The Annals of Applied Probability

A Liouville theorem for elliptic systems with degenerate ergodic coefficients

Peter Bella, Benjamin Fehrman, and Felix Otto

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Abstract

We study the behavior of second-order degenerate elliptic systems in divergence form with random coefficients which are stationary and ergodic. Assuming moment bounds like Chiarini and Deuschel (2014) on the coefficient field $a$ and its inverse, we prove an intrinsic large-scale $C^{1,\alpha}$-regularity estimate for $a$-harmonic functions and obtain a first-order Liouville theorem for $a$-harmonic functions.

Article information

Source
Ann. Appl. Probab., Volume 28, Number 3 (2018), 1379-1422.

Dates
Received: May 2016
Revised: July 2017
First available in Project Euclid: 1 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1527840022

Digital Object Identifier
doi:10.1214/17-AAP1332

Mathematical Reviews number (MathSciNet)
MR3809467

Zentralblatt MATH identifier
06919728

Subjects
Primary: 35B53: Liouville theorems, Phragmén-Lindelöf theorems 35B65: Smoothness and regularity of solutions 35J70: Degenerate elliptic equations 60H25: Random operators and equations [See also 47B80]
Secondary: 60K37: Processes in random environments

Keywords
Degenerate elliptic equation degenerate elliptic system stochastic homogenization large-scale regularity Liouville theorem

Citation

Bella, Peter; Fehrman, Benjamin; Otto, Felix. A Liouville theorem for elliptic systems with degenerate ergodic coefficients. Ann. Appl. Probab. 28 (2018), no. 3, 1379--1422. doi:10.1214/17-AAP1332. https://projecteuclid.org/euclid.aoap/1527840022


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