Abstract
The asymptotic behavior of an anisotropic Cahn–Hilliard functional with prescribed mass and Dirichlet boundary condition is studied when the parameter \(\varepsilon \) that determines the width of the transition layers tends to zero. The double-well potential is assumed to be even and equal to \(|s-1|^\beta \) near \(s=1\), with \(1<\beta <2\). The first order term in the asymptotic development by \(\Gamma \)-convergence is well-known, and is related to a suitable anisotropic perimeter of the interface. Here it is shown that, under these assumptions, the second order term is zero, which gives an estimate on the rate of convergence of the minimum values.
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Acknowledgments
The authors warmly thank the Center for Nonlinear Analysis (NSF Grant No. DMS-0635983), where part of this research was carried out. The research of I. Fonseca was partially funded by the National Science Foundation under Grant No. DMS-0905778 and that of G. Leoni under Grant No. DMS-1007989. I. Fonseca and G. Leoni also acknowledge support of the National Science Foundation under the PIRE Grant No. OISE-0967140. The research of G. Dal Maso was also supported by by the Italian Ministry of Education, University, and Research under the Project “Variational Problems with Multiple Scales” 2008 and by the European Research Council under Grant No. 290888 “Quasistatic and Dynamic Evolution Problems in Plasticity and Fracture”. The authors wish to thank Matteo Focardi for several discussions on the subject of this paper and Michael Goldman, who called their attention to reference [1]. The results of this paper led to a dramatic simplification in the proof of the \(\Gamma \)-limsup inequality and were crucial in the proof of the \(\Gamma \)-liminf inequality in the anisotropic case.
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Communicated by L. Ambrosio.
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Dal Maso, G., Fonseca, I. & Leoni, G. Second order asymptotic development for the anisotropic Cahn–Hilliard functional. Calc. Var. 54, 1119–1145 (2015). https://doi.org/10.1007/s00526-015-0819-0
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DOI: https://doi.org/10.1007/s00526-015-0819-0