---
OA_place: publisher
OA_type: gold
_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.'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Julian L
full_name: Fischer, Julian L
id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87
last_name: Fischer
orcid: 0000-0002-0479-558X
- first_name: Felix
full_name: Otto, Felix
last_name: Otto
citation:
ama: Fischer JL, Otto F. A higher-order large scale regularity theory for random
elliptic operators. Communications in Partial Differential Equations. 2016;41(7):1108-1148.
doi:10.1080/03605302.2016.1179318
apa: Fischer, J. L., & Otto, F. (2016). A higher-order large scale regularity
theory for random elliptic operators. Communications in Partial Differential
Equations. Taylor & Francis. https://doi.org/10.1080/03605302.2016.1179318
chicago: Fischer, Julian L, and Felix Otto. “A Higher-Order Large Scale Regularity
Theory for Random Elliptic Operators.” Communications in Partial Differential
Equations. Taylor & Francis, 2016. https://doi.org/10.1080/03605302.2016.1179318.
ieee: J. L. Fischer and F. Otto, “A higher-order large scale regularity theory for
random elliptic operators,” Communications in Partial Differential Equations,
vol. 41, no. 7. Taylor & 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.
mla: Fischer, Julian L., and Felix Otto. “A Higher-Order Large Scale Regularity
Theory for Random Elliptic Operators.” Communications in Partial Differential
Equations, vol. 41, no. 7, Taylor & Francis, 2016, pp. 1108–48, doi:10.1080/03605302.2016.1179318.
short: J.L. Fischer, F. Otto, Communications in Partial Differential Equations 41
(2016) 1108–1148.
date_created: 2018-12-11T11:51:20Z
date_published: 2016-06-16T00:00:00Z
date_updated: 2026-05-06T07:05:16Z
day: '16'
doi: 10.1080/03605302.2016.1179318
extern: '1'
external_id:
arxiv:
- '1503.07578'
fulldoi: https://doi.org/10.1080/03605302.2016.1179318
intvolume: ' 41'
issue: '7'
keyword:
- Ck
- α regularity
- higher-ordercorrectors
- Liouville principle
- random elliptic operator
- regularity theory
- stochastic homogenization
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.1080/03605302.2016.1179318
month: '06'
oa: 1
oa_version: Published Version
page: 1108 - 1148
publication: Communications in Partial Differential Equations
publication_identifier:
eissn:
- 1532-4133
issnl:
- 0360-5302
publication_status: published
publisher: Taylor & Francis
publist_id: '5953'
quality_controlled: '1'
status: public
title: A higher-order large scale regularity theory for random elliptic operators
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 41
year: '2016'
...