Laplaciano de Dirichlet e Dirichlet-Neumann em faixas ilimitadas - Amorim, Rafael Toledo (2023)
Let $Ω$ be an unbounded two dimensional strip on a ruled surface in $\mathbb{R}^d$, $d≥2$. Consider $-Δ_{Ω}^{D}$ the Dirichlet Laplacian operator restricted to $Ω$ and $-Δ_{Ω}^{DN}$ the Laplacian in $Ω$ with Dirichlet and Neumann boundary conditions on opposite sides of $Ω$. In this work, we performed a detailed spectral study of $-Δ_{Ω}^j$, $j\in\{D,\, DN\}$. 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 $\partial Ω$. Furthermore, in some situations, we find an asymptotic behavior for the eigenvalues of $-Δ_{Ω}^j$, $j\in\{D,\, DN\}$, when the width of $Ω$ is small enough.