Espectro do Laplaciano de Dirichlet-Neumann em faixas estreitas

Abstract

Let $\Omega_\varepsilon$ be a thin strip in $\mathbb{R}^2$ and $-\Delta_{DN}^{\Omega_\varepsilon}$ the Dirichlet-Neumann Laplacian in $\Omega_\varepsilon$. In this work, we study the spectral problem of $-\Delta_{DN}^{\Omega_\varepsilon}$. The asymptotic behaviour for the non-decreasing sequence of numbers $\{\lambda_j(-\Delta_{DN}^{\Omega_\varepsilon})\}_{j=1}^{\infty}$ given by Max–Min Principle will be obtained, under the condition that $\Omega_\varepsilon$ is thin enough. Furthermore, we study the spectral properties of the Dirichlet-Neumann Laplacian in a thin strip of a fixed width.