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Free boundary problems as parabolic Integro-differential equations Guillen, Nestor
Description
We demonstrate that a class of one and two phase free boundary problems can be recast as nonlocal parabolic equations on a codimension one submanifold. The canonical examples would be one-phase Hele-Shaw and Laplacian growth. 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) and nonlinear parabolic equation in Euclidean space. 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, a propagation of modulus of continuity for weak solutions of the free boundary flow. This is based on joint works with Hector Chang-Lara and Russell Schwab.
Item Metadata
Title |
Free boundary problems as parabolic Integro-differential equations
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2018-06-19T11:00
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Description |
We demonstrate that a class of one and two phase free boundary problems can be
recast as nonlocal parabolic equations on a codimension one submanifold. The
canonical examples would be one-phase Hele-Shaw and Laplacian growth. 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) and nonlinear parabolic equation in
Euclidean space. 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, a propagation of modulus of continuity for weak solutions of the
free boundary flow. This is based on joint works with Hector Chang-Lara and
Russell Schwab.
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Extent |
31.0
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: University of Massachusetts Amherst
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Series | |
Date Available |
2019-03-15
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0376900
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Researcher
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International