Some free boundary problems recast as nonlocal parabolic equations

TL;DR

This paper shows that certain free boundary problems, including Hele-Shaw flows, can be reformulated as nonlocal parabolic equations on submanifolds, providing new insights into their dynamics and regularity properties.

Contribution

It introduces a novel framework connecting free boundary problems to nonlocal parabolic equations with the Global Comparison Property, enabling unified analysis of various free boundary flows.

Findings

Free boundary problems can be recast as nonlocal parabolic equations.

The free boundary condition defines a nonlocal operator with the Global Comparison Property.

Propagation of modulus of continuity for solutions is established across problems.

Abstract

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.