Geometric evolutions driven by threshold dynamics

where S(·) is a continuous function taking values in the space S of n × n symmetric matrices and v is real-valued. Special cases of (1.2) are motion by curvature, corresponding to S(n) ≡ I and v(n) ≡ 0, and curvature independent motions, corresponding to S(n) ≡ 0. Solutions of these geometric evolution problems typically develop singularities, regardless of the smoothness of the initial data (see [12], [14]). To overcome such difficulties, the level set approach was introduced by Chen, Giga, and Goto [8] and independently by Evans and Spruck [10]. In [10] motion by mean curvature is dealt with while in [8] more general geometric motions are discussed. Their ideas are based on considering interfaces as level sets of the solution of a degenerate parabolic partial differential equation. One approach to (1.1) is that some classes of dynamics which model the microscopic behavior of physical phenomena (for example, threshold dynamics, cellular automata) provide approximate schemes for (1.1). Among others, Bence, Merriman, and Osher [6] first proposed a simple approximation scheme (BMO) for motion by mean curvature. Later, Evans [9] and Barles and Georgelin [2] gave the first analysis of the BMO algorithm. More recently, Ishii, Pires,