Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-05T11:12:00.666Z Has data issue: false hasContentIssue false

Nonlocal problems in perforated domains

Published online by Cambridge University Press:  25 January 2019

Marcone C. Pereira
Affiliation:
Dpto. de Matemática Aplicada, IME, Universidade de São Paulo, Rua do Matão 1010, São Paulo - SP, Brazil ([email protected])
Julio D. Rossi
Affiliation:
Dpto. de Matemáticas, FCEyN, Universidad de Buenos Aires, Ciudad Universitaria Pab 1, 1428 Buenos Aires, Argentina ([email protected])

Abstract

In this paper, we analyse nonlocal equations in perforated domains. We consider nonlocal problems of the form $f(x) = \int \nolimits _{B} J(x-y) (u(y) - u(x)) {\rm d}y$ with x in a perforated domain $\Omega ^\epsilon \subset \Omega $. Here J is a nonsingular kernel. We think about $\Omega ^\epsilon $ as a fixed set Ω from where we have removed a subset that we call the holes. We deal both with the Neumann and Dirichlet conditions in the holes and assume a Dirichlet condition outside Ω. In the latter case we impose that u vanishes in the holes but integrate in the whole ℝN (B = ℝN) and in the former we just consider integrals in ℝN minus the holes ($B={\open R} ^N \setminus (\Omega \setminus \Omega ^\epsilon )$). Assuming weak convergence of the holes, specifically, under the assumption that the characteristic function of $\Omega ^\epsilon $ has a weak limit, $\chi _{\epsilon } \rightharpoonup {\cal X}$ weakly* in L(Ω), we analyse the limit as ε → 0 of the solutions to the nonlocal problems proving that there is a nonlocal limit problem. In the case in which the holes are periodically removed balls, we obtain that the critical radius is of the order of the size of the typical cell (that gives the period). In addition, in this periodic case, we also study the behaviour of these nonlocal problems when we rescale the kernel in order to approximate local PDE problems.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Allaire, G.. Homogenization of the Stokes flow in a connected porous medium. Asymp. Anal. 2 (1989), 203222.Google Scholar
2Andreu-Vaillo, F., Mazón, J. M., Rossi, J. D. and Toledo, J.. Nonlocal diffusion problems. Mathematical surveys and monographs, vol. 165 (AMS, 2010).CrossRefGoogle Scholar
3Arrieta, J. M. and Bruschi, S. M.. Very rapidly varying boundaries in equations with nonlinear boundary conditions. The case of non uniform Lispschitz deformation. Discrete. Cont. Dyn. Syst. Ser. B. 14 (2010), 327351.Google Scholar
4Barbosa, P. S., Pereira, A. L. and Pereira, M. C.. Continuity of attractors for a family of ${\cal {C}}^1$ perturbations of the square. Annali di Matematica Pura ed Applicata 196 (2017), 13651398.CrossRefGoogle Scholar
5Barles, G., Chasseigne, E. and Imbert, C.. On the Dirichlet problem for second-order elliptic integro-differential equations. Indiana Univ. Math. J. 57 (2008), 213246.CrossRefGoogle Scholar
6Brillard, A., Gómez, D., Lobo, M., Pérez, E. and Shaposhnikovad, T. A.. Boundary homogenization in perforated domains for adsorption problems with an advection term. Appl. Anal. 95 (2016), 15171533.CrossRefGoogle Scholar
7Caffarelli, L. A. and Mellet, A.. Random homogenization of fractional obstacle problems. Netw. Heterog. Media 3 (2008), 523554.CrossRefGoogle Scholar
8Calvo-Jurado, C., Casado-Díaz, J. and Luna-Laynez, M.. Homogenization of nonlinear Dirichlet problems in random perforated domains. Nonlinear Anal. 133 (2016), 250274.CrossRefGoogle Scholar
9Cardone, G. and Khrabustovskyi, A.. Neumann spectral problem in a domain with very corrugated boundary. J. of Diff. Equ. 259 (2015), 23332367.CrossRefGoogle Scholar
10Cazeaux, P. and Grandmont, C.. Homogenization of a multiscale viscoelastic model with nonlocal damping, application to the human lungs. Math. Models Methods Appl. Sci. 25 (2015), 11251177.CrossRefGoogle Scholar
11Cioranescu, D. and Donato, P.. An introduction to homogenization. Oxford lecture series in mathematics and its applications, vol. 17 (Oxford University Press, 1999).Google Scholar
12Cioranescu, D. and Murat, F.. A strange term coming from nowhere. Progress in Nonl. Diff. Eq. and Their Appl. 31 (1997), 4593.Google Scholar
