In this paper we obtain several results concerning the optimization of higher Steklov eigenvalues both in two and higher dimensional cases. We first show that the normalized (by boundary length) k-th Steklov eigenvalue on the disk is not maximized for a smooth metric on the disk for \(k\ge 3\). For \(k=1\) the classical result of Weinstock (J Ration Mech Anal 3:745–753, 1954) shows that \(\sigma _1\) is maximized by the standard metric on the round disk. For \(k=2\) it was shown by Girouard and Polterovich (Funct Anal Appl 44(2):106–117, 2010) that \(\sigma _2\) is not maximized for a smooth metric. We also prove a local rigidity result for the critical catenoid and the critical Möbius band as free boundary minimal surfaces in a ball under \(C^2\) deformations. We next show that the first k Steklov eigenvalues are continuous under certain degenerations of Riemannian manifolds in any dimension. Finally we show that for \(k\ge 2\) the supremum of the k-th Steklov eigenvalue on the annulus over all metrics is strictly larger that that over \(S^1\)-invariant metrics. We prove this same result for metrics on the Möbius band.

在本文中，我们获得了有关在二维和较高维情况下优化Steklov特征值的一些结果。我们首先显示，对于\（k \ ge 3 \）而言，磁盘上的标准化度量（按边界长度）的第k个Steklov特征值对于磁盘上的平滑度量没有最大化。对于\（k = 1 \），Weinstock的经典结果（J Ration Mech Anal 3：745–753，1954年）表明，\（\ sigma _1 \）被圆盘上的标准度量最大化。对于\（k = 2 \），Girouard和Polterovich（Funct Anal Appl 44（2）：106–117，2010）证明\（\ sigma _2 \）没有为平滑指标最大化。我们还证明了临界悬链线和临界莫比乌斯带的局部刚度结果为\（C ^ 2 \）变形下球中的自由边界最小曲面。接下来我们表明，在任何维数的黎曼流形某些退化下，前k个Steklov特征值都是连续的。最后，我们证明，对于\（k \ ge 2 \），所有度量上的环上第k个Steklov特征值的绝对值严格大于\（S ^ 1 \）不变度量上的值。对于莫比乌斯乐队的指标，我们证明了同样的结果。