Some qualitative properties for geometric flows and its Euler implicit discretization

Abstract

We study the geometric flow parabolic equation and its implicit discretization which yield a family of nonlinear elliptic problems. We show that there are important differences in the study of those equations which concerns the propagation of level sets of data. Our study is based on the previous study of radially symmetric solutions of the corresponding equation. Curiously, in radial coordinates both equations reduce to suitable singular Hamilton-Jacobi first order equations. After considering the case of monotone data we point out a new peculiar behavior for non-monotone data with a profile of Batman type (g=min{g1,g2},g1(r) increasing, g2(r) decreasing and g1(rd)=g2(rd) for some rd>0). In the parabolic regime, and when the velocity of the convexity part of the level sets is greater than the velocity of the concavity part, we show that the level set {u=g(rd)} develops a non-empty interior set for any t>0. Nothing similar occurs in the stationary regime. We also present some numerical experiences.