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

Abstract

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