Some free boundary problems recast as nonlocal parabolic equations

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 ph...

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Bibliographic Details
Published in:Nonlinear analysis 2019-12, Vol.189, p.111538, Article 111538
Main Authors: Chang-Lara, Héctor A., Guillen, Nestor, Schwab, Russell W.
Format: Article
Language:English
Subjects:
Boundary conditions
Dirichlet-to-Neumann
Free boundaries
Fully nonlinear equations
Global comparison principle
Hele-Shaw
Integro-differential operators
Manifolds (mathematics)
Mathematical analysis
Operators (mathematics)
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Summary: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 Rd, 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 Rd. 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.
ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2019.05.019