On the isoperimetric problem for the higher eigenvalues of the Robin and Wentzell Laplacians

We consider the problem of minimising the kth eigenvalue, k >= 2, of the (p-)Laplacian with Robin boundary conditions with respect to all domains in R(N) of given volume. When k = 2, we prove that the second eigenvalue of the p-Laplacian is minimised by the domain consisting of the disjoint union of two balls of equal volume, and that this is the unique domain with this property. For p = 2 and k >= 3, we prove that in many cases a minimiser cannot be independent of the value of the constant in the boundary condition, or equivalently of the domain's volume. We obtain similar results for the Laplacian with generalised Wentzell boundary conditions.