1. We begin by giving a simple example of a partial differential inequality that occurs in an elementary physics problem. We consider a fluid with pressure u(x t) at the point x at the instant t that 3 occupies a region Q oflR bounded by a membrane r of negligible thickness that however is semi-permeable i. e. a membrane that permits the fluid to enter Q freely but that prevents all outflow of fluid. One can prove then (cf. the details in Chapter 1 Section 2.2.1) that au (aZu azu aZu) (1) in Q t>o -a - du = g du = -a z + -a z + -a z t Xl X X3 z l g a given function with boundary conditions in the form of inequalities u(X t»o => au(x t)/an=O XEr (2) u(x t)=o => au(x t)/an?: O XEr to which is added the initial condition (3) u(x O)=uo(x). We note that conditions (2) are non linear; they imply that at each fixed instant t there exist on r two regions r and n where u(x t) =0 and au (x t)/an = 0 respectively. These regions are not prescribed; thus we deal with a free boundary problem.