Anisotropic motion of an interface relaxed by the formation of infinitesimal wrinkles
Consider a region \(\Omega (t)\) in the plane whose boundary curve evolves according to the equation \(B (\Theta) V = G (\Theta) K - U\). Here \(V\) denotes the normal velocity, \(K\) the curvature (measured w.r.t. the outward normal), and \(B (\Theta)\), \(G (\Theta)\) are given functions of the normal-angle \(\Theta\), which is the counterclockwise angle from a fixed axis to the outward normal of \(\partial \Omega (t)\), and \(U\) is a given constant. Assuming that \(B\) and \(G\) are continuous and strictly positive functions, the above equation is of parabolic type with well- behaved solutions. In the present paper the authors relax the assumptions imposed on \(G\): now \(G\) is piecewise continuous, \(\geq 0\) and continuous on any interval of strict positivity. They then show a theorem of existence and local uniqueness and a global comparison theorem for level- set solutions.
scientific article; zbMATH DE number 768999