We investigate the stability of the evolution by anysotropic and crystalline curvature starting from an initial surface equal to the Wulff shape. It is well known that the Wulff shape evolves selfsimilarly according to the law $V=-\kappa_\phi n_\phi$. Here the index $\phi$ refers to the underlying anisotropy described by the Wulff shape, so that $\kappa_\phi$ is the relative mean curvature and $n_\phi$ is the Cahn-Hoffmann conormal vector field. Such selfsimilar evolution is also known to be stable under small perturbations of the initial surface in the isotropic setting (the Wulff shape is a sphere) or in 2D if the underlying anisotropy is symmetric. We show that this evolution is unstable for some specific choices of the Wulff shape both rotationally symmetric and fully crystalline.

Paolini, M., Pasquarelli, F., Unstable crystalline Wulff shapes in 3D, in Progress in Nonlinear Differential Equations and Their Applications 51, (Cernobbio, 04-06 July 2001), Birkhäuser Verlag, Basilea 2002:2002;(51) 141-153 [http://hdl.handle.net/10807/20433]

Unstable crystalline Wulff shapes in 3D

Abstract

We investigate the stability of the evolution by anysotropic and crystalline curvature starting from an initial surface equal to the Wulff shape. It is well known that the Wulff shape evolves selfsimilarly according to the law $V=-\kappa_\phi n_\phi$. Here the index $\phi$ refers to the underlying anisotropy described by the Wulff shape, so that $\kappa_\phi$ is the relative mean curvature and $n_\phi$ is the Cahn-Hoffmann conormal vector field. Such selfsimilar evolution is also known to be stable under small perturbations of the initial surface in the isotropic setting (the Wulff shape is a sphere) or in 2D if the underlying anisotropy is symmetric. We show that this evolution is unstable for some specific choices of the Wulff shape both rotationally symmetric and fully crystalline.
Scheda breve Scheda completa Scheda completa (DC)
Inglese
Progress in Nonlinear Differential Equations and Their Applications 51
Variational methods for discontinuous structures
Cernobbio
4-lug-2001
6-lug-2001
3-7643-6913-2
Paolini, M., Pasquarelli, F., Unstable crystalline Wulff shapes in 3D, in Progress in Nonlinear Differential Equations and Their Applications 51, (Cernobbio, 04-06 July 2001), Birkhäuser Verlag, Basilea 2002:2002;(51) 141-153 [http://hdl.handle.net/10807/20433]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10807/20433
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