Laplaciano de Dirichlet e Dirichlet-Neumann em faixas ilimitadas

Abstract

Let $\Omega$ be an unbounded two dimensional strip on a ruled surface in $\mathbb{R}^d$, $d\geq2$. Consider $-\Delta_{\Omega}^{D}$ the Dirichlet Laplacian operator restricted to $\Omega$ and $-\Delta_{\Omega}^{DN}$ the Laplacian in $\Omega$ with Dirichlet and Neumann boundary conditions on opposite sides of $\Omega$. In this work, we performed a detailed spectral study of $-\Delta_{\Omega}^j$, $j\in\{D,\, DN\}$. In particular, we find information about the essential and discrete spectrum of operators; these results are influenced by the geometry of strip and the boundary conditions on $\partial \Omega$. Furthermore, in some situations, we find an asymptotic behavior for the eigenvalues of $-\Delta_{\Omega}^j$, $j\in\{D,\, DN\}$, when the width of $\Omega$ is small enough.