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

Catherine Bandle; Alfred Wagner

Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

Catherine Bandle

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

Alfred Wagner

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

Keywords

Rayleigh-Faber Krahn inequality
Robin boundary conditions
domain variation
harmonic transplantation

Mathematics Subject Classification

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

Suggested citation

C. Bandle, A. Wagner. “Isoperimetric Inequalities for the Principal Eigenvalue of a Membrane and the Energy of Problems with Robin Boundary Conditions.” Journal of Convex Analysis 22 (2015), No. 3, 627–640.