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

  • Jongchon Kim
  • Marius LemmEmail author
Article

Abstract

We consider a divergence-form elliptic difference operator on the lattice \(\mathbb {Z}^d\), 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 \(-2d+\epsilon \) to \(-3d+\epsilon \). (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.

Notes

Acknowledgements

The authors would like to thank Wilhelm Schlag and Tom Spencer for helpful discussions. The authors are grateful to the Institute for Advanced Study for its hospitality during the 2017-2018 academic year. They also thank Alexis Drouot, Antoine Gloria, Felix Otto and Israel Michal Sigal for useful comments on a preprint version of this paper. This material is based upon work supported by the National Science Foundation under Grant No. DMS - 1638352.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  2. 2.Department of MathematicsHarvard UniversityCambridgeUSA

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