Solution of nonlinear curvature driven evolution of plane convex curves

The numerical approximation scheme for solving the nonlinear initial value problem ∂tb(v) = ∂xxβ(x, v) + β(x, v) with periodic boundary conditions is presented. Local existence and uniqueness of a solution and convergence of approximations is a consequence of the results of Mikula and Kačur (1996), where the anisotropic curvature driven evolution of plane convex curves is studied. The considered problem is a nonlinear generalization of plane convex curves evolution depending on curvature, known as curve shortening flow. It corresponds to the evolution equation ν = β(θ, k), where ν is the normal velocity of the curve, k its curvature and θ the angle of the tangent to the curve with horizontal axis. It arises in the theory of image and shape multiscale analysis introduced by Alvarez, Guichard, Lions and Morel and Sapiro and Tannenbaum, and also in anisotropic interface motions proposed by Angenent and Gurtin.  1997 Elsevier Science B.V.