{"month":"04","day":"09","type":"preprint","year":"2021","oa":1,"user_id":"D865714E-FA4E-11E9-B85B-F5C5E5697425","article_processing_charge":"No","date_published":"2021-04-09T00:00:00Z","oa_version":"Preprint","language":[{"iso":"eng"}],"date_created":"2021-10-23T10:50:55Z","title":"Quantitative nonlinear homogenization: control of oscillations","publication_status":"submitted","status":"public","external_id":{"arxiv":["2104.04263"]},"citation":{"ista":"Clozeau N, Gloria A. Quantitative nonlinear homogenization: control of oscillations. arXiv, 2104.04263.","ama":"Clozeau N, Gloria A. Quantitative nonlinear homogenization: control of oscillations. <i>arXiv</i>.","short":"N. Clozeau, A. Gloria, ArXiv (n.d.).","chicago":"Clozeau, Nicolas, and Antoine Gloria. “Quantitative Nonlinear Homogenization: Control of Oscillations.” <i>ArXiv</i>, n.d.","mla":"Clozeau, Nicolas, and Antoine Gloria. “Quantitative Nonlinear Homogenization: Control of Oscillations.” <i>ArXiv</i>, 2104.04263.","apa":"Clozeau, N., &#38; Gloria, A. (n.d.). Quantitative nonlinear homogenization: control of oscillations. <i>arXiv</i>.","ieee":"N. Clozeau and A. Gloria, “Quantitative nonlinear homogenization: control of oscillations,” <i>arXiv</i>. ."},"author":[{"first_name":"Nicolas","id":"fea1b376-906f-11eb-847d-b2c0cf46455b","full_name":"Clozeau, Nicolas","last_name":"Clozeau"},{"first_name":"Antoine","full_name":"Gloria, Antoine","last_name":"Gloria"}],"department":[{"_id":"JuFi"}],"acknowledgement":"The authors warmly thank Mitia Duerinckx for discussions on annealed estimates, and Mathias Schäffner for pointing out that the conditions of [14] apply to  ̄a in the setting of Theorem 2.2 and for discussions on regularity theory for operators with non-standard growth conditions. The authors received financial support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement n◦ 864066).","article_number":"2104.04263","main_file_link":[{"url":"https://arxiv.org/abs/2104.04263","open_access":"1"}],"date_updated":"2021-10-28T15:44:05Z","publication":"arXiv","_id":"10174","abstract":[{"lang":"eng","text":"Quantitative stochastic homogenization of linear elliptic operators is by now well-understood. In this contribution we move forward to the nonlinear setting of monotone operators with p-growth. This first work is dedicated to a quantitative two-scale expansion result. Fluctuations will be addressed in companion articles. By treating the range of exponents 2≤p<∞ in dimensions d≤3, we are able to consider genuinely nonlinear elliptic equations and systems such as −∇⋅A(x)(1+|∇u|p−2)∇u=f (with A random, non-necessarily symmetric) for the first time. When going from p=2 to p>2, the main difficulty is to analyze the associated linearized operator, whose coefficients are degenerate, unbounded, and depend on the random input A via the solution of a nonlinear equation. One of our main achievements is the control of this intricate nonlinear dependence, leading to annealed Meyers' estimates for the linearized operator, which are key to the quantitative two-scale expansion result."}]}