Extremal problems for Steklov eigenvalues on annuli

Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems

“…For odd j = 2m − 1, m ∈ N, from the results of Fan, Tam, and Yu [12], we have that the extremum is attained at the crossings of the two length-normalized eigenvalues with associated L 2 (∂Ω, ρ)-normalized eigenfunctions…”

“…For even j, Fan, Tam, and Yu [12] show the following. For j = 2, the extremal value is not attained among rotationally symmetric annuli and for even j ≥ 4, the extremal value is attained.…”

“…In [12], Fan, Tam, and Yu extended the study of (1.3) to higher values of j on the cylinder (γ = 0, b = 2) among rotationally symmetric conformal metrics. They obtained different results for even and odd eigenvalues.…”

“…Here, we discuss the Steklov eigenvalues of rotationally symmetric annuli, the critical catenoid, and coverings of the critical catenoid. These results are also discussed in [12,14] using cylindrical coordinates, but it is useful to review these computations and have them written in annular coordinates for comparison and discussion; see also [10,29].…”

Existence and Classification of $$\pmb {\mathbb {S}}^1$$-Invariant Free Boundary Minimal Annuli and Möbius Bands in $$\pmb {\mathbb {B}}^n$$

“…We may assume α ≥ 1, in which case we have 0 < β ≤ 1 with β = 1 if and only if α = 1. It follows from [FTY,Lemmas 2.4,2.5] that the critical points (local maxima and minima) of the normalized Steklov eigenvaluesσ i (β, T ) among metrics with β fixed are T = t m,n (β) (see [FTY,Definition 2.1]) for all m, n with m/n > α. However, by [FTY,Proof of Lemma 2.6(i)…”