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Journal of Convex Analysis 22 (2015), No. 3, 627--640
Copyright Heldermann Verlag 2015

Isoperimetric Inequalities for the Principal Eigenvalue of a Membrane and the Energy of Problems with Robin Boundary Conditions

Catherine Bandle
Mathematische Institut, Universität Basel, Rheinsprung 21, 4051 Basel, Switzerland

Alfred Wagner
Institut für Mathematik, RWTH Aachen, Templergraben 55, 52062 Aachen, Germany

An inequality for the reverse Bossel-Daners inequality is derived by means of the harmonic transplantation. The first and second shape derivatives are computed for the ball. The same method is then applied to elliptic boundary value problems with inhomogeneous Neumann conditions. The first variation is computed and an isoperimetric inequality is derived for the minimal energy.

Keywords: Rayleigh-Faber Krahn inequality, Robin boundary conditions, domain variation, harmonic transplantation.

MSC: 49K20, 49R05, 15A42, 35J20, 35N25

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