# American Institute of Mathematical Sciences

September  2021, 16(3): 341-375. doi: 10.3934/nhm.2021009

## Convergence rates for the homogenization of the Poisson problem in randomly perforated domains

 Imperial College London, Department of Mathematics, London, UK

Received  December 2020 Revised  March 2021 Published  September 2021 Early access  April 2021

In this paper we provide converge rates for the homogenization of the Poisson problem with Dirichlet boundary conditions in a randomly perforated domain of $\mathbb{R}^d$, $d \geqslant 3$. We assume that the holes that perforate the domain are spherical and are generated by a rescaled marked point process $(\Phi, \mathcal{R})$. The point process $\Phi$ generating the centres of the holes is either a Poisson point process or the lattice $\mathbb{Z}^d$; the marks $\mathcal{R}$ generating the radii are unbounded i.i.d random variables having finite $(d-2+\beta)$-moment, for $\beta > 0$. We study the rate of convergence to the homogenized solution in terms of the parameter $\beta$. We stress that, for low values of $\beta$, the balls generating the holes may overlap with overwhelming probability.

Citation: Arianna Giunti. Convergence rates for the homogenization of the Poisson problem in randomly perforated domains. Networks and Heterogeneous Media, 2021, 16 (3) : 341-375. doi: 10.3934/nhm.2021009
##### References:
 [1] G. Allaire, Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes, Arch. Rational Mech. Anal., 113 (1990), 209-259.  doi: 10.1007/BF00375065. [2] 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. [3] L. A. Caffarelli and A. Mellet, Random homogenization of an obstacle problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 375-395.  doi: 10.1016/j.anihpc.2007.09.001. [4] C. Calvo-Jurado, J. Casado-Díaz and M. Luna-Laynez, Homogenization of nonlinear Dirichlet problems in random perforated domains, Nonlinear Anal., 133 (2016), 250-274.  doi: 10.1016/j.na.2015.12.008. [5] K. Carrapatoso and M. Hillairet, On the derivation of a Stokes-Brinkman problem from Stokes equations around a random array of moving spheres, Comm. Math. Phys., 373 (2020), 265-325.  doi: 10.1007/s00220-019-03637-8. [6] D. Cioranescu and F. Murat, A strange term coming from nowhere, Topics in the Mathematical Modelling of Composite Materials, Progr. Nonlinear Differential Equations Appl., 31, Birkhäuser Boston, Boston, MA, 1997, 45–93. doi: 10.1007/978-1-4612-2032-9_4. [7] D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes. Vol.II. General Theory and Structures, Probability and Its Applications, Springer, New York, 2008. doi: 10.1007/978-0-387-49835-5. [8] L. Desvillettes, F. Golse and V. Ricci, The mean-field limit for solid particles in a {N}avier-{S}tokes flow, J. Stat. Phys., 131 (2008), 941-967.  doi: 10.1007/s10955-008-9521-3. [9] R. Figari, E. Orlandi and S. Teta, The Laplacian in regions with many small obstacles: Fluctuations around the limit operator, J. Statist. Phys., 41 (1985), 465-487.  