TY - JOUR

T1 - Elliptic Regularity and Quantitative Homogenization on Percolation Clusters

AU - Armstrong, Scott

AU - Dario, Paul

PY - 2018/9

Y1 - 2018/9

N2 - We establish quantitative homogenization, large-scale regularity, and Liouville results for the random conductance model on a supercritical (Bernoulli bond) percolation cluster. The results are also new in the case that the conductivity is constant on the cluster. The argument passes through a series of renormalization steps: first, we use standard percolation results to find a large scale above which the geometry of the percolation cluster behaves (in a sense, made precise) like that of euclidean space. Then, following the work of Barlow [8], we find a succession of larger scales on which certain functional and elliptic estimates hold. This gives us the analytic tools to adapt the quantitative homogenization program of Armstrong and Smart [7] to estimate the yet larger scale on which solutions on the cluster can be well-approximated by harmonic functions on ℝd. This is the first quantitative homogenization result in a porous medium, and the harmonic approximation allows us to estimate the scale on which a higher-order regularity theory holds. The size of each of these random scales is shown to have at least a stretched exponential moment. As a consequence of this regularity theory, we obtain a Liouville-type result that states that, for each k ∊ ℕ, the vector space of solutions growing at most like o(|x|k+1) as |x| → ∞ has the same dimension as the set of harmonic polynomials of degree at most k, generalizing a result of Benjamini, Duminil-Copin, Kozma, and Yadin from k ≤ 1 to k ∊ ℕ.

AB - We establish quantitative homogenization, large-scale regularity, and Liouville results for the random conductance model on a supercritical (Bernoulli bond) percolation cluster. The results are also new in the case that the conductivity is constant on the cluster. The argument passes through a series of renormalization steps: first, we use standard percolation results to find a large scale above which the geometry of the percolation cluster behaves (in a sense, made precise) like that of euclidean space. Then, following the work of Barlow [8], we find a succession of larger scales on which certain functional and elliptic estimates hold. This gives us the analytic tools to adapt the quantitative homogenization program of Armstrong and Smart [7] to estimate the yet larger scale on which solutions on the cluster can be well-approximated by harmonic functions on ℝd. This is the first quantitative homogenization result in a porous medium, and the harmonic approximation allows us to estimate the scale on which a higher-order regularity theory holds. The size of each of these random scales is shown to have at least a stretched exponential moment. As a consequence of this regularity theory, we obtain a Liouville-type result that states that, for each k ∊ ℕ, the vector space of solutions growing at most like o(|x|k+1) as |x| → ∞ has the same dimension as the set of harmonic polynomials of degree at most k, generalizing a result of Benjamini, Duminil-Copin, Kozma, and Yadin from k ≤ 1 to k ∊ ℕ.

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U2 - 10.1002/cpa.21726

DO - 10.1002/cpa.21726

M3 - Article

AN - SCOPUS:85050502809

VL - 71

SP - 1717

EP - 1849

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 9

ER -