Modica type gradient estimates for an inhomogeneous variant of the normalized p-laplacian evolution

2014/11/12 by Banerjee, Agnid, Garofalo, Nicola
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1411.3154

Abstract

In this paper, we study an inhomogeneous variant of the normalized p-Laplacian evolution which has been recently treated in \citeBG1, \citeDo, \citeMPR and \citeJu. We show that if the initial datum satisfies the pointwise gradient estimate \eqrefe:main1 a.e., then the unique solution to the Cauchy problem \eqrefmain5 satisfies the same gradient estimate a.e. for all later times, see \eqrefe:main below. A general pointwise gradient bound for the entire bounded solutions of the elliptic counterpart of equation \eqrefmain5 was first obtained in \citeCGS. 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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