Free boundary problems and Viscosity solutions.

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Proposal DMS-0401436PIs: David Jerison and Christina Kim (MIT)Title: Free boundary problems and viscosity solutionsABSTRACTThe main focus of this project is on problems in nonlinear differentialequations in which the boundary is unknown and has to be determined: aso-called 'free boundary'. The PI is interested in the free boundaryproblems with nonlocal structure, where velocities of the free boundariesdepend on global characteristics of the solutions. The goal is to gain abetter understanding of the following aspects of such problems: theexistence and uniqueness of solutions in global time; asymptotic behaviorof the free boundary as time reaches zero or infinity; waiting timephenomena; and the regularity properties of the free boundaries. The PIemploys the notion of viscosity solutions to address the behavior ofaformentioned free boundary problems. This approach has been succesfullyused to study nonlinear PDEs and local laws of motions such as 'motion bymean curvature'. The great advantage of this approach is its seamlesshandling of topological transitions such as pinch-off. The key-step of thetheory of viscosity solutions is to establish a comparison principle. Thisproperty leads, in turn, to the existence of solutions and allows for thefurther study of their properties. Techniques coming from quantitativeversions of the comparison principle and harmonic analysis are employed tostudy further properties of solutions.The classical Stefan problem of melting ice is an example of the freeboundary problems. In the Stefan problem, one is modeling the evolution ofthe polar ice caps, and the question of interest is the location, as afunction of time, of the interface between water and ice. The particularproblems to which the methods of the present proposal apply also include:theHele-Shaw problem which models fluid motion in a narrow cell betweentwo parallel plates; flame fronts; and the interface between oil andwater in a flow. This is an area where modeling and computation are farahead of mathematical analysis. The main obstacle for developing awell-defined notion for general motion of interfaces is that initiallysmooth boundaries moving under smooth velocities may develop singularitiesin finite time. The hitting and splitting of interfaces in flamepropagation is an example. It is natural to wonder whether thecontinuation of the solutions is uniquely determined after pinch-off, orwhether additional constitutive information might be required at thesingular time. The impact of the singularity to the other part of the freeboundary is another interesting question. The PI plans to investigatethese questions in local and global perspectives.