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

TL;DR

This paper investigates the maximization of the first Dirichlet eigenvalue of the $p$-Laplacian on symmetric hypersurfaces, revealing conditions under which Simons' cone or other surfaces serve as maximizers.

Contribution

It characterizes the maximizing hypersurfaces for the first eigenvalue of the $p$-Laplacian under symmetry constraints, identifying when Simons' cone is optimal or not.

Findings

Simons' cone maximizes the eigenvalue for certain $p$ and $n$.

For large $p$, the maximizer is not Simons' cone.

Simons' cone does not maximize the eigenvalue when $p=2$ and $n \\le 5.

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.