Eigenvalue problems on disconnected domains

Dear friends and colleagues,

in this informal working seminar on topics in Analysis and PDE
our colleague Zeaiter is providing us with a glimpse into his research interest.

He is speaking about

Eigenvalue problems on disconnected domains

Abstract

In this presentation, we will investigate elliptic eigenvalue problems on disconnected
domains. Of particular interest will be the case, where we consider an eigenvalue problem
with a parameterized potential

        $-\mathrm{\Delta }{u}_{\lambda }+\lambda m{u}_{\lambda }=\mu (\lambda )u_{\lambda }.$

Here, $m\in L^{\mathrm{\infty }}$ with a disconnected zero set.

We will investigate, in some simple cases the limit as $\lambda \to \mathrm{\infty }$ and show that there is
an eigenpair $(u_{\mathrm{\infty }},{\lambda }_{\mathrm{\infty }})$ which is the limit of the sequence of eigenpairs $(u_{\lambda },\mu (\lambda ))$ as
$\lambda \to \mathrm{\infty }$, and the pair $(u_{\mathrm{\infty }},{\lambda }_{\mathrm{\infty }})$ is a solution of the equation

        $-\mathrm{\Delta }{u}_{\mathrm{\infty }}=\mu (\mathrm{\infty })u_{\mathrm{\infty }}.$

Everyone interested in Analysis and PDEs is warmly invited to attend this seminar.