A mathematical model of the wearing process of a nonconvex stone (Q2782958)

In [Mathematika 21, 1--11 (1974; Zbl 0311.52003)], \textit{W. J. Firey} proposed a mathematical model to describe the wearing of a convex stone rolling on a beach. In this model the boundary of the stone evolves by the well known Gauss curvature flow, which was subsequently studied by \textit{K. Tso} [Commun. Pure Appl. Math. 38, 867--882 (1985; Zbl 0612.53005)], \textit{B. Andrews} [Invent. Math. 138, 151--161 (1999; Zbl 0936.35080)], and others.NEWLINENEWLINEHere the authors formulate a model to describe the wearing of a nonconvex stone, along the lines proposed by Firey for the convex case. The nonconvexity of the stone has the consequence that the evolution equation obtained is nonlocal, which makes the analysis of the equation difficult. They also derive a comparison result, an existence theorem and some stability properties for viscosity solutions of the flow under the simplifying (but unrealistic) assumption that the boundary of the evolving stone is the graph of a function \(u:{\mathbb R}^n\times[0,\infty)\rightarrow{\mathbb R}\).