{"publication_status":"published","extern":1,"title":"A higher-order large scale regularity theory for random elliptic operators","status":"public","month":"07","type":"journal_article","year":"2016","volume":41,"publisher":"Taylor & Francis","day":"02","oa":1,"date_created":"2018-12-11T11:51:20Z","date_published":"2016-07-02T00:00:00Z","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1503.07578"}],"quality_controlled":0,"date_updated":"2021-01-12T06:49:50Z","publication":"Communications in Partial Differential Equations","_id":"1318","abstract":[{"lang":"eng","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."}],"publist_id":"5953","author":[{"full_name":"Julian Fischer","last_name":"Fischer","first_name":"Julian L","id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-0479-558X"},{"last_name":"Otto","full_name":"Otto, Felix","first_name":"Felix"}],"intvolume":"        41","citation":{"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>.","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>.","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>","short":"J.L. Fischer, F. Otto, Communications in Partial Differential Equations 41 (2016) 1108–1148."},"issue":"7","doi":"10.1080/03605302.2016.1179318","page":"1108 - 1148"}