论文标题
通过边界均化的steklov特征值的灵活性
Flexibility of Steklov eigenvalues via boundary homogenisation
论文作者
论文摘要
最近，D。Bucur和M. Nahon使用边界均匀化来显示平面域特征值的显着灵活性。在本文中，我们将结果扩展到更高的维度，并将其与边界的任意流形扩展到任意歧管，尽管在这种情况下，边界通常不会表现出任何周期性结构。我们的论点使用各种特征值的框架，并提供了原始结果的不同证明。此外，我们提出了这种灵活性在外围约束下优化Steklov特征值的应用。事实证明，零属表面的归一化steklov特征值的最佳上限和任何固定数量的边界组件始终可以被平面域饱和。即使任何实际的最大化器（除了简单的连接表面除外）始终远非平面本身，却是这种情况。特别是，它产生了双重连接的平面域的第一个steklov特征值的尖锐上限。
Recently, D. Bucur and M. Nahon used boundary homogenisation to show the remarkable flexibility of Steklov eigenvalues of planar domains. In the present paper we extend their result to higher dimensions and to arbitrary manifolds with boundary, even though in those cases the boundary does not generally exhibit any periodic structure. Our arguments use framework of variational eigenvalues and provides a different proof of the original results. Furthermore, we present an application of this flexibility to the optimisation of Steklov eigenvalues under perimeter constraint. It is proved that the best upper bound for normalised Steklov eigenvalues of surfaces of genus zero and any fixed number of boundary components can always be saturated by planar domains. This is the case even though any actual maximiser (except for simply connected surfaces) is always far from being planar themselves. In particular, it yields sharp upper bound for the first Steklov eigenvalue of doubly connected planar domains.
