Modica type gradient estimates for an inhomogeneous variant of the normalized p-laplacian evolution
Abstract
In this paper, we study an inhomogeneous variant of the normalized p-Laplacian evolution which has been recently treated in BG1, Do, MPR and Ju. We show that if the initial datum satisfies the pointwise gradient estimate e:main1 a.e., then the unique solution to the Cauchy problem main5 satisfies the same gradient estimate a.e. for all later times, see e:main below. A general pointwise gradient bound for the entire bounded solutions of the elliptic counterpart of equation main5 was first obtained in CGS. Such estimate generalizes one obtained by L. Modica for the Laplacian, and it has connections to a famous conjecture of De Giorgi.
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