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Generalization of unfolding operator for highly oscillating smooth boundary domains and homogenization

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Abstract

Unfolding operators have been introduced and used to study homogenization problems. Initially, they were introduced for problems with rapidly oscillating coefficients and porous domains. Later, this has been developed for domains with oscillating boundaries, typically with rectangular or pillar type boundaries which are classified as non-smooth. In this article, we develop new unfolding operators, where the oscillations can be smooth and hence they have wider applications. We have demonstrated by developing unfolding operators for circular domains with rapid oscillations with high amplitude of O(1) to study the homogenization of an elliptic problem.

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Acknowledgements

The authors are very grateful to the referee for his fruitful remarks that helped us to improve the paper. The first author would like to thank National Board for Higher Mathematics (NBHM), Department of Atomic Energy, India, for the financial support. The project was partially supported by Science and Engineering Board (SERB), Department of Science and Technology, Government of India under the Project No. EMR/2016/006018 dtd 8.9.17 and the first two authors would like to thank SERB. The third author would like to thank the Facultad de Ciencias Fsicas y Matemticas, Universidad de Concepción (Chile), for their financial support through PROYECTOS VRID INICIACIÓN n 216.013.0.41-1.0IN.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Science, Bangalore, India

    S. Aiyappan & A. K. Nandakumaran

  2. Department of Mathematics, Faculty of Physical Sciences and Mathematics, University of Concepcion, Concepción, Chile

    Ravi Prakash

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  1. S. Aiyappan
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  2. A. K. Nandakumaran
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Correspondence to Ravi Prakash.

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Communicated by A. Malchiodi.

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Aiyappan, S., Nandakumaran, A.K. & Prakash, R. Generalization of unfolding operator for highly oscillating smooth boundary domains and homogenization. Calc. Var. 57, 86 (2018). https://doi.org/10.1007/s00526-018-1354-6

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