The Pólya conjecture for the Neumann eigenvalues holds for the second eigenvalue

You need to know: Euclidean space ${\mathbb R}^n$ , open subsets of ${\mathbb R}^n$ , bounded subsets of ${\mathbb R}^n$ , notation $|\Omega|$ for the volume of $\Omega \subset {\mathbb R}^n$ , differentiable function on ${\mathbb R}^n$ , its gradient $\nabla u(x)=\left(\frac{\partial u}{\partial x_1}(x), \dots, \frac{\partial u}{\partial x_n}(x) \right)$ , integral over $\Omega \subset {\mathbb R}^n$ , normed space, subspace, dimension of a subspace, compactly embedded space.

Background: Let $n\geq 2$ , and let $\Omega \subset {\mathbb R}^n$ be a bounded open set. Let $L^2(\Omega)$ be the set of functions $u:\Omega\to{\mathbb R}$ with norm $||u||_2:=\left(\int_\Omega |u(x)|^2 dx\right)^{1/2}<\infty$ . Let $H^1(\Omega)$ be the set of differentiable functions $u:\Omega\to{\mathbb R}$ with norm $||u||_H := \left(\int_\Omega\left(|\nabla u(x)|^2 + |u(x)|^2\right)dx\right)^{1/2}<\infty$ . We say that domain $\Omega$ is regular if $H^1(\Omega)$ is compactly embedded in $L^2(\Omega)$ . For every integer $k\geq 1$ , let ${\cal S}_k$ be the family of all subspaces of dimension k in $\{u\in H^1(\Omega) :\int_\Omega u(x) dx = 0\}$ , and let $\mu_k(\Omega)=\min\limits_{S \in {\cal S}_k} \max\limits_{u\in S}\frac{\int_\Omega |\nabla u(x)|^2 dx}{\int_\Omega |u(x)|^2 dx}$ . If $B \subset {\mathbb R}^n$ is a ball, the quantity $\mu_2^*=2^{2/n}|B|^{2/n}\mu_1(B)$ is a constant which does not depend on B.

The Theorem: On 22nd January 2018, Dorin Bucur and Antoine Henrot submitted to Acta Mathematica a paper in which they proved that inequality $|\Omega|^{2/n}\mu_2(\Omega) \leq \mu_2^*$ holds for every regular set $\Omega \subset {\mathbb R}^n$ , with equality if $\Omega$ is the union of two disjoint, equal balls.

Short context: The sequence $\mu_k(\Omega)$ is known as “the spectrum of the Laplace operator with Neumann boundary conditions”, and is well-studied in mathematics with applications in physics. In 1950-th, Szegő and Weinberger proved that $|\Omega|^{2/n}\mu_1(\Omega)$ is maximised when $\Omega$ is the ball in ${\mathbb R}^n$ . The Theorem solves the maximization problem for $|\Omega|^{2/n}\mu_k(\Omega)$ for $k=2$ . As one of the applications, it immediately implies the $k=2$ case of important Polya conjecture for the Neumann eigenvalues, which states that $\mu_k(\Omega) \leq 4\pi^2\left(\frac{k}{w_n|\Omega|}\right)^{2/n}$ , where $w_n$ is the volume of the unit ball in ${\mathbb R}^n$ .