doi: 10.1007/BF01009018. [10] A. Giunti, Derivation of Darcy's law in randomly punctured domains, preprint, arXiv: 2101.01046. [11] A. Giunti and R. M. Höfer, Convergence of the pressure in the homogenization of the Stokes equations in randomly perforated domains, preprint, arXiv: 2003.04724. [12] A. Giunti and R. M. Höfer, Homogenisation for the Stokes equations in randomly perforated domains under almost minimal assumptions on the size of the holes, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 1829-1868.  doi: 10.1016/j.anihpc.2019.06.002. [13] A. Giunti, R. Höfer and J. J. L. Velàzquez, Homogenization for the Poisson equation in randomly perforated domains under minimal assumptions on the size of the holes, Comm. Partial Differential Equations, 43 (2018), 1377-1412.  doi: 10.1080/03605302.2018.1531425. [14] M. Hillairet, On the homogenization of the Stokes problem in a perforated domain, Arch. Ration. Mech. Anal., 230 (2018), 1179-1228.  doi: 10.1007/s00205-018-1268-7. [15] M. Hillairet, A. Moussa and F. Sueur, On the effect of polydispersity and rotation on the Brinkman force induced by a cloud of particles on a viscous incompressible flow, Kinet. Relat. Models, 12 (2019), 681-701.  doi: 10.3934/krm.2019026. [16] R. M. Höfer and J. Jansen, Fluctuations in the homogenization of the Poisson and Stokes equations in perforated domains, preprint, arXiv: 2004.04111. [17] W. Jing, A unified homogenization approach for the Dirichlet problem in perforated domains, SIAM J. Math. Anal., 52 (2020), 1192-1220.  doi: 10.1137/19M1255525. [18] H. Kacimi and F. Murat, Estimation de l'erreur dans des problèmes de Dirichlet où apparait un terme étrange, in Partial Differential Equations and the Calculus of Variations, Vol. II, Progr. Nonlinear Differential Equations Appl., 2, Birkhäuser Boston, Boston, MA, 1989,661–696. doi: 10.1007/978-1-4615-9831-2_6. [19] R. V. Kohn and M. Vogelius, A new model for thin plates with rapidly varying thickness. II. A convergence proof, Quart. Appl. Math., 43 (1985), 1-22.  doi: 10.1090/qam/782253. [20] V. A. Marchenko and E. Y. Khruslov, Homogenization of Partial Differential Equations, Progress in Mathematical Physics, 46, Birkhäuser Boston, Inc., Boston, MA, 2006. doi: 10.1007/978-0-8176-4468-0. [21] G. C. Papanicolaou and S. R. S. Varadhan, Diffusion in regions with many small holes, in Stochastic Differential Systems, Lecture Notes in Control and Information Sci., 25, Springer, Berlin-New York, 1980,190–206. doi: 10.1007/BFb0004010. [22] E. Sánchez-Palencia, On the asymptotics of the fluid flow past an array of fixed obstacles, Internat. J. Engrg. Sci., 20 (1982), 1291-1301.  doi: 10.1016/0020-7225(82)90055-6. [23] L. Tartar, The General Theory of Homogenization. A personalized Introduction, Lecture Notes of the Unione Matematica Italiana, 7, Springer-Verlag, Berlin; UMI, Bologna, 2009. doi: 10.1007/978-3-642-05195-1. [24] V. V. Zhikov and M. E. Rychago, Homogenization of non-linear second-order elliptic equations in perforated domains, Izv. Ross. Akad. Nauk Ser. Mat., 61 (1997), 69-88.  doi: 10.1070/im1997v061n01ABEH000105.

