Title: Sloshing in containers with vertical walls: isoperimetric inequalities for the fundamental eigenvalue Show Chinese title

Title: 容器中具有垂直壁的晃动：基本特征值的等周不等式

Abstract: One isoperimetric inequality for the fundamental sloshing eigenvalue is derived under the assumption that containers have vertical side walls and either finite or infinite depth. It asserts that among all such containers, whose free surfaces are convex, have two axes of symmetry and a given perimeter length, this eigenvalue is maximized by infinitely deep ones provided the free surface is either the square or the equilateral triangle. The proof is based on the recent isoperimetric result obtained by A. Henrot, A. Lemenant and I.~Lucardesi for the first nonzero eigenvalue of the two-dimensional Neumann Laplacian under the perimeter constraint. Another isoperimetric inequality for the fundamental eigenvalue, which describes sloshing in containers with vertical walls, is a consequence of the classical result due to G. Szeg\H o concerning the first nonzero eigenvalue of the free membrane problem. Show Chinese abstract

Abstract: 关于基本晃动特征值的一个等周不等式是在假设容器具有垂直侧壁且深度有限或无限的情况下推导出来的。 它断言，在所有这样的容器中，自由表面为凸形、具有两个对称轴且给定周长的所有容器里，只要自由表面是正方形或等边三角形，这一特征值在深度无限时达到最大。 证明基于A. Henrot, A. Lemenant和I.~Lucardesi最近对于二维Neumann拉普拉斯算子第一个非零特征值在周长约束下的等周结果。 另一个关于基本特征值的等周不等式，用于描述具有垂直墙壁的容器中的晃动情况，是由于G. Szegő关于自由膜问题第一个非零特征值的经典结果所导致的。