Optimization of the principal eigenvalue in various geometries

Search for shapes which optimize the principle eigenvalue of a PDE problem is a trademark problem of mathematical physics. In this talk we shall address questions of that type for Schrödinger operators with an attractive singular ‘potential’, supported by a manifold or a geometric complex, which can be formally written as $-\Delta-\alpha \delta(x-\Gamma)$ with $\alpha>0$. Various classes of the interaction support $\Gamma$ will be discussed: loops, manifolds homothetic to a sphere, cones and stars; while most attention will be paid to the case $\mathrm{codim}\,\Gamma=1$, we will consider also situations where the interaction is strongly singular, either of the $\delta'$ type or with the support of codimension two.