Isoperimetric inequalities for the first Aharonov-Bohm eigenvalue of the Neumann and Steklov problems

We discuss isoperimetric inequalities for the magnetic Laplacian on bounded domains of R endowed with an Aharonov-Bohm potential. When the flux of the potential around the pole is not an integer, the lowest eigenvalue for the Neumann and the Steklov problems is positive. We generalize the classical inequalities of Szegö-Weinberger, Brock and Weistock to the lowest eigenvalue of this particular magnetic operator, the model domain being a disk with the pole at its center. We consider more generally domains in the plane endowed with a rotationally invariant metric, which include the spherical and the hyperbolic case.

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