We consider the problem of geometric optimisation of the lowest eigenvalue of the Laplacian in the exterior of a compact set in any dimension, subject to attractive Robin boundary conditions. As an improvement upon our previous work (Krejčiřík and Lotoreichik J. Convex Anal. 25, 319–337, 2018), we show that under either a constraint of fixed perimeter or area, the maximiser within the class of exteriors of simply connected planar sets is always the exterior of a disk, without the need of convexity assumption. Moreover, we generalise the result to disconnected compact planar sets. Namely, we prove that under a constraint of fixed average value of the perimeter over all the connected components, the maximiser within the class of disconnected compact planar sets, consisting of finitely many simply connected components, is again a disk. In higher dimensions, we prove a completely new result that the lowest point in the spectrum is maximised by the exterior of a ball among all sets exterior to bounded convex sets satisfying a constraint on the integral of a dimensional power of the mean curvature of their boundaries. Furthermore, it follows that the critical coupling at which the lowest point in the spectrum becomes a discrete eigenvalue emerging from the essential spectrum is minimised under the same constraint by the critical coupling for the exterior of a ball.

中文翻译：

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紧集外部的最低Robin特征值的优化，II：非凸域和高维

我们考虑在任何维度上的紧凑集的外部，在有吸引力的Robin边界条件下，拉普拉斯算子的最低特征值的几何优化问题。正如在我们以前的工作（Krejčiřík和Lotoreichik J.凸肛门的改善。25（319-337，2018年），我们表明，在固定周长或面积的约束下，简单连接的平面集的外部类之内的最大化器始终是磁盘的外部，而无需凸性假设。此外，我们将结果推广到不连续的紧凑平面集。即，我们证明，在所有连接的组件的周长的平均值固定的约束下，由有限多个简单连接的组件组成的，不连续的紧凑平面集类别中的最大化器还是一个圆盘。在更高的维度上，我们证明了一个全新的结果：在有界凸集外部的所有集合中，球的外部最大化了频谱中的最低点，从而满足了对它们边界平均曲率的维数积分的约束。