For a wedge-like membrane, Payne and Weinberger proved in 1960 an isoperimetric inequality for the fundamental eigenvalue which in some cases improves the classical isoperimetric inequality of Faber- Krahn. In this work, we introduce "relative torsional rigidity" for this type of membrane and prove new isoperimetric inequalities in the spirit of Saint-Venant, Pólya-Szeg?, Payne, Payne-Rayner, Chiti, and Talenti, which link the eigenvalue problem with the boundary value problem in a fundamental way.
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