no. G8
Équations aux dérivées partielles, Probabilités
A short proof of Gevrey regularity for homogenized coefficients of the Poisson point process
Duerinckx, Mitia1 ; Gloria, Antoine2
1 Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d’Orsay, 91405 Orsay, France & Université Libre de Bruxelles, Département de Mathématique, 1050 Brussels, Belgium
2 Sorbonne Université, CNRS, Université de Paris, Laboratoire Jacques-Louis Lions, 75005 Paris, France & Institut Universitaire de France & Université Libre de Bruxelles, Département de Mathématique, 1050 Brussels, Belgium
Comptes Rendus. Mathématique, Tome 360 (2022) no. G8, pp. 909-918.

In this short note we capitalize on and complete our previous results on the regularity of the homogenized coefficients for Bernoulli perturbations by addressing the case of the Poisson point process, for which the crucial uniform local finiteness assumption fails. In particular, we strengthen the qualitative regularity result first obtained in this setting by the first author to Gevrey regularity of order 2. The new ingredient is a fine application of properties of Poisson point processes, in a form recently used by Giunti, Gu, Mourrat, and Nitzschner.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.354
Classification : 35R60, 60G55
Duerinckx, Mitia 1 ; Gloria, Antoine 2

1 Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d’Orsay, 91405 Orsay, France & Université Libre de Bruxelles, Département de Mathématique, 1050 Brussels, Belgium
2 Sorbonne Université, CNRS, Université de Paris, Laboratoire Jacques-Louis Lions, 75005 Paris, France & Institut Universitaire de France & Université Libre de Bruxelles, Département de Mathématique, 1050 Brussels, Belgium 
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Duerinckx, Mitia; Gloria, Antoine. A short proof of Gevrey regularity for homogenized coefficients of the Poisson point process. Comptes Rendus. Mathématique, Tome 360 (2022) no. G8, pp. 909-918. doi : 10.5802/crmath.354. http://www.numdam.org/articles/10.5802/crmath.354/

[1] Anantharaman, Arnaud; Le Bris, Claude A numerical approach related to defect-type theories for some weakly random problems in homogenization, Multiscale Model. Simul., Volume 9 (2011) no. 2, pp. 513-544 | DOI | MR | Zbl

[2] Anantharaman, Arnaud; Le Bris, Claude Elements of mathematical foundations for numerical approaches for weakly random homogenization problems, Commun. Comput. Phys., Volume 11 (2012) no. 4, pp. 1103-1143 | DOI | MR | Zbl

[3] Duerinckx, Mitia Topics in the Mathematics of Disordered Media, Ph. D. Thesis, Université Libre de Bruxelles & Université Pierre et Marie Curie (2017) | Numdam

[4] Duerinckx, Mitia; Gloria, Antoine Analyticity of homogenized coefficients under Bernoulli perturbations and the Clausius-Mossotti formulas, Arch. Ration. Mech. Anal., Volume 220 (2016) no. 1, pp. 297-361 | DOI | MR | Zbl

[5] Giunti, Arianna; Gu, Chenlin; Mourrat, Jean-Christophe Quantitative homogenization of interacting particle systems (2011) | arXiv

[6] Giunti, Arianna; Gu, Chenlin; Mourrat, Jean-Christophe; Nitzschner, Maximilian Smoothness of the diffusion coefficients for particle systems in continuous space (2021) | arXiv

[7] Gloria, Antoine; Habibi, Zakaria Reduction in the resonance error in numerical homogenization II: Correctors and extrapolation, Found. Comput. Math., Volume 16 (2016) no. 1, pp. 217-296 | DOI | MR | Zbl

[8] Gloria, Antoine; Otto, Felix Quantitative results on the corrector equation in stochastic homogenization, J. Eur. Math. Soc., Volume 19 (2017) no. 11, pp. 3489-3548 | DOI | MR | Zbl

[9] Last, Günter; Penrose, Mathew Lectures on the Poisson process, Institute of Mathematical Statistics Textbooks, 7, Cambridge University Press, 2018

[10] Mourrat, Jean-Christophe First-order expansion of homogenized coefficients under Bernoulli perturbations, J. Math. Pures Appl., Volume 103 (2015) no. 1, pp. 68-101 | DOI | MR | Zbl

[11] Torquato, Salvatore Random heterogeneous materials. Microstructure and macroscopic properties, Interdisciplinary Applied Mathematics, 16, Springer, 2002 | DOI

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