On the asymptotics of a Robin eigenvalue problem

The considered Robin problem can formally be seen as a small perturbation of a Dirichlet problem. However, due to the sign of the impedance value, its associated eigenvalues converge point-wise to $-\infty$ as the perturbation goes to zero. We prove that in this case, Dirichlet eigenpairs are the only accumulation points of the Robin eigenpairs with normalized eigenvectors. We then provide a criteria to select accumulating sequences of eigenvalues and eigenvectors and exhibit their full asymptotic with respect to the small parameter

On the Robin eigenvalues of the Laplacian in the exterior of a convex polygon

Let $\Omega\subset \mathbb{R}^2$ be the exterior of a convex polygon whose side lengths are $\ell_1,...,\ell_M$. For $\alpha>0$, let $H^\Omega_\alpha$ denote the Laplacian in $\Omega$, $u\mapsto -\Delta u$, with the Robin boundary conditions $\partial u/\partial\nu =\alpha u$, where $\nu$ is the exterior unit normal at the boundary of $\Omega$. We show that, for any fixed $m\in\mathbb{N}$, the $m$th eigenvalue $E^\Omega_m(\alpha)$ of $H^\Omega_\alpha$ behaves as E^\Omega_m(\alpha)=-\alpha^2+\mu^D_m +\mathcal{O}\Big(\dfrac{1}{\sqrt\alpha}\Big) \quad {as $\alpha$ tends to $+\infty$}, where $\mu^D_m$ stands for the $m$th eigenvalue of the operator $D_1\oplus...\oplus D_M$ and $D_n$ denotes the one-dimensional Laplacian $f\mapsto -f"$ on $(0,\ell_n)$ with the Dirichlet boundary conditions.Comment: 10 pages. To appear in Nanosystems: Physics, Chemistry, Mathematics. Minor revision: misprints corrected, references update

On the principal eigenvalue of a Robin problem with a large parameter

We study the asymptotic behaviour of the principal eigenvalue of a Robin (or generalised Neumann) problem with a large parameter in the boundary condition for the Laplacian in a piecewise smooth domain. We show that the leading asymptotic term depends only on the singularities of the boundary of the domain, and give either explicit expressions or two-sided estimates for this term in a variety of situations.Comment: 16 pages; no figures; replaces math.SP/0403179; completely re-writte

On the discrete spectrum of Robin Laplacians in conical domains

We discuss several geometric conditions guaranteeing the finiteness or the infiniteness of the discrete spectrum for Robin Laplacians on conical domains.Comment: 12 page

Tunneling between corners for Robin Laplacians

We study the Robin Laplacian in a domain with two corners of the same opening, and we calculate the asymptotics of the two lowest eigenvalues as the distance between the corners increases to infinity.Comment: 27 pages, 5 figure