Sloshing in Containers with Vertical Walls: Isoperimetric Inequalities for the Fundamental Eigenvalue

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. We assert that, among all containers such that their free surfaces are convex and the containers have two symmetry axes and a given perimeter length, this eigenvalue is maximized by infinitely deep ones provided that the free surface is either the square or the equilateral triangle. Another isoperimetric inequality for the fundamental eigenvalue describing sloshing in containers with vertical walls is a consequence of the classical result due to G. Szegő concerning the first nonzero eigenvalue of the free membrane problem.