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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. E. Payne, “Isoperimetric inequalities and their applications,” SIAM Rev. 9, 453–488 (1967).
C. Bandle, Isoperimetric Inequalities and Applications, Pitman, Boston etc. (1980).
A. Henrot, A. Lemenant, and I. Lucardesi, “An isoperimetric problem with two distinct solutions,” Trans. Am. Math. Soc. https://doi.org/10.1090/tran/9074 (February 13, 2024).
H. Lamb, Hydrodynamics, Cambridge Univ. Press, Cambridge (1932).
P. Henrici, B. A. Troesch, and L. Wuytack, “Sloshing frequencies for a half-space with circular or strip-like aperture,” Z. Angew. Math. Phys. 21, 285–318 (1970).
N. N. Moiseev, “Introduction to the theory of oscillations of liquid-containing bodies,” Adv. Appl. Mech. 8, 233–289 (1964).
A. N. Komarenko, I. A. Lukovskii, and S. F. Feshchenko, “On the eigenvalue problem with a parameter in the boundary conditions” [in Russian], Ukr. Mat. Zh. 17, No. 6, 22–30 (1965).
B. A. Troesch, “An isoperimetric sloshing problem,” Commun. Pure Appl. Math. 18, 319–338 (1965).
G. Szegő, “Inequalities for certain eigenvalues of a membrane of given area,” J. Ration. Mech. Anal. 3, 343–356 (1954).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Problemy Matematicheskogo Analiza 126, 2024, pp. 21-25.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Kuznetsov, N. Sloshing in Containers with Vertical Walls: Isoperimetric Inequalities for the Fundamental Eigenvalue. J Math Sci 279, 472–477 (2024). https://doi.org/10.1007/s10958-024-07026-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-024-07026-y