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‘High spots’ theorems for sloshing problems
Author(s) -
Kulczycki Tadeusz,
Kuznetsov Nikolay
Publication year - 2009
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdp021
Subject(s) - eigenfunction , mathematics , slosh dynamics , monotonic function , eigenvalues and eigenvectors , domain (mathematical analysis) , function (biology) , mathematical analysis , graph , simple (philosophy) , cross section (physics) , interval (graph theory) , geometry , combinatorics , physics , mechanics , quantum mechanics , philosophy , epistemology , evolutionary biology , biology
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 ⊂ R 2 (canal's cross‐section), where ∂ W = F ∪ 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 u 1 of the sloshing problem (the corresponding eigenvalue is simple) has monotonic traces on F and B ; moreover, u 1 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 u 1 (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.