Abstract
In this paper we prove convergence results for the homogenization of the Dirichlet problem for elliptic equations in divergence form with rapidly oscillating boundary data and non oscillating coefficients in convex polygonal domains. Our analysis is based on integral representation of solutions. Under a certain Diophantine condition on the boundary of the domain and smooth coefficients we prove pointwise, as well as \(L^p\) convergence results. For larger exponents \(p\) we prove that the \(L^p\) convergence rate is close to optimal. We also suggest several directions of possible generalization of the results in this paper.
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Acknowledgments
H. Shahgholian has been supported by Swedish Research Council. H. Aleksanyan thanks Göran Gustafsson Foundation for visiting appointment to KTH. The authors thank the referees for constructive suggestions.
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Communicated by Luis Vega.
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Aleksanyan, H., Shahgholian, H. & Sjölin, P. Applications of Fourier Analysis in Homogenization of Dirichlet Problem III: Polygonal Domains. J Fourier Anal Appl 20, 524–546 (2014). https://doi.org/10.1007/s00041-014-9327-4
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DOI: https://doi.org/10.1007/s00041-014-9327-4