Some free boundary problems recast as nonlocal parabolic equations. (December 2019)

• Record Type: Journal Article
• Authors: Chang-Lara, Héctor A.; Guillen, Nestor; Schwab, Russell W.
• Is Part Of: Nonlinear analysis. Volume 189(2019)
• Publication Date: 2019-12
• Subjects: 35B51 -- 35R09 -- 35R35 -- 45K05 -- 47G20 -- 49L25 -- 60J75 -- 76D27 -- 76S05; Global comparison principle -- Integro-differential operators -- Dirichlet-to-Neumann -- Free boundaries -- Hele-Shaw -- Fully nonlinear equations; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.05.019
• Languages: English
• ISSNs: 0362-546X

Abstract

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 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 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.