Modica type gradient estimates for an inhomogeneous variant of the normalized pp-Laplacian evolution

In this paper, we study an inhomogeneous variant of the normalized pp-Laplacian evolution which has been recently treated in Banerjee and Garofalo (2013), Does (2011), Manfredi et al. (2010), and Juutinen (2014). We show that if the initial datum satisfies the pointwise gradient estimate (1.6) a.e., then the unique solution to the Cauchy problem (1.2) satisfies the same gradient estimate a.e. for all later times, see (1.7). A general pointwise gradient bound for the entire bounded solutions of the elliptic counterpart of (1.2) was first obtained in Caffarelli et al. (1994). Such estimate generalizes one obtained by L. Modica for the Laplacian, and it has connections to a famous conjecture of De Giorgi.