We show, using symmetrization techniques, that it is possible to prove a comparison principle (we are mainly focused on L1 comparison) between solutions to an elliptic partial differential equation on a smooth bounded set Ω with a rather general boundary condition, and solutions to a suitable related problem defined on a ball having the same volume as Ω. This includes for instance mixed problems.

The Bossel-Daners is a Faber-Krahn type inequality for the first Laplacian eigenvalue with Robin boundary conditions. We prove a stability result for such inequality.