分野別セミナー

In this talk I will give an overview of the notion of viscosity solutions for the crystalline mean curvature flow in three or more dimensions, introduced in joint work with Yoshikazu Giga from the University of Tokyo. This problem serves as a model of crystal growth but it also has applications in image processing and related fields. Its level set formulation leads to a nonlocal, very singular parabolic equation with non-smooth, faceted solutions to which the standard viscosity theory does not apply. I will also show how the ODE crystalline algorithm can be used to generate interesting explicit viscosity solutions, in particular self-similar shrinking compact surfaces of positive genus in three dimensions.