*Preprint*

**Inserted:** 2 sep 2012

**Last Updated:** 2 sep 2012

**Year:** 2012

**Notes:**

Work in progress

**Abstract:**

We are interested in the following one-phase Stefan problem: \(\partial_t(u+\chi)=\triangle u+f\chi\), \(u(0)=g\), \(\chi\in \mathcal{H}(u(t))\), where \(u\) is defined on \([0, +\infty)\times\mathbb{R}^n\), both \(g\) and \(f\) are non-negative time-independent functions and \(\mathcal{H}:\mathbb{R}\rightarrow\mathcal{P}([0, 1])\) is the usual maximal monotone Heavyside graph. We introduce the notion of w-weak and of weak solution. Adapting suitably the variational techniques introduced in a paper of Tilli and Scianna (see $[5]$) we get a discrete in time approximation scheme involving, step by step, one-phase Hele-Shaw problems, the approximate solutions having a-priori estimates which pass to the limit; existence, uniqueness, monotonicity and regularity results for w-weak and weak solutions are then obtained, at last.