On a one-phase Stefan problem

In this paper we introduce a new way to approach the one-phase Stefan problem partial_t(u +chi) = Delta u + fchi, u(0) = g, chiin H(u(t)), with both g and f non-negative and where H : R ightarrow P([0; 1]) is the usual maximal monotone Heavyside graph. Our idea is to introduce a discrete in time approximation scheme involving, step by step, suitable one-phase Hele-Shaw problems. With natural regularity hypotheses on data we are able to solve them, by means of results similar to a joint work of the second author with P. Tilli. Using the qualitative behavior of these solutions we can then construct a discrete in time approximate solution for the Stefan problem endowed with suitable a-priory estimates. So we can ¯finally pass to the limit, getting existence of solutions, uniqueness, monotonicity and regularity results.