The dam problem for nonlinear Darcy's laws and Dirichlet boundary conditions.

Abstract

The subject of this paper is the study of a free boundary problem for a steady fluid flow through a porous medium, in which the classical Darcy law (1) −!v = ar(p(x)+xn), x = (x1, · · · , xn) 2 Rn, a > 0, is replaced by the nonlinear law (2) |−!v |m−1−!v = ar(p(x)+xn), x = (x1, · · · , xn) 2
Rn, a, m > 0, where −!v and p are, respectively, the velocity and the pressure of the fluid. This approach is particularly interesting because Darcy’s law was established on a purely experimental basis; but it is not clear why, from a physical point of view, the specific form of (2) gives a better model for the dam problem. The authors first reduce the problem to a variational inequality
involving the degenerate Laplacian operator for the hydrostatic head u(x) = p(x) + xn. Then,using a perturbation argument, they prove existence of weak solutions. The remaining portion of the paper is devoted to the study of the qualitative properties of the solutions. In particular, it is proven that the free boundary is a lower semicontinuous curve of the form xn = (x1, · · · , xn−1),and that there is a unique minimal solution.
Moreover, in the two-dimensional case the authors show that is actually continuous, and that there is a unique S3-connected solution.