
-
Previous Article
Multiple patterns formation for an aggregation/diffusion predator-prey system
- NHM Home
- This Issue
- Next Article
Convergence rates for the homogenization of the Poisson problem in randomly perforated domains
Imperial College London, Department of Mathematics, London, UK |
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.
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. |
show all references
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. |

[1] |
Patrizia Donato, Florian Gaveau. Homogenization and correctors for the wave equation in non periodic perforated domains. Networks and Heterogeneous Media, 2008, 3 (1) : 97-124. doi: 10.3934/nhm.2008.3.97 |
[2] |
M. M. Cavalcanti, V.N. Domingos Cavalcanti, D. Andrade, T. F. Ma. Homogenization for a nonlinear wave equation in domains with holes of small capacity. Discrete and Continuous Dynamical Systems, 2006, 16 (4) : 721-743. doi: 10.3934/dcds.2006.16.721 |
[3] |
Gregory A. Chechkin, Tatiana P. Chechkina, Ciro D’Apice, Umberto De Maio. Homogenization in domains randomly perforated along the boundary. Discrete and Continuous Dynamical Systems - B, 2009, 12 (4) : 713-730. doi: 10.3934/dcdsb.2009.12.713 |
[4] |
Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. Homogenization of variational functionals with nonstandard growth in perforated domains. Networks and Heterogeneous Media, 2010, 5 (2) : 189-215. doi: 10.3934/nhm.2010.5.189 |
[5] |
Hakima Bessaih, Yalchin Efendiev, Florin Maris. Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition. Networks and Heterogeneous Media, 2015, 10 (2) : 343-367. doi: 10.3934/nhm.2015.10.343 |
[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] |
Valeria Chiado Piat, Sergey S. Nazarov, Andrey Piatnitski. Steklov problems in perforated domains with a coefficient of indefinite sign. Networks and Heterogeneous Media, 2012, 7 (1) : 151-178. doi: 10.3934/nhm.2012.7.151 |
[8] |
Li-Ming Yeh. Pointwise estimate for elliptic equations in periodic perforated domains. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1961-1986. doi: 10.3934/cpaa.2015.14.1961 |
[9] |
Wenjia Jing, Olivier Pinaud. A backscattering model based on corrector theory of homogenization for the random Helmholtz equation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5377-5407. doi: 10.3934/dcdsb.2019063 |
[10] |
Leonid Berlyand, Petru Mironescu. Two-parameter homogenization for a Ginzburg-Landau problem in a perforated domain. Networks and Heterogeneous Media, 2008, 3 (3) : 461-487. doi: 10.3934/nhm.2008.3.461 |
[11] |
Oliver Knill. Singular continuous spectrum and quantitative rates of weak mixing. Discrete and Continuous Dynamical Systems, 1998, 4 (1) : 33-42. doi: 10.3934/dcds.1998.4.33 |
[12] |
Tobias Sutter, David Sutter, John Lygeros. Capacity of random channels with large alphabets. Advances in Mathematics of Communications, 2017, 11 (4) : 813-835. doi: 10.3934/amc.2017060 |
[13] |
Peter Hinow, Ami Radunskaya. Ergodicity and loss of capacity for a random family of concave maps. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2193-2210. doi: 10.3934/dcdsb.2016043 |
[14] |
Luis Caffarelli, Antoine Mellet. Random homogenization of fractional obstacle problems. Networks and Heterogeneous Media, 2008, 3 (3) : 523-554. doi: 10.3934/nhm.2008.3.523 |
[15] |
Jie Zhao. Convergence rates for elliptic reiterated homogenization problems. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2787-2795. doi: 10.3934/cpaa.2013.12.2787 |
[16] |
Zhimin Zhang, Yang Yang, Chaolin Liu. On a perturbed compound Poisson model with varying premium rates. Journal of Industrial and Management Optimization, 2017, 13 (2) : 721-736. doi: 10.3934/jimo.2016043 |
[17] |
Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311 |
[18] |
Wei Liu, Shiji Song, Ying Qiao, Han Zhao. The loss-averse newsvendor problem with random supply capacity. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1417-1429. doi: 10.3934/jimo.2016080 |
[19] |
Angkana Rüland, Mikko Salo. Quantitative approximation properties for the fractional heat equation. Mathematical Control and Related Fields, 2020, 10 (1) : 1-26. doi: 10.3934/mcrf.2019027 |
[20] |
Can Zhang. Quantitative unique continuation for the heat equation with Coulomb potentials. Mathematical Control and Related Fields, 2018, 8 (3&4) : 1097-1116. doi: 10.3934/mcrf.2018047 |
2020 Impact Factor: 1.213
Tools
Metrics
Other articles
by authors
[Back to Top]