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Uniqueness of local minimizers for crystalline variational problems

Hold Date
2021-04-13 12:00〜2021-04-13 13:00
Zoom meeting
Object person
Miyuki Koiso (IMI, Kyushu University)

Abstract: An anisotropic surface energy is the integral of an energy density that depends on the normal at each point over the considered surface, which is a generalization of the area of surfaces. The minimizer of any such an energy among all closed surfaces enclosing the same volume is unique and it is (up to rescaling) so-called the Wulff shape. When the Wulff shape is a polyhedron, local minimizers for volume-preserving variations serve as a mathematical model of single crystals, and problems on such surfaces are called crystalline variational problems. In crystalline variational problems, the energy density function is not differentiable at the points corresponding to the faces and edges of the Wulff shape, and equilibrium surfaces also have singular points such as edges and vertices. These facts make it impossible to apply the standard variational methods to crystalline variational problems. In this talk, by using “multi-valued Gauss map” and “multi-valued Cahn-Hoffman field”, we show some results on the uniqueness for local minimizers of crystalline variational problems. This is joint work with Dr. Kento Okuda (IMI).