Some remarks on a problem of percolation in gently sloping beaches

We study a free boundary problem related to a model of unsteady flow of a fluid through a special porous medium, precisely a gently sloping beach, when an asymptotic formulation is considered (see a paper by J. M. Aitchison, C. M. Elliott and J. R. Ockendon [I.M.A. J. Appl. Math. 30 (1983), no. 3, 269–287; MR0719980] for details). We achieve a variational formulation of the problem for which we show existence and uniqueness of the solution.