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arXiv · 1807.02714

Some free boundary problems recast as nonlocal parabolic equations

Hector A. Chang-Lara·Nestor Guillen·Russell W. Schwab
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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.

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Hector A. Chang-Lara, Nestor Guillen, Russell W. Schwab. 2018-07-07. Some free boundary problems recast as nonlocal parabolic equations. https://arxiv.org/abs/1807.02714

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