Two examples of fattening for the curvature flow with a driving force
The authors consider the (mean) curvature flow with a driving force \(g(t)\) for curves in the plane. They give two examples of a bounded Lipschitz function \(u_0 : \mathbb{R}^2 \to \mathbb{R}\) whose level curves evolve in time, in the generalized viscosity sense, such that a certain regular initial level curve develops, after a positive finite time, a full two-dimensional interior part. This behavior is called fattening, and can be interpreted as a phenomenon of non-uniqueness of the level-set flow. For \(g \equiv 0\) it is known that all smooth compact level curves of \(u_0\) never develop an interior. One of the examples is \(g(t) = 1 - t\) where the initial level curve is the union of two disjoint circles having the same radius. The second example is \(g \equiv 1\). Here, the initial level curve is the union of two disjoint circles having different radii.
scientific article; zbMATH DE number 722299