A class of parabolic quasi-variational inequalities,II

We consider a quasi-variational inequality (q.v.i.) introduced by A. Friedman and D. Kinderlehrer. A q.v.i. of this form gives rise, at least formally, to a Stefan problem of melting of water, where the relation ?v_x(x, t) = ?a(x, t)·(t) + b(x, t) holds on the free boundary x = s(t), and a > 0, b ? 0; the water temperature, v(x, t), is not necessarily nonnegative. In the standard Stefan problem a ≡ 1, b ≡ 0, and v ? 0. Friedman and Kinderlehrer proved the existence of a solution of the q.v.i. by a fixed point theorem for monotone mappings. Here we prove the existence of a solution by an entirely different method, based on finite difference approximations. The solution is shown to be smoother than that constructed by Friedman and Kinderlehrer.