Remarks on the one-dimensional sloshing problem involving the $p$-Laplacian operator

In this paper, we study the inverse nodal problem and the eigenvalue gap for the one-dimensional sloshing problem with the $p$-Laplacian operator. By applying the Prüfer substitution, we first derive the reconstruction formula of the depth function by using the information of the nodal data. Furthermore, we employ the Tikhonov regularization method to consider how to reconstruct the depth function using only zeros of one eigenfunction. Finally, we investigate the eigenvalue gap under the restriction of symmetric single-well depth functions. We show the gap attains its minimum when the depth function is constant.

Keywords:

Inverse nodal problem, Tikhonov regularization method eigenvalue gap, $p$-Laplacian,

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