{"publication_identifier":{"eissn":["1432-0673"],"issn":["0003-9527"]},"abstract":[{"text":"We derive optimal-order homogenization rates for random nonlinear elliptic PDEs with monotone nonlinearity in the uniformly elliptic case. More precisely, for a random monotone operator on \\mathbb {R}^d with stationary law (that is spatially homogeneous statistics) and fast decay of correlations on scales larger than the microscale \\varepsilon >0, we establish homogenization error estimates of the order \\varepsilon in case d\\geqq 3, and of the order \\varepsilon |\\log \\varepsilon |^{1/2} in case d=2. Previous results in nonlinear stochastic homogenization have been limited to a small algebraic rate of convergence \\varepsilon ^\\delta . We also establish error estimates for the approximation of the homogenized operator by the method of representative volumes of the order (L/\\varepsilon )^{-d/2} for a representative volume of size L. Our results also hold in the case of systems for which a (small-scale) C^{1,\\alpha } regularity theory is available.","lang":"eng"}],"ddc":["530"],"year":"2021","issue":"1","publisher":"Springer Nature","article_type":"original","status":"public","department":[{"_id":"JuFi"}],"oa_version":"Published Version","type":"journal_article","citation":{"mla":"Fischer, Julian L., and Stefan Neukamm. “Optimal Homogenization Rates in Stochastic Homogenization of Nonlinear Uniformly Elliptic Equations and Systems.” <i>Archive for Rational Mechanics and Analysis</i>, vol. 242, no. 1, Springer Nature, 2021, pp. 343–452, doi:<a href=\"https://doi.org/10.1007/s00205-021-01686-9\">10.1007/s00205-021-01686-9</a>.","ieee":"J. L. Fischer and S. Neukamm, “Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems,” <i>Archive for Rational Mechanics and Analysis</i>, vol. 242, no. 1. Springer Nature, pp. 343–452, 2021.","ama":"Fischer JL, Neukamm S. Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems. <i>Archive for Rational Mechanics and Analysis</i>. 2021;242(1):343-452. doi:<a href=\"https://doi.org/10.1007/s00205-021-01686-9\">10.1007/s00205-021-01686-9</a>","ista":"Fischer JL, Neukamm S. 2021. Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems. Archive for Rational Mechanics and Analysis. 242(1), 343–452.","short":"J.L. Fischer, S. Neukamm, Archive for Rational Mechanics and Analysis 242 (2021) 343–452.","apa":"Fischer, J. L., &#38; Neukamm, S. (2021). Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems. <i>Archive for Rational Mechanics and Analysis</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00205-021-01686-9\">https://doi.org/10.1007/s00205-021-01686-9</a>","chicago":"Fischer, Julian L, and Stefan Neukamm. “Optimal Homogenization Rates in Stochastic Homogenization of Nonlinear Uniformly Elliptic Equations and Systems.” <i>Archive for Rational Mechanics and Analysis</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s00205-021-01686-9\">https://doi.org/10.1007/s00205-021-01686-9</a>."},"date_created":"2021-12-16T12:12:33Z","has_accepted_license":"1","scopus_import":"1","article_processing_charge":"Yes (via OA deal)","keyword":["Mechanical Engineering","Mathematics (miscellaneous)","Analysis"],"isi":1,"acknowledgement":"Open access funding provided by Institute of Science and Technology (IST Austria). SN acknowledges partial support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – project number 405009441.","month":"06","date_updated":"2023-08-17T06:23:21Z","title":"Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"oa":1,"language":[{"iso":"eng"}],"quality_controlled":"1","license":"https://creativecommons.org/licenses/by/4.0/","intvolume":"       242","file":[{"content_type":"application/pdf","creator":"cchlebak","date_updated":"2021-12-16T14:58:08Z","file_size":1640121,"relation":"main_file","access_level":"open_access","file_name":"2021_ArchRatMechAnalysis_Fischer.pdf","checksum":"cc830b739aed83ca2e32c4e0ce266a4c","file_id":"10558","date_created":"2021-12-16T14:58:08Z","success":1}],"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","file_date_updated":"2021-12-16T14:58:08Z","page":"343-452","day":"30","publication_status":"published","publication":"Archive for Rational Mechanics and Analysis","date_published":"2021-06-30T00:00:00Z","external_id":{"arxiv":["1908.02273"],"isi":["000668431200001"]},"_id":"10549","volume":242,"author":[{"last_name":"Fischer","full_name":"Fischer, Julian L","orcid":"0000-0002-0479-558X","id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87","first_name":"Julian L"},{"last_name":"Neukamm","full_name":"Neukamm, Stefan","first_name":"Stefan"}],"doi":"10.1007/s00205-021-01686-9"}