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

Some free boundary problems recast as nonlocal parabolic equations