The Dam Problem

Abstract

Among the applications of Variational Inequalities with obstacles, the most famous one is probably that of the Dam Problem. The modern approach started with the pioneering work of C. Baiocchi (see [9], [10]). Many developments have followed, many of them due to the Pavia school, (see [11], [12], [16], [17] and [13], [15] where a complete bibliography can be found). Unfortunately this approach is only possible in the case of porous media with vertical walls. For instance, the case of a rectangular dam (as in the figure (4.1(B)) is suitable in dimension 2 (see [76], [102] for the case of porous media in three dimensions). To overcome this restriction several attempts have been made including formulations via Quasivariational Inequalities (see [11], [12], [15]) and the use of a “maximal” solution (see [5]) (see also [109] for a very interesting approach). More recently H. Brezis, D. Kinderlehrer, and G. Stampacchía in [35] and H. W. Alt in [7] introduced a new formulation which can treat the case of general domains. We shall adopt the setting of [35] and we will show how it can be reduced to a Variational Inequality with obstacle in the rectangular case. Let us first begin by presenting the physical problem (see [32], [49]).