Asymptotic behavior of thin ferroelectric models

• Published in: Nonlinear analysis: real world applications Vol. 85; p. 104379
• Main Authors: Hamdache, Kamel, Hamroun, Djamila
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.10.2025
• Subjects: Dimension reduction; Elasticity equations; Electrostatic equations; Polarization; Thin Ferroelectric models; 74-10; Polarization; Dimension reduction; 35K61; Thin Ferroelectric models; Electrostatic equations; 35M13; 35B40; Elasticity equations; 35K45; 74K35
• ISSN: 1468-1218
• DOI: 10.1016/j.nonrwa.2025.104379

It was pointed out in Shaw et al. (2000) that the boundary conditions satisfied by the polarization play an important role in the description of the thin-limit of ferroelectric materials. In this work we confirm the importance of this choice. In the present work, we consider the limiting process as the thickness h of a thin cylinder of R3 goes to 0, and when the polarization satisfies two different boundary conditions. The first type of boundary conditions leads to an in-plane model or 2d−2d configuration while the second one leads to an (in-plane)-(out-of-plane) model for the displacement and the polarization namely a 2d−3d configuration. Moreover, the thin-limit process in both cases induces a change of the Lamé coefficients in the displacement equation, the coupling coefficient between the displacement and polarization equations, the double wells potential of the polarization together to a new contribution in the equation of the out-of-plane component of the polarization for the 2d−3d model. The techniques used rely on a rescaling method that penalizes the out-of-plane variable. Uniform bounds with respect to h are established, compactness techniques are employed, and the limits of the penalized terms are identified.