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Homogenization with strong contrasting diffusivity in a circular oscillating domain with \(L^1\) source term

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Abstract

In this article, we study the homogenization of an elliptic variational form with oscillating coefficients in a circular, highly oscillating domain, where the oscillatory part is made of two materials with high contrasting conductivity (or diffusivity) with the source term in \(L^1\). We incorporate this phenomenon, namely, highly oscillating boundary, rapid oscillating coefficient, and the oscillating part made of high contrasting materials, which leads to non-uniform ellipticity as the oscillating parameter goes to 0. Further, due to the \(L ^1\) source term, the solutions are interpreted as renormalized solutions. To achieve our primary goal, we have proved the strong convergence results in the context of the \(L^2\) source term in the first part (corrector results). In the second part, we have homogenized the renormalized variational form and established the relation between the \(\varepsilon\)-stage renormalized solution and the limit renormalized solution via convergence results. The unfolding operator for the polar coordinates is a central tool for the analysis.

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Acknowledgements

We wish to express our appreciation to the referees for their fruitful comments and suggestions in the first version of the paper, which helped us to improve our article’s quality. The first and third authors would like to thank the Department of Mathematics, IISc, Bangalore, India, and the second author would like to thank TIFR Center for Applicable Mathematics, Bangalore, India, for the academic support. The third author would like to acknowledge the National Board for Higher Mathematics (NBHM), Department of Atomic Energy(DAE), India, for the financial support. The first author would also like to thank Department of Science and Technology (DST), Government of India for the partial financial support under Project No. CRG/2021/000458.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Science, Bangalore, 560012, India

    A. K. Nandakumaran & Renjith Thazhathethil

  2. TIFR Center for Applicable Mathematics, Bangalore, 560065, India

    Abu Sufian

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  1. A. K. Nandakumaran
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  3. Renjith Thazhathethil
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Correspondence to A. K. Nandakumaran.

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Nandakumaran, A.K., Sufian, A. & Thazhathethil, R. Homogenization with strong contrasting diffusivity in a circular oscillating domain with \(L^1\) source term. Annali di Matematica 202, 763–786 (2023). https://doi.org/10.1007/s10231-022-01259-x

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  • DOI: https://doi.org/10.1007/s10231-022-01259-x

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