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Homogenization of an Elliptic Equation in a Domain with Oscillating Boundary with Non-homogeneous Non-linear Boundary Conditions

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Abstract

While considering boundary value problems with oscillating coefficients or in oscillating domains, it is important to associate an asymptotic model which accounts for the average behaviour. This model permits to obtain the average behaviour without costly numerical computations implied by the fine scale of oscillations in the original model. The asymptotic analysis of boundary value problems in oscillating domains has been extensively studied and involves some key issues such as: finding uniformly bounded extension operators for function spaces on oscillating domains, the choice of suitable sequences of test functions for passing to the limit in the variational formulation of the model equations etc. In this article, we study a boundary value problem for the Laplacian in a domain, a part of whose boundary is highly oscillating (periodically), involving non-homogeneous non-linear Neumann or Robin boundary condition on the periodically oscillating boundary. The non-homogeneous Neumann condition or the Robin boundary condition on the oscillating boundary adds a further difficulty to the limit analysis since it involves taking the limits of surface integrals where the surface changes with respect to the parameter. Previously, some model problems have been studied successfully in Gaudiello (Ricerche Mat 43(2):239–292, 1994) and in Mel’nyk (Math Methods Appl Sci 31(9):1005–1027, 2008) by converting the surface term into a volume term using auxiliary boundary value problems. Some problems of this nature have also been studied using an extension of the notion of two-scale convergence (Allaire et al. in Proceedings of the international conference on mathematical modelling of flow through porous media, Singapore, 15–25, 1996, Neuss-Radu in C R Acad Sci Paris Sr I Math 322:899–904, 1996). In this article, we use a different approach to handle of such terms based on the unfolding operator.

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Acknowledgements

The first author would like to thank the financial support of CONICYT-CHILE through the grant FONDECYT 1130595. The work was partially supported by the project (No. SR/S4/MS: 855/13 dtd 17.10.14) and the second author would like to acknowledge the support of Department of Science Technology (DST), India for the project. The third author wishes to thank CONICYT for the financial support through FONDECYT POSTDOCTORADO NO. 3140138. He would also like to thank the Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción (Chile) for their financial support through PROYECTOS VRID INICIACIÓN NO. 216.013.0.41-1.0IN.

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Authors and Affiliations

  1. Depto. de Matemática, FCFM, Univ. de Concepción, Concepción, Chile

    Rajesh Mahadevan & Ravi Prakash

  2. Department of Mathematics, Indian Institute of Science, Bengaluru, India

    A. K. Nandakumaran

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  1. Rajesh Mahadevan
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Correspondence to Rajesh Mahadevan.

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Mahadevan, R., Nandakumaran, A.K. & Prakash, R. Homogenization of an Elliptic Equation in a Domain with Oscillating Boundary with Non-homogeneous Non-linear Boundary Conditions. Appl Math Optim 82, 245–278 (2020). https://doi.org/10.1007/s00245-018-9499-4

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