Comptes Rendus
Équations aux dérivées partielles, Probabilités
A short proof of Gevrey regularity for homogenized coefficients of the Poisson point process
Mitia Duerinckx1 ; Antoine Gloria2
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, Volume 360 (2022), 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.

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Accepté le :
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DOI : 10.5802/crmath.354
Classification : 35R60, 60G55
Mitia Duerinckx 1 ; Antoine Gloria 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
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Mitia Duerinckx; Antoine Gloria. A short proof of Gevrey regularity for homogenized coefficients of the Poisson point process. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 909-918. doi : 10.5802/crmath.354. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.354/

[1] Arnaud Anantharaman; Claude Le Bris 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 | Zbl

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

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

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

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

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

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

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

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

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

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

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