‘high Spots’ Theorems for Sloshing Problems

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 (canal’s crosssection), 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 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.