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Generalized motion by mean curvature with Neumann conditions and the Allen-Cahn model for phase transitions
Authors:Markos Katsoulakis  Georgios T Kossioris  Fernando Reitich
Institution:1. Department of Mathematics, Michigan State University, 48824-1027, East Lansing, Michigan
2. Department of Mathematics, University of Crete, P.O. Box 470, 71409, Hraklion, Crete, Greece
3. Department of Mathematics, North Carolina State University, Box 8205, 27695-8205, Raleigh, NC
Abstract:We study asharpinterface model for phase transitions which incorporates the interaction of the phase boundaries with the walls of a container Ω. In this model, the interfaces move by their mean curvature and are normal to δΩ. We first establish local-in-time existence and uniqueness of smooth solutions for the mean curvature equation with a normal contact angle condition. We then discuss global solutions by interpreting the equation and the boundary condition in a weak (viscosity) sense. Finally, we investigate the relation of the aforementioned model with atransitionlayer model. We prove that if Ω isconvex, the transition-layer solutions converge to the sharp-interface solutions as the thickness of the layer tends to zero. We conclude with a discussion of the difficulties that arise in establishing this result in nonconvex domains. Communicated by David Kinderlehrer
Keywords:Math Subject Classification" target="_blank">Math Subject Classification  35A05  53A10
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