Geodesic
Mixed Stekloff eigenvalue problem and new extremal properties of the Gr\"otzsch ring
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 28, Tome 270 (2000), pp. 51-79

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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. 
@article{ZNSL_2000_270_a2,
     author = {B. Dittmar and A. Yu. Solynin},
     title = {Mixed {Stekloff} eigenvalue problem and new extremal properties of the {Gr\"otzsch} ring},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {51--79},
     publisher = {mathdoc},
     volume = {270},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2000_270_a2/}
}
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B. Dittmar; A. Yu. Solynin. Mixed Stekloff eigenvalue problem and new extremal properties of the Gr\"otzsch ring. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 28, Tome 270 (2000), pp. 51-79. http://geodesic.mathdoc.fr/item/ZNSL_2000_270_a2/