Two examples of nonconvex self-similar solution curves for a crystalline curvature flow
The main goal of the present paper consists in the construction of examples of nonconvex self-similar solutions for a crystalline curvature flow with an interfacial energy of which the Wulff shape is a regular triangle or a square. Let \(\mathcal{P}\) a \(K\)-admissible curve. For each edge \(S_k\), a crystalline curvature \(H_k\) is defined in terms of the length of \(S_k\) and the length of the \(n\)-th edge of the Wulff shape. Under a crystalline curvature flow, each edge \(S_k\) keeps the same direction but moves in the outward normal direction with the velocity \(V_k\) determined by a homogeneous function of some degree \(\alpha>0\) in the crystalline curvature \(H_k\): \[ V_k=\sigma(\theta_k)| H_k| ^{\alpha-1}H_k, \] where \(\sigma\) defines the interfacial energy density and \(\theta_k\) is the exterior normal angle of the \(k\)-th edge \(S_k\). Among other results, the authors show that, for \(\alpha\in(0,1)\), a nonconvex self-similar solution exists even in the case of free-orientation motions, i.e., \(\sigma(\theta+\pi)=\sigma(\theta)\). This result is in contrast with that obtained in the case of motion by smooth interfacial density, where the curve becomes convex in a finite time.
scientific article; zbMATH DE number 2151799