Evolution for a constrained problem related to the one dimensional alt Caffarelli functional

This paper deals with the evolution for the one dimensional Alt Caffarelli functional I(u) = ∫ 01 |u'(x)|2 dx + meas([u > 0]), with respect to the L2 metric, subject to an obstacle condition u ≥ φu on [u ≥ 0], where φu(x) ≃ ε dist(x, [u ≤ 0]). This leads to solving a parabolic equation with a free boundary condition on the set where u equals zero and a variational imequality where u = φu. The introduction of the obstacle condition makes it easier to apply the framework of the "curves of maximal slope". It is expected that this approach could allow to treat the unconstraned problem if the initial data does not touch the obstacle.