Locally periodic unfolding operator for highly oscillating rough domains
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This article aims to understand the locally periodic oscillating domain via unfolding operators. A three-dimensional rough domain \(\Omega _\varepsilon \), \(\varepsilon >0\) a small parameter, has been considered for the study where the boundary is rapidly oscillating with high amplitude. Though there are some articles with locally periodic boundary oscillations with small amplitude we do not see any literature with high-amplitude (O(1)) locally periodic oscillating domains. In this article, we attempt to study a problem in locally periodic rough domains with an eye towards the general oscillating domains without periodicity. With our experience of handling such domains and unfolding operators, we develop locally periodic unfolding operators to study our problems. We consider a nonlinear inhomogeneous Robin boundary value problem posed on this domain to demonstrate the utility of the newly defined operator.
KeywordsAsymptotic analysis Unfolding operator Locally periodic oscillating boundary domain Homogenization
Mathematics Subject Classification80M35 80M40 35B27
We thank the referee for their meticulous reading of the article and their comments which have improved the paper substantially. The first and second authors would like to thank Department of Science and Technology (DST), Government of India, as the work was partially supported by the Project No. EMR/2016/005018 dtd 8.8.17. The third author wishes to thank CONICYT for the financial support through FONDECYT INICIACIÓN NO. 11180551. He would also acknowledge the support from the Facultad de Ciencias Fisicas y Matemáticas, Universidad de Concepción (Chile).
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