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Existence of solutions to an anisotropic phase‐field model
Author(s) -
Burman Erik,
Rappaz Jacques
Publication year - 2003
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.405
Subject(s) - mathematics , convexity , lipschitz continuity , anisotropy , isothermal process , operator (biology) , mathematical analysis , field (mathematics) , order (exchange) , phase (matter) , uniqueness , pure mathematics , thermodynamics , physics , biochemistry , chemistry , finance , repressor , quantum mechanics , gene , transcription factor , financial economics , economics
We consider an anisotropic phase‐field model for the isothermal solidification of a binary alloy due to Warren–Boettinger ( Acta. Metall. Mater . 1995; 43 (2):689). Existence of weak solutions is established under a certain convexity condition on the strongly non‐linear second‐order anisotropic operator and Lipschitz and boundedness assumptions for the non‐linearities. A maximum principle holds that guarantees the existence of a solution under physical assumptions on the non‐linearities. The qualitative properties of the solutions are illustrated by a numerical example. Copyright © 2003 John Wiley & Sons, Ltd.