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

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.