Theory of pattern selection in three-dimensional nonaxisymmetric dendritic growth

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Abstract
We study the selection of the shape and growth velocity of three-dimensional dendritic crystals in cubically anisotropic materials. In the framework of asymptotics beyond all orders we derive the inner boundary-layer equation for the nonaxisymmetric shape correction to the Ivantsov paraboloid shape. The solvability condition for this equation provides selection for both the velocity and the shape. The comparison with available numerical and experimental results is reasonably good.