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 and the first shape derivative. This 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. 
Catherine Bandle, Alfred Wagner
math.OC