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

Abstract

We consider an optimization problem for the first Dirichlet eigenvalue of the $p$-Laplacian on a hypersurface in ${\mathbb{R}}^{2n}$, with $n\ge 2$. If $p\ge 2n-1$, then among hypersurfaces in ${\mathbb{R}}^{2n}$ which are $O(n)\times O(n)$-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 $C^1$ 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 $n\le 5$, then Simons' cone does not maximize the first eigenvalue.

References in corpus (2)

• A disc maximizes Laplace eigenvalues among isoperimetric surfaces of revolution
• Eigenvalue maximization for surfaces of revolution with prescribed boundary