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##### References:
 [1] G. Allaire, Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes, Arch. Rational Mech. Anal., 113 (1990), 209-259.  doi: 10.1007/BF00375065. [2] 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. [3] L. A. Caffarelli and A. Mellet, Random homogenization of an obstacle problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 375-395.  doi: 10.1016/j.anihpc.2007.09.001. [4] C. Calvo-Jurado, J. Casado-Díaz and M. Luna-Laynez, Homogenization of nonlinear Dirichlet problems in random perforated domains, Nonlinear Anal., 133 (2016), 250-274.  doi: 10.1016/j.na.2015.12.008. [5] K. Carrapatoso and M. Hillairet, On the derivation of a Stokes-Brinkman problem from Stokes equations around a random array of moving spheres, Comm. Math. Phys., 373 (2020), 265-325.  doi: 10.1007/s00220-019-03637-8. [6] D. Cioranescu and F. Murat, A strange term coming from nowhere, Topics in the Mathematical Modelling of Composite Materials, Progr. Nonlinear Differential Equations Appl., 31, Birkhäuser Boston, Boston, MA, 1997, 45–93. doi: 10.1007/978-1-4612-2032-9_4. [7] D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes. Vol.II. General Theory and Structures, Probability and Its Applications, Springer, New York, 2008. doi: 10.1007/978-0-387-49835-5. [8] L. Desvillettes, F. Golse and V. Ricci, The mean-field limit for solid particles in a {N}avier-{S}tokes flow, J. Stat. Phys., 131 (2008), 941-967.  doi: 10.1007/s10955-008-9521-3. [9] R. Figari, E. Orlandi and S. Teta, The Laplacian in regions with many small obstacles: Fluctuations around the limit operator, J. Statist. Phys., 41 (1985), 465-487.  doi: 10.1007/BF01009018. [10] A. Giunti, Derivation of Darcy's law in randomly punctured domains, preprint, arXiv: 2101.01046. [11] A. Giunti and R. M. Höfer, Convergence of the pressure in the homogenization of the Stokes equations in randomly perforated domains, preprint, arXiv: 2003.04724. [12] A. Giunti and R. M. Höfer, Homogenisation for the Stokes equations in randomly perforated domains under almost minimal assumptions on the size of the holes, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 1829-1868.  doi: 10.1016/j.anihpc.2019.06.002. [13] A. Giunti, R. Höfer and J. J. L. Velàzquez, Homogenization for the Poisson equation in randomly perforated domains under minimal assumptions on the size of the holes, Comm. Partial Differential Equations, 43 (2018), 1377-1412.  doi: 10.1080/03605302.2018.1531425. [14] M. Hillairet, On the homogenization of the Stokes problem in a perforated domain, Arch. Ration. Mech. Anal., 230 (2018), 1179-1228.  doi: 10.1007/s00205-018-1268-7. [15] M. Hillairet, A. Moussa and F. Sueur, On the effect of polydispersity and rotation on the Brinkman force induced by a cloud of particles on a viscous incompressible flow, Kinet. Relat. Models, 12 (2019), 681-701.  doi: 10.3934/krm.2019026. [16] R. M. Höfer and J. Jansen, Fluctuations in the homogenization of the Poisson and Stokes equations in perforated domains, preprint, arXiv: 2004.04111. [17] W. Jing, A unified homogenization approach for the Dirichlet problem in perforated domains, SIAM J. Math. Anal., 52 (2020), 1192-1220.  doi: 10.1137/19M1255525. [18] H. Kacimi and F. Murat, Estimation de l'erreur dans des problèmes de Dirichlet où apparait un terme étrange, in Partial Differential Equations and the Calculus of Variations, Vol. II, Progr. Nonlinear Differential Equations Appl., 2, Birkhäuser Boston, Boston, MA, 1989,661–696. doi: 10.1007/978-1-4615-9831-2_6. [19] R. V. Kohn and M. Vogelius, A new model for thin plates with rapidly varying thickness. II. A convergence proof, Quart. Appl. Math., 43 (1985), 1-22.  doi: 10.1090/qam/782253. [20] V. A. Marchenko and E. Y. Khruslov, Homogenization of Partial Differential Equations, Progress in Mathematical Physics, 46, Birkhäuser Boston, Inc., Boston, MA, 2006. doi: 10.1007/978-0-8176-4468-0. [21] G. C. Papanicolaou and S. R. S. Varadhan, Diffusion in regions with many small holes, in Stochastic Differential Systems, Lecture Notes in Control and Information Sci., 25, Springer, Berlin-New York, 1980,190–206. doi: 10.1007/BFb0004010. [22] E. Sánchez-Palencia, On the asymptotics of the fluid flow past an array of fixed obstacles, Internat. J. Engrg. Sci., 20 (1982), 1291-1301.  doi: 10.1016/0020-7225(82)90055-6. [23] L. Tartar, The General Theory of Homogenization. A personalized Introduction, Lecture Notes of the Unione Matematica Italiana, 7, Springer-Verlag, Berlin; UMI, Bologna, 2009. doi: 10.1007/978-3-642-05195-1. [24] V. V. Zhikov and M. E. Rychago, Homogenization of non-linear second-order elliptic equations in perforated domains, Izv. Ross. Akad. Nauk Ser. Mat., 61 (1997), 69-88.  doi: 10.1070/im1997v061n01ABEH000105.
The construction of $K_{ \varepsilon, z}$ from the cube $Q_{k, \varepsilon}$. The dashed grey area corresponds to the set $K_{ \varepsilon, z}$, while $Q_{ \varepsilon, z}$ is the square bounded by the thick black line. The green dots are the points of $\Phi^ \varepsilon_\delta$ that fall inside the set $Q_{k-1, z}$ (here bounded by the dashed blue line). The red dots are the points that are outside of $Q_{k, z}$ but whose associated cube intersects $\partial Q_{k, z}$. The black dots are the points that are in $Q_{k, z} \backslash Q_{k-1, z}$. Note that the cubes associated to the black and red dots are typically smaller than the ones associated to the green dots due to the cut-off $\tilde R_{ \varepsilon, z}$
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