Simons' cone and equivariant maximization of the first p-Laplace eigenvalue

We consider an optimization problem for the first Dirichlet eigenvalue of the p-Laplacian on a hypersurface in R2n, with n≥2. If p≥2n−1, then among hypersurfaces in R2n which are O(n)×O(n)-invariant and have one fixed boundary component, there is a surface which maximizes the first Dirichlet eigenvalue of the p-Laplacian. This surface is either Simons' cone or a C1 hypersurface, depending on p and n. If n is fixed and p is large, then the maximizing surface is not Simons' cone. If p=2 and n≤5, then Simons' cone does not maximize the first eigenvalue.