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Eigenvalue Fluctuations for Random Elliptic Operators in Homogenization Regime

Abstract

This work is devoted to the asymptotic behavior of eigenvalues of an elliptic operator with rapidly oscillating random coefficients on a bounded domain with Dirichlet boundary conditions. A sharp convergence rate is obtained for eigenvalues towards those of the homogenized problem, as well as a quantitative two-scale expansion result for eigenfunctions. Next, a quantitative central limit theorem is established for fluctuations of isolated eigenvalues; more precisely, a pathwise characterization of eigenvalue fluctuations is obtained in terms of the so-called homogenization commutator, in parallel with the recent fluctuation theory for the solution operator.

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Notes

  1. Throughout, we use Einstein’s convention of summation on repeated indices, here on \(1\le \alpha ,\beta \le d\).

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Acknowledgements

We thank Antoine Gloria and Christopher Shirley for motivating discussions on the topic, and we acknowledge financial support from the CNRS-Momentum program.

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Correspondence to Mitia Duerinckx.

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Communicated by Eric A. Carlen.

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Duerinckx, M. Eigenvalue Fluctuations for Random Elliptic Operators in Homogenization Regime. J Stat Phys 187, 32 (2022). https://doi.org/10.1007/s10955-022-02918-2

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  • DOI: https://doi.org/10.1007/s10955-022-02918-2

Keywords

  • Eigenvalue fluctuations
  • Homogenization theory
  • Linear elliptic operator
  • Quantitative central lilmit theorem