Homogenization for nonlocal problems with smooth kernels

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

doi:10.3934/dcds.2020385

Article Contents

2021, Volume 41, Issue 6: 2777-2808. Doi: 10.3934/dcds.2020385

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

Homogenization for nonlocal problems with smooth kernels

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
^* Corresponding author: Julio D. Rossi

Received: May 2020

Revised: October 2020

Early access: November 2020

Published: June 2021

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)

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Abstract

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.

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Mathematics Subject Classification: Primary: 45A05, 45C05; Secondary: 45M05.

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[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.