Mixed Stekloff eigenvalue problem and new extremal properties of the Grötzsch ring

We study the mixed Stekloff eigenvalue problem in doubly-connected domains. Using circular symmetrization and a distortion theorem on conformal mapping of an annulus, we find a lower bound for the first eigenvalue that is sharp for the Grötzsch ring. We solve also an extremal problem for some polygonal doubly-connected domains and prove some results concerning the existence of a closed nodal line.