Abstract
Abstract The paper deals with the Robin function and eigenvalue problems generated by the Robin operator. First we show that Green's function of an 𝑛-fold connected domain is the Robin function of an appropriate simply connected domain. The main part of the paper deals with eigenvalue problems for the Robin operator: the mixed Stekloff eigenvalue problem and the membrane problem with mixed boundary conditions. Isoperimetric inequalities are proved for the sum of reciprocal eigenvalues.
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