The Saint-Venant Inequality for the Laplace Operator with Robin Boundary Conditions

This survey paper is focused on the Saint-Venant inequality for the Laplace operator with Robin boundary conditions. In a larger context, we make the point on the recent advances concerning isoperimetric inequalities of Faber-Krahn type for elastically supported membranes and describe the main ideas of their proofs in both contexts of rearrangement and free discontinuity techniques.