[{"article_processing_charge":"No","day":"16","publication_status":"published","extern":"1","date_updated":"2026-05-06T07:05:16Z","arxiv":1,"title":"A higher-order large scale regularity theory for random elliptic operators","publication":"Communications in Partial Differential Equations","author":[{"id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87","first_name":"Julian L","orcid":"0000-0002-0479-558X","last_name":"Fischer","full_name":"Fischer, Julian L"},{"last_name":"Otto","full_name":"Otto, Felix","first_name":"Felix"}],"page":"1108 - 1148","abstract":[{"text":"We develop a large-scale regularity theory of higher order for divergence-form elliptic equations with heterogeneous coefficient fields a in the context of stochastic homogenization. The large-scale regularity of a-harmonic functions is encoded by Liouville principles: The space of a-harmonic functions that grow at most like a polynomial of degree k has the same dimension as in the constant-coefficient case. This result can be seen as the qualitative side of a large-scale Ck,α-regularity theory, which in the present work is developed in the form of a corresponding Ck,α-“excess decay” estimate: For a given a-harmonic function u on a ball BR, its energy distance on some ball Br to the above space of a-harmonic functions that grow at most like a polynomial of degree k has the natural decay in the radius r above some minimal radius r0. Though motivated by stochastic homogenization, the contribution of this paper is of purely deterministic nature: We work under the assumption that for the given realization a of the coefficient field, the couple (φ, σ) of scalar and vector potentials of the harmonic coordinates, where φ is the usual corrector, grows sublinearly in a mildly quantified way. We then construct “kth-order correctors” and thereby the space of a-harmonic functions that grow at most like a polynomial of degree k, establish the above excess decay, and then the corresponding Liouville principle.","lang":"eng"}],"year":"2016","_id":"1318","publication_identifier":{"eissn":["1532-4133"],"issnl":["0360-5302"]},"oa":1,"citation":{"ieee":"J. L. Fischer and F. Otto, “A higher-order large scale regularity theory for random elliptic operators,” <i>Communications in Partial Differential Equations</i>, vol. 41, no. 7. Taylor &#38; Francis, pp. 1108–1148, 2016.","ista":"Fischer JL, Otto F. 2016. A higher-order large scale regularity theory for random elliptic operators. Communications in Partial Differential Equations. 41(7), 1108–1148.","ama":"Fischer JL, Otto F. A higher-order large scale regularity theory for random elliptic operators. <i>Communications in Partial Differential Equations</i>. 2016;41(7):1108-1148. doi:<a href=\"https://doi.org/10.1080/03605302.2016.1179318\">10.1080/03605302.2016.1179318</a>","apa":"Fischer, J. L., &#38; Otto, F. (2016). A higher-order large scale regularity theory for random elliptic operators. <i>Communications in Partial Differential Equations</i>. Taylor &#38; Francis. <a href=\"https://doi.org/10.1080/03605302.2016.1179318\">https://doi.org/10.1080/03605302.2016.1179318</a>","mla":"Fischer, Julian L., and Felix Otto. “A Higher-Order Large Scale Regularity Theory for Random Elliptic Operators.” <i>Communications in Partial Differential Equations</i>, vol. 41, no. 7, Taylor &#38; Francis, 2016, pp. 1108–48, doi:<a href=\"https://doi.org/10.1080/03605302.2016.1179318\">10.1080/03605302.2016.1179318</a>.","short":"J.L. Fischer, F. Otto, Communications in Partial Differential Equations 41 (2016) 1108–1148.","chicago":"Fischer, Julian L, and Felix Otto. “A Higher-Order Large Scale Regularity Theory for Random Elliptic Operators.” <i>Communications in Partial Differential Equations</i>. Taylor &#38; Francis, 2016. <a href=\"https://doi.org/10.1080/03605302.2016.1179318\">https://doi.org/10.1080/03605302.2016.1179318</a>."},"article_type":"original","issue":"7","date_created":"2018-12-11T11:51:20Z","publisher":"Taylor & Francis","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","tmp":{"short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"fulldoi":"https://doi.org/10.1080/03605302.2016.1179318","publist_id":"5953","doi":"10.1080/03605302.2016.1179318","OA_place":"publisher","status":"public","type":"journal_article","date_published":"2016-06-16T00:00:00Z","intvolume":"        41","oa_version":"Published Version","language":[{"iso":"eng"}],"keyword":["Ck","α regularity","higher-ordercorrectors","Liouville principle","random elliptic operator","regularity theory","stochastic homogenization"],"external_id":{"arxiv":["1503.07578"]},"volume":41,"month":"06","main_file_link":[{"open_access":"1","url":"https://doi.org/10.1080/03605302.2016.1179318"}],"quality_controlled":"1","OA_type":"gold"}]