A higher-order large-scale regularity theory for random elliptic\n operators

2015/03/25 by Julian Fischer, Felix Otto, Félix Otto +2
Computer Science · Engineering · Mathematics · #35B27 #35B53 #35B65 #35J15 #35J47 #60K37 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Composite Material Mechanics #FOS: Mathematics #Geometry #Harmonic function #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Partial Differential Equations #Scalar (mathematics) #math.AP #msc:35B27 #msc:35B53 #msc:35B65 #msc:35J15 #msc:35J47 #msc:60K37

paper · pdf · doi:10.48550/arxiv.1503.07578

published in arXiv (Cornell University) (Cornell University) · 37 pages; revised version, now includes the regularity theory of arbitrary order

openalex publication_date 2015/03/25 · arxiv created 2015/08/24 · arxiv updated 2015/08/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

We develop a large-scale regularity theory of higher order for\ndivergence-form elliptic equations with heterogeneous coefficient fields a in\nthe context of stochastic homogenization. The large-scale regularity of\na-harmonic functions is encoded by Liouville principles: The space of\na-harmonic functions that grow at most like a polynomial of degree k has\nthe same dimension as in the constant-coefficient case. This result can be seen\nas the qualitative side of a large-scale C^k,\α-regularity theory,\nwhich in the present work is developed in the form of a corresponding\nC^k,\α-"excess decay" estimate: For a given a-harmonic function u\non a ball B_R, its energy distance on some ball B_r to the above space of\na-harmonic functions that grow at most like a polynomial of degree k has\nthe natural decay in the radius r above some minimal radius r₀.\n Though motivated by stochastic homogenization, the contribution of this paper\nis of purely deterministic nature: We work under the assumption that for the\ngiven realization a of the coefficient field, the couple (\φ,\σ) of\nscalar and vector potentials of the harmonic coordinates, where \φ is the\nusual corrector, grows sublinearly in a mildly quantified way. We then\nconstruct "kth-order correctors" and thereby the space of a-harmonic\nfunctions that grow at most like a polynomial of degree k, establish the\nabove excess decay and then the corresponding Liouville principle.\n

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