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Bulletin of the London Mathematical Society Advance Access originally published online on March 19, 2009
Bulletin of the London Mathematical Society 2009 41(3):495-505; doi:10.1112/blms/bdp021
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© 2009 London Mathematical Society

‘High spots’ theorems for sloshing problems

Tadeusz Kulczycki

Institute of Mathematics
Polish Academy of Sciences
ul. Kopernika 18
51-617 Wroclaw
Poland
Institute of Mathematics and Computer Science
Technical University of Wroclaw
Wybrzeze Wyspianskiego 27
50-370 Wroclaw
Poland

Nikolay Kuznetsov

Laboratory for Mathematical Modelling of Wave Phenomena
Institute for Problems in Mechanical Engineering
Russian Academy of Sciences
V.O., Bol'shoy pr. 61
199178 St. Petersburg
Russian Federation
ngk@wave.ipme.ru

Received 28 April 2008. Revision received 11 November 2008.

We investigate several 2D and 3D cases of the classical eigenvalue problem that arises in hydrodynamics and is referred to as the sloshing problem. In particular, for a domain W sub R2 (canal's cross-section), where {partial}W = F {cup} B and F (cross-section of the free surface of fluid) is an interval of the x-axis, whereas B (bottom's cross-section) is the graph of a negative function, the following result is proved. The fundamental eigenfunction u1 of the sloshing problem (the corresponding eigenvalue is simple) has monotonic traces on F and B; moreover, u1 attains its maximum and minimum values at the end-points of F. It is established that for the 2D (3D) ice-fishing problem with a single (circular) hole, the function u1 (both fundamental eigenfunctions) attains its maximum value at an inner point of F. A relationship between the high spots and hot spots theorems is considered.

2000 Mathematics Subject Classification 35Q35, 76B10, 35P99.

The first author was partially supported by KBN grant.


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