Spectral analysis on ruled surfaces with combined Dirichlet and Neumann boundary conditions

Let $\mathrm{\Omega }$ be an unbounded two dimensional strip on a ruled surface in ${\mathbb{R}}^d$, $d\ge 2$. Consider the Laplacian operator in $\mathrm{\Omega }$ with Dirichlet and Neumann boundary conditions on opposite sides of $\mathrm{\Omega }$. We prove some results on the existence and absence of the discrete spectrum of the operator; which are influenced by the twisted and bent effects of $\mathrm{\Omega }$. Provided that $\mathrm{\Omega }$ is thin enough, we show an asymptotic behavior of the eigenvalues. The interest in those considerations lies on the difference from the purely Dirichlet case. Finally, we perform an appropriate dilatation in $\mathrm{\Omega }$ and we compare the results.