Approximation to driven motion by crystalline curvature in two dimensions

We study the approximation of driven motion by crystalline curvature in two dimensions with a reaction-diffusion type differential inclusion. A quasi-optimal O(" 2 j log "j 2 ) and an optimal O(" 2 ) error bound between the original flow and the zero level set of the approximate solution are proved, for the regular and the double obstacle potential respectively. This result is valid before the onset of singularities, and applies when the driving force g does not depend on the space variable x. A comparison principle between crystalline flows and a notion of weak solution for crystalline evolutions, for suitable g(x; t), are also obtained. 1 Introduction The interest in anisotropic fronts evolutions is motivated by many physical examples where an interface propagation with preferred directions is evident [13], [14], [12], [10]. Anisotropic motion by mean curvature is also strictly related to the geometry of convex bodies [45], [41] and with the theory of Minkowskian and Finsler s