A Stefan problem on an evolving surface.

We formulate a Stefan problem on an evolving hypersurface and study the well posedness of weak solutions given L(1) data. To do this, we first develop function spaces and results to handle equations on evolving surfaces in order to give a natural treatment of the problem. Then, we consider the existence of solutions for L(∞) data; this is done by regularization of the nonlinearity. The regularized problem is solved by a fixed point theorem and then uniform estimates are obtained in order to pass to the limit. By using a duality method, we show continuous dependence, which allows us to extend the results to L(1) data.