Mathematical existence of crystal growth with Gibbs-Thomson curvature effects
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 equations above, in the sense of distributions. When \(\phi\) is smooth and elliptic, the Gibbs-Thomson condition holds exactly along the crystal interface at almost every time, and in two and three dimensions the crystal interfaces are Hölder continuously differentiable hypersurfaces.
Hölder continuously differentiable hypersurface
scientific article; zbMATH DE number 1542072
scientific article