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We investigate the classical eigenvalue problem that arises in hydrodynamics and is referred to as the sloshing problem. It describes free liquid oscillations in a liquid container  W ⊂ℝ 3 . The Cartesian coordinates ( x ,  y ,  z ) are chosen so that the mean free surface of the liquid  F  lies in the ( x ,  z )‐plane and the  y ‐axis is directed upwards. We study the case when  W  is an axially symmetric, convex, bounded domain such that  W ⊂ F ×(−∞, 0). Our first result states that the fundamental eigenvalue has multiplicity 2 and for each fundamental eigenfunction φ, there is a change of  x ,  z ‐coordinates by a rotation around the  y ‐axis so that φ is odd in  x ‐variable.  The second result of the paper gives the following monotonicity property of the fundamental eigenfunction φ. If φ is odd in  x ‐variable, then it is strictly monotonic in  x ‐variable. This property has the following hydrodynamical meaning. If the liquid oscillates freely with the fundamental frequency according to φ, then the free surface elevation of the liquid is increasing along each line parallel to the  x ‐axis during one half‐period of time and decreasing during the other half‐period. The proof of the second result is based on the method developed by Jerison and Nadirashvili for the hot‐spots problem for the Neumann–Laplacian.

On high spots of the fundamental sloshing eigenfunctions in axially symmetric domains