High-Frequency Vibrations of Systems with Concentrated Masses Along Planes

Abstract

Let Ω be an open bounded domain of ℝ³ with a smooth boundary $\partial\Omega$. Weassume that Ω is divided into two parts Ω₊ and Ω_- by the plane $\gamma: \Omega = \Omega_+ \cup \Omega_- \cup \gamma$ .For simplicity, we assume that the plane { x ₃ = 0} cuts Ω and $\gamma = \Omega \cap \{x_3 = 0\}$. Let ε be a small positive parameter that tends to zero. We denote by ω_ε the ε-neighborhood of γ, i.e., $\omega_\varepsilon = \Omega \cup \{|x_3| < \varepsilon\}$; for ε sufficiently small, we assume that $\omega_\varepsilon = \gamma \times (-\varepsilon, \varepsilon)$) (see Figure 15.1). Note that this conditions the geometry of Ω near γ. Let us denote by $\bar{x}$ the two first components of any x = (x ₁, x ₂, x ₃) ε ℝ³, that is,$\bar{x} = (x_1, x_2)$