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Dirichlet vs Neumann

Part of: Partial differential equations

Published online by Cambridge University Press:  22 November 2022

Eduard Marušić-Paloka
Eduard Marušić-Paloka*
Affiliation:
Department of mathematics, Faculty of Science and Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia (emarusic@math.hr)
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Abstract

We study the asymptotic behaviour of the periodically mixed Zaremba problem. We cover the part of the boundary by a chess board with a small period (square size) $\varepsilon$ and impose the Dirichlet condition on black and the Neumann condition on white squares. As $\varepsilon \to 0$, we get the effective boundary condition which is always of the Dirichlet type. The Dirichlet data on the boundary, however, depend on the ratio between the magnitudes of the two boundary values.

Keywords

periodic boundary conditionhomogenizationboundary layerLaplace equationthe Zaremba problem

MSC classification

Primary: 35B27: Homogenization; equations in media with periodic structure 35B25: Singular perturbations 35B40: Asymptotic behavior of solutions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 65 , Issue 4 , November 2022 , pp. 1063 - 1074
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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