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BIRS Workshop Lecture Videos

Free boundary problems as parabolic Integro-differential equations Guillen, Nestor


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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