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Remarks on optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form

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Abstract

We study and characterize the optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form. We obtain that the optimal rate of convergence is either \(O(\varepsilon )\) or \(O(\varepsilon ^2)\) depending on the diffusion matrix A, source term f, and boundary data g. Moreover, we show that the set of diffusion matrices A that give optimal rate \(O(\varepsilon )\) is open and dense in the set of \(C^2\) periodic, symmetric, and positive definite matrices, which means that generically, the optimal rate is \(O(\varepsilon )\).

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Acknowledgements

We would like to thank Fanghua Lin for some very useful discussions.

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Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin Madison, Van Vleck hall, 480 Lincoln drive, Madison, WI, 53706, USA

    Xiaoqin Guo & Hung V. Tran

  2. Department of Mathematics, University of California, Irvine, 410G Rowland Hall, Irvine, CA, 92697, USA

    Yifeng Yu

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  1. Xiaoqin Guo
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  2. Hung V. Tran
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  3. Yifeng Yu
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Correspondence to Hung V. Tran.

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Dedicated to Professor Hitoshi Ishii with our admiration.

The work of HT is partially supported by NSF grant DMS-1664424 and NSF CAREER award DMS-1843320.

This article is part of the topical collection “Viscosity solutions - Dedicated to Hitoshi Ishii on the award of the 1st Kodaira Kunihiko Prize” edited by Kazuhiro Ishige, Shigeaki Koike, Tohru Ozawa, and Senjo Shimizu

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Guo, X., Tran, H.V. & Yu, Y. Remarks on optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form. SN Partial Differ. Equ. Appl. 1, 15 (2020). https://doi.org/10.1007/s42985-020-00017-z

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  • DOI: https://doi.org/10.1007/s42985-020-00017-z

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