, Paola Mannucci. On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations. Communications on Pure and Applied Analysis, 2006, 5 (4) : 709-731. doi: 10.3934/cpaa.2006.5.709
[17]	Goro Akagi. Energy solutions of the Cauchy-Neumann problem for porous medium equations. Conference Publications, 2009, 2009 (Special) : 1-10. doi: 10.3934/proc.2009.2009.1
[18]	Masato Hashizume, Chun-Hsiung Hsia, Gyeongha Hwang. On the Neumann problem of Hardy-Sobolev critical equations with the multiple singularities. Communications on Pure and Applied Analysis, 2019, 18 (1) : 301-322. doi: 10.3934/cpaa.2019016
[19]	Silvia Frassu. Nonlinear Dirichlet problem for the nonlocal anisotropic operator $ L_K $. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1847-1867. doi: 10.3934/cpaa.2019086
[20]	Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101

American Institute of Mathematical Sciences

Homogenization for nonlocal problems with smooth kernels

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Homogenization for nonlocal problems with smooth kernels

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors