A variational approach to the Hele-Shaw flow with injection

We investigate a variational approach to the Hele–Shaw flow \partial_t\chi = \Delta u + f\chi, f ≥ 0 in R^n, where \chi is the characteristic function of an open set \Omega(t)\in\R^n and u(\cdot, t)\in H^1_0(\Omega(t)) solves −\Delta u(\cdot, t)= f in \Omega(t). By iteratively solving a variational problem in R^n, we construct a staircase family of opens sets and a corresponding family of functions: both sets and functions converge increasingly, at fixed time, to a weak solution of the problem. When the latter is not unique, the solution thus obtained is characterized by a minimality property, with respect to set inclusion, at fixed time. We also prove several monotonicity results of the solutions thus obtained, with respect to both the initial set and the forcing term f.