Sloshing in containers with vertical walls: isoperimetric inequalities for the fundamental eigenvalue
Nikolay Kuznetsov
TL;DR
The paper investigates isoperimetric properties of the fundamental sloshing eigenvalue $\nu_1$ in containers with vertical walls and convex free surfaces, establishing two sharp inequalities under symmetry and constraint assumptions. By connecting the sloshing problem to the Neumann Laplacian eigenvalue $\mu_1(F)$ via a perimeter constraint (and leveraging the Henrot–Lemenant–Lucardesi result) and to the classical Szeg\H o bound for the membrane problem, it derives that $P(F)\,\nu_1(W) \le 4\pi$ with equality only in the infinite-depth, square or equilateral-triangle free surface case, and $\sqrt{|F|}\,\nu_1(W)\le \sqrt{\pi}\, j'_{1,1}$ with equality only for infinite depth and a disk free surface. These results rely on domain monotonicity of sloshing eigenvalues, the variational characterization of $\nu_1$, and established isoperimetric inequalities for related eigenvalue problems. The findings clarify how extremal container shapes arise under perimeter or area constraints and within symmetry classes, highlighting the role of infinite depth as an asymptotic extremal regime.
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.