Abstract
We consider the limit of sequences of normalized (s, 2)-Gagliardo seminorms with an oscillating coefficient as \(s\rightarrow 1\). In a seminal paper by Bourgain et al. (Another look at Sobolev spaces. In: Optimal control and partial differential equations. IOS, Amsterdam, pp 439–455, 2001) it is proven that if the coefficient is constant then this sequence \(\Gamma \)-converges to a multiple of the Dirichlet integral. Here we prove that, if we denote by \(\varepsilon \) the scale of the oscillations and we assume that \(1-s<\!<\varepsilon ^2\), this sequence converges to the homogenized functional formally obtained by separating the effects of s and \(\varepsilon \); that is, by the homogenization as \(\varepsilon \rightarrow 0\) of the Dirichlet integral with oscillating coefficient obtained by formally letting \(s\rightarrow 1\) first.
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Acknowledgements
This paper is based on work supported by the National Research Project PRIN 2022J4FYNJ “Variational methods for stationary and evolution problems with singularities and interfaces”, funded by the Italian Ministry of University and Research. Andrea Braides and Davide Donati are members of GNAMPA, INdAM.
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Braides, A., Brusca, G.C. & Donati, D. Another Look at Elliptic Homogenization. Milan J. Math. 92, 1–23 (2024). https://doi.org/10.1007/s00032-023-00389-y
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DOI: https://doi.org/10.1007/s00032-023-00389-y