Extremal Steklov–Neumann eigenvalues on the disk

Let $\Omega=\mathbb{D}$ be the unit disk, let $\Gamma=\partial\Omega=\mathbb{S}^1$, and let $\mathcal{I}_\alpha^n\subset\mathbb{S}^1$ be the union of $n$ evenly spaced intervals of equal length, as defined by

$$\mathcal{I}_\alpha^n=\bigcup_{\ell\in\{1,\ldots,n\}}\left[\frac{\pi(2\ell-\alpha)}{n},\frac{\pi(2\ell+\alpha)}{n}\right].$$

For a density $\rho$ on $\Gamma$, write $\lambda_k(\rho)$ for the corresponding relaxed Steklov–Neumann eigenvalue and $\sigma_k$ for the associated eigenvalue in the unrelaxed problem. Disk extremal eigenvalue conjecture. For $k\geq1$ and $\alpha\in(0,1]$, the unique minimizer of $\lambda_k$ up to rotation is $\rho_k^\bigtriangledown=\mathds{1}_{\mathcal{I}_\alpha^{k+1}}$; for odd $k$ the multiplicity is one, and for even $k$ it is two. Consequently, the unique minimizer of $\sigma_k$ up to rotation is $\mathcal{I}_\alpha^{k+1}$. For $k=1$ and $\alpha\in(0,1]$, the constant density $\rho_1^\triangle=\alpha$ is a maximizer of $\lambda_1$. For $k\geq2$, the unique maximizer of $\lambda_k$ up to rotation has the described $k$-repeated structure: it equals $1$ on a first subinterval and is nontrivial, strictly positive, and convex on a second subinterval. For even $k$ its multiplicity is one, with the eigenfunction constant on the second subintervals where $\rho_k^\triangle<1$, while for odd $k$ its multiplicity is two. For every $k\geq1$, rapidly oscillating Steklov/Neumann boundary configurations can arbitrarily well approximate the supremum in the unrelaxed problem.