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Fully Nonlinear Phase Field Equations and Generalized Mean Curvature Motion

1995; Taylor & Francis; Volume: 20; Issue: 1-2 Linguagem: Inglês

10.1080/03605309508821092

1532-4133

Robert L. Jerrard,

Solidification and crystal growth phenomena

A number of the ideas we will use were introduced by Barles, Soner, and Souganidis, who used a nonlinear change of variables to prove global convergence of fairly general phase-field equations to geometric motions in the semilinear case. The averaging effects require that we combine these methods with techniques developed by L.C. Evans in the context of periodic homogenization. Making these two arguments work simultaneously is the key technical issue of this paper. In this paper we give notation that will be used throughout this paper and we state our main theorem. Section 2 contains a brief discussion of existence and uniqueness issues. In Section 3 we construct supersolutions that enable us to derive local uniform bounds on the functions z. Using these local uniform bounds and various technical constructions, we can pass to limits in a weak sense. This is carried out in Section 4. In Section 5 we complete the proof of our main theorem, using the results of Sections 3 and 4 and the earlier work of Barles, Soner, and Souganidis. Finally, Section 6 contains an appendix in which we prove certain facts about ODEs which are used in the body of the paper.