Some regularity results for anisotropic motion of fronts. (Q1850191)

This paper concerns the regularity of fronts propagating in the environment being anisotropic and inhomogeneous. It is proved that under appropriate assumptions there is at most one normal direction at each point of the front. With the help of this result the author shows that convex fronts are \(C^{1,1}\). The level-set approach, together with the theory of viscosity solutions, is used. The fronts are regarded as level sets of viscosity solutions of the anisotropic mean curvature equations. In particular, the author uses the results of \textit{Y. Giga, S. Goto, H. Ishii}, and \textit{M.-H. Sato} [Indiana Univ. Math. J. 40, 443--470 (1991; Zbl 0836.35009)] concerning existence, uniqueness, and convexity of viscosity solutions.