  We propose a method to determine the smoothness of sufficiently flat
solutions of one phase Hele-Shaw problems. The novelty is the observation that
under a flatness assumption the free boundary --represented by the hodograph
transform of the solution- solves a nonlinear integro-differential equation.
This nonlinear equation is linearized to a (nonlocal) parabolic equation with
bounded measurable coefficients, for which regularity estimates are available.
This fact is used to prove a regularity result for the free boundary of a weak
solution near points where the solution looks sufficiently flat. More
concretely, flat means that in a parabolic neighborhood of the point the
solution lies between the solutions corresponding to two parallel flat fronts a
small distance apart --a condition that only depends on the the local behavior
of the solution. In a neighborhood of such a point, the free boundary is given
by the graph of a function whose spatial gradient enjoys a universal H\"older
estimate in both space and time.
