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Large-scale Regularity of Nearly Incompressible Elasticity in Stochastic Homogenization

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Abstract

In this paper, we systematically study the regularity theory of the linear system of nearly incompressible elasticity. In the setting of stochastic homogenization, we develop new techniques to establish the large-scale estimates of displacement and pressure, which are uniform in both the scale parameter and the incompressibility parameter. In particular, we obtain the boundary estimates in a new class of Lipschitz domains whose boundaries are smooth at large scales and bumpy at small scales.

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Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Notes

  1. Without ambiguity, we write \(u^\varepsilon _\lambda \), instead of \(u^\varepsilon _{\lambda _0}\), for short.

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Acknowledgements

Both of the authors would like to thank Professor Fanghua Lin for the helpful comments after the second author reporting the results of this paper in the SUSTech PDE Workshop in Shenzhen. Both of the authors would like to thank Professor Hongjie Dong for pointing out a mistake in an early version of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Fordham University, New York, NY, 10458, USA

    Shu Gu

  2. Department of Mathematics, University of Chicago, Chicago, IL, 60637, USA

    **** Zhuge

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  1. Shu Gu
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  2. **** Zhuge
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Correspondence to **** Zhuge.

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Communicated by N. Masmoudi.

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Gu, S., Zhuge, J. Large-scale Regularity of Nearly Incompressible Elasticity in Stochastic Homogenization. Arch Rational Mech Anal 244, 1311–1372 (2022). https://doi.org/10.1007/s00205-022-01772-6

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  • DOI: https://doi.org/10.1007/s00205-022-01772-6

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