13Cioranescu, D. and Saint Jean Paulin, J.. Homogenization in open sets with holes. J. Math. Anal. Appl. 71 (1979), 590607.CrossRefGoogle Scholar
14Cioranescu, D. and Saint Jean Paulin, J.. Homogenization of reticulated structures. Applied mathematical sciences, vol. 136 (New York: Springer-Verlag, 1999).CrossRefGoogle Scholar
15Cioranescu, D., Damlamian, A., Donato, P., Griso, G. and Zaki, R.. The periodic unfolding method in domains with holes. SIAM J. Math. Anal. 44 (2012), 718760.CrossRefGoogle Scholar
16Chasseigne, E., Felmer, P., Rossi, J. D. and Topp, E.. Fractional decay bounds for nonlocal zero order heat equations. Bull. Lond. Math. Soc. 46 (2014), 943952.CrossRefGoogle Scholar
17Chourabi, I. and Donato, P.. Homogenization and correctors of a class of elliptic problems in perforated domains. Asymptotic Anal. 92 (2015), 143.CrossRefGoogle Scholar
18Cortazar, C., Elgueta, M., Rossi, J. D. and Wolanski, N.. How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems. Arch. Ration. Mech. Anal. 187 (2008), 137156.CrossRefGoogle Scholar
19Cortazar, C., Elgueta, M. and Rossi, J. D.. Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions. Israel J. Math. 170 (2009), 5360.CrossRefGoogle Scholar
20Courant, R. and Hilbert, D.. Methods of mathematical physics, vol. I (New York: Interscience, 1953).Google Scholar
21Du, Q., Gunzburger, M., Lehoucq, R. B. and Zhou, K.. A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws. Math. Models Methods Appl. Sci. 23 (2013), 493540.CrossRefGoogle Scholar
22García Melián, J. and Rossi, J. D.. On the principal eigenvalue of some nonlocal diffusion problems. J. Differ. Equ. 246 (2009), 2138.CrossRefGoogle Scholar
23Ignat, L. I., Pinasco, D., Rossi, J. D. and San Antolin, A.. Decay estimates for nonlinear nonlocal diffusion problems in the whole space. J. Anal. Math. 122 (2014), 375401.CrossRefGoogle Scholar
24Ignat, L. I., Ignat, T. and Stancu-Dumitru, D.. A compactness tool for the analysis of nonlocal evolution equations. SIAM J. Math. Anal. 47 (2015), 13301354.CrossRefGoogle Scholar
25Lehoucq, R. B. and Silling, S. A.. Force flux and the peridynamic stress tensor. J. Mech. Phys. Solids 56 (2008), 15661577.CrossRefGoogle Scholar
26Mocenni, C., Sparacino, E. and Zubelli, J. P.. Effective rough boundary parametrization for reaction-diffusion systems. Appl. Anal. Discr. Math. 8 (2014), 3359.CrossRefGoogle Scholar
27Nandakumaran, A. K., Prakash, R. and Sardar, B. C.. Periodic Controls in an Oscillating Domain: Controls via Unfolding and Homogenization. SIAM J. Contr. Optim. 53 (2015), 32453269.CrossRefGoogle Scholar
28Necas, J.. Les méthodes directes en théorie des équations elliptiques (Paris: Masson, 1967).Google Scholar
29Nguetseng, G.. A General Convergence Result for a Functional Related to the Theory of Homogenization. SIAM J. Math. Anal. 20 (1989), 608623.CrossRefGoogle Scholar
30Nguetseng, G.. Homogenization in perforated domains beyond the periodic setting. J. Math. Anal. Appl. 289 (2004), 608628.CrossRefGoogle Scholar
31Pereira, M. C. and Rossi, J. D.. Nonlocal problems in thin domains. J. Differ. Equ. 263 (2017), 17251754.CrossRefGoogle Scholar
32Pereira, M. C. and Rossi, J. D.. An Obstacle Problem for Nonlocal Equations in Perforated Domains. Potential Anal. 49 (2018), 361373.CrossRefGoogle Scholar
33Pérez, M. E., Shaposhnikova, T. A. and Zubova, M. N.. A homogenization problem in a domain perforated by tiny isoperimetric holes with nonlinear Robin type boundary conditions. Dokl. Math. 90 (2014), 489494.CrossRefGoogle Scholar
34Rauch, J. and Taylor, M.. Potential and scattering theory on wildly perturbed domains. J. Funct. Anal. 18 (1975), 2759.CrossRefGoogle Scholar
35Rodríguez-Bernal, A. and Sastre-Gómez, S.. Linear nonlocal diffusion problems in metric measure spaces. The R. Soc. Edinb. Proc. A 146 (2016), 833863.CrossRefGoogle Scholar
36Sanchez-Palencia, E.. Non homogeneous media and vibration theory. Lecture Notes in Physics, vol. 127 (Berlin: Springer, 1980).Google Scholar
37Schwab, R. W.. Periodic homogenization for nonlinear integro-differential equations. SIAM J. Math. Anal. 42 (2010), 26522680.CrossRefGoogle Scholar
38Waurick, M.. Homogenization in fractional elasticity. SIAM J. Math. Anal. 46 (2014), 15511576.CrossRefGoogle Scholar