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

@article{Fischer2015AHL, title={A higher-order large-scale regularity theory for random elliptic operators}, author={Julian Fischer and Felix Otto}, journal={Communications in Partial Differential Equations}, year={2015}, volume={41}, pages={1108 - 1148} }

ABSTRACT 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… Expand

#### 45 Citations

Liouville Principles and a Large-Scale Regularity Theory for Random Elliptic Operators on the Half-Space

- Mathematics, Computer Science
- SIAM J. Math. Anal.
- 2017

This work considers the large-scale regularity of solutions to second-order linear elliptic equations with random coefficient fields and derives an associated C^{1,\alpha}$-type large- scale regularity theory in the form of a corresponding decay estimate for the homogenization-adapted tilt-excess. Expand

A REGULARITY THEORY FOR RANDOM ELLIPTIC OPERATORS AND HOMOGENIZATION

- 2018

The qualitative theory of stochastic homogenization of uniformly elliptic linear (but possibly non-symmetric) systems in divergence form is well-understood. Quantitative results on the speed of… Expand

A Large-Scale Regularity Theory for Random Elliptic Operators on the Half-Space with Homogeneous Neumann Boundary Data

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- 2017

In this note we derive large-scale regularity properties of solutions to second-order linear elliptic equations with random coefficients on the half- space with homogeneous Neumann boundary data; it… Expand

A Regularity Theory for Random Elliptic Operators

- Mathematics
- 2014

Since the seminal results by Avellaneda & Lin it is known that elliptic operators with periodic coefficients enjoy the same regularity theory as the Laplacian on large scales. In a recent inspiring… Expand

Some Large-Scale Regularity Results for Linear Elliptic Equations with Random Coefficients and on the Well-Posedness of Singular Quasilinear SPDEs

- Mathematics
- 2019

This thesis is split into two parts, the rst one is concerned with some problems in stochastic homogenization and the second addresses a problem in singular SPDEs. In the part on stochastic… Expand

Mesoscopic Higher Regularity and Subadditivity in Elliptic Homogenization

- Mathematics
- 2015

We introduce a new method for obtaining quantitative results in stochastic homogenization for linear elliptic equations in divergence form. Unlike previous works on the topic, our method does not use… Expand

The additive structure of elliptic homogenization

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- 2016

One of the principal difficulties in stochastic homogenization is transferring quantitative ergodic information from the coefficients to the solutions, since the latter are nonlocal functions of the… Expand

Quantitative Stochastic Homogenization and Large-Scale Regularity

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- 2019

This is a preliminary version of a book which presents the quantitative homogenization and large-scale regularity theory for elliptic equations in divergence-form. The self-contained presentation… Expand

Stochastic Homogenization of Linear Elliptic Equations: Higher-Order Error Estimates in Weak Norms Via Second-Order Correctors

- Mathematics, Computer Science
- SIAM J. Math. Anal.
- 2017

For symmetric coefficient fields with only short-range correlations, quantified through a logarithmic Sobolev inequality for the ensemble, it is proved that when measured in weak spatial norms, the solution to the homogenized equation provides a higher-order approximation of the solution of the equation with oscillating coefficients. Expand

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Consider an ergodic stationary random field A on the ambient space R d. We are interested in the concentration of measure phenomenon for nonlinear functions X(A) in terms of assumptions on A. In… Expand

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