On the Averaged Green’s Function of an Elliptic Equation with Random Coefficients

We consider a divergence-form elliptic difference operator on the lattice Zd, with a coefficient matrix that is an i.i.d. perturbation of the identity matrix. Recently, Bourgain introduced novel techniques from harmonic analysis to prove the convergence of the Feshbach-Schur perturbation series related to the averaged Green’s function of this model. Our main contribution is a refinement of Bourgain’s approach which improves the key decay rate from −2d+ϵ to −3d+ϵ. (The optimal decay rate is conjectured to be −3d.) As an application, we derive estimates on higher derivatives of the averaged Green’s function which go beyond the second derivatives considered by Delmotte–Deuschel and related works.


Published in:
Archive for Rational Mechanics and Analysis, 234, 3, 1121-1166
Year:
Dec 01 2019
ISSN:
0003-9527
1432-0673
Laboratories:




 Record created 2020-10-01, last modified 2020-10-26


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