Lipschitz estimate for elliptic equations with oscillatory coefficients

Li-Ming Yeh; Li-Ming Yeh

doi:10.3934/math.20241413

AIMS Mathematics

2024, Volume 9, Issue 10: 29135-29166. doi: 10.3934/math.20241413

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Research article

Lipschitz estimate for elliptic equations with oscillatory coefficients

Li-Ming Yeh ^,

Department of Applied Mathematics, National Yang Ming Chiao Tung University, Hsinchu 30050, Taiwan

Received: 26 August 2024 Revised: 30 September 2024 Accepted: 01 October 2024 Published: 15 October 2024
MSC : 35J05, 35J15, 35J25, 35J70

Regularity for elliptic equations with oscillatory coefficients was concerned. Problem domains were periodic and consisted of a connected region with normal permeability and a disconnected matrix block subset with high permeability. Coefficients of the elliptic equations depending on the permeability of the domains were highly oscillatory. Let $ {\epsilon}\in(0, 1) $ be the periodic size of domain, $ {\epsilon}\mu\in(0, 1) $ the size ratio of a matrix block to the whole domain, and $ \omega^2\in(1, \infty) $ the permeability ratio of the disconnected matrix block subset to the connected sub-region of the domains. This work presented Lipschitz estimate uniformly in $ {\epsilon}, \mu, \omega $ for the Green's functions and the solutions of the elliptic equations.
- Lipschitz estimate,
- permeability,
- periodic size,
- Green's function,
- oscillatory coefficient
Citation: Li-Ming Yeh. Lipschitz estimate for elliptic equations with oscillatory coefficients[J]. AIMS Mathematics, 2024, 9(10): 29135-29166. doi: 10.3934/math.20241413

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Abstract

Regularity for elliptic equations with oscillatory coefficients was concerned. Problem domains were periodic and consisted of a connected region with normal permeability and a disconnected matrix block subset with high permeability. Coefficients of the elliptic equations depending on the permeability of the domains were highly oscillatory. Let $ {\epsilon}\in(0, 1) $ be the periodic size of domain, $ {\epsilon}\mu\in(0, 1) $ the size ratio of a matrix block to the whole domain, and $ \omega^2\in(1, \infty) $ the permeability ratio of the disconnected matrix block subset to the connected sub-region of the domains. This work presented Lipschitz estimate uniformly in $ {\epsilon}, \mu, \omega $ for the Green's functions and the solutions of the elliptic equations.

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