Mathematical existence of crystal growth with Gibbs-Thomson curvature effects
type: Item
instance of: scholarly article
title: Mathematical existence of crystal growth with Gibbs-Thomson curvature effects
zbMATH Open document ID: 0981.74041
DOI: 10.1007/BF02921806
author: Lihe Wang
author: Frederick J. jun. Almgren
published in: The Journal of Geometric Analysis
publication date: 2000-12-11T00:00:00Z
review text: In this impressive paper the authors rigorously examine a process which models a certain type of crystal growth. It incorporates the Gibbs-Thomson relation between the interface surface tension and temperatures, and allows heat capacities and nonisotropic heat conductivites which differ in crystal and in melt. The process is a combination of a heat flow with a ``crystal moving step'' based on a variational argument. The authors first construct approximating evolutions: on a small interval \(\Delta t_{j}\), they assume that the crystal \(K_{j}\) remains constant while the temperature field \(T_{j}\) and heat distribution \(Q_{j}\) evolve according to the heat equations \(c_{K }T_{j} = Q_{j}, \;{\partial \over \partial t} Q_{j} = \text{div} (\kappa_{K} \nabla T_{j}).\) Here \(\kappa_{K}\) is heat conductivity, and \(c_{K}\) is heat capacity. They prove existence for this flow, together with a regularity theorem when the boundary is a Hölder continuously differentiable hypersurface. At the end of the time interval \(\Delta t_{j}\), the authors stop the clock and proceed with the variational argument. Using techniques from geometric measure theory, they minimize the quantity \[ \phi(\partial {{L}}) + \int_{\Omega} c_{L} F(T) + {1 \over \Delta t^{\alpha}}||P-Q|| \] to determine the new crystal position \(L\) and heat distribution \(Q\). Here, \(\phi\) is the surface energy density, \(P\) is the heat distribution at the end of previous time interval, \(||P-Q||\) is the Monge-Kantorovich distance between \(P\) and \(Q\), \(\alpha\) is a suitable small positive constant, and \(F(T)\) is the bulk energy density determined by a prescribed Gibbs-Thomson function. The ambient space \(\Omega\) is taken to be a flat torus. The evolution is now the limit as \(j \rightarrow \infty\) of these approximating evolutions (here \(\Delta t_{j} \rightarrow 0\)). The authors prove that for suitable initial and boundary conditions, \(K(t)\), \(Q(t)\), and \(T(t)\) exist and solve the heat eq…
reviewed by: Christine Guenther
Mathematics Subject Classification ID: 74N05
Mathematics Subject Classification ID: 74G65
Mathematics Subject Classification ID: 80A22
Mathematics Subject Classification ID: 35K05
zbMATH DE Number: 1542072
zbMATH Keywords: crystal growth
zbMATH Keywords: Gibbs-Thomson curvature effects
zbMATH Keywords: heat flow
zbMATH Keywords: crystal moving step
zbMATH Keywords: existence theorem
zbMATH Keywords: regularity theorem
zbMATH Keywords: Hölder continuously differentiable hypersurface
zbMATH Keywords: geometric measure theory
zbMATH Keywords: surface energy density
zbMATH Keywords: Monte-Kantorovich distance
zbMATH Keywords: bulk energy density
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1542072