Some regularity results for anisotropic motion of fronts

Abstract

We study the regularity of propagating fronts whose motion is anisotropic. We prove that there is at most one normal direction at each point of the front; as an application, we prove that convex fronts are $C^{1,1}.$ These results are by-products of some necessary conditions for viscosity solutions of quasilinear elliptic equations. These conditions are of independent interest; for instance they imply some regularity for viscosity solutions of nondegenerate quasilinear elliptic equations.