  In this work we demonstrate that a class of some one and two phase free
boundary problems can be recast as nonlocal parabolic equations on a
submanifold. The canonical examples would be one-phase Hele Shaw flow, as well
as its two-phase analog. We also treat nonlinear versions of both one and two
phase problems. In the special class of free boundaries that are graphs over
$\mathbb{R}^d$, we give a precise characterization that shows their motion is
equivalent to that of a solution of a nonlocal (fractional), nonlinear
parabolic equation for functions on $\mathbb{R}^d$. Our main observation is
that the free boundary condition defines a nonlocal operator having what we
call the Global Comparison Property. A consequence of the connection with
nonlocal parabolic equations is that for free boundary problems arising from
translation invariant elliptic operators in the positive and negative phases,
one obtains, in a uniform treatment for all of the problems (one and two
phase), a propagation of modulus of continuity for viscosity solutions of the
free boundary flow.
