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

TL;DR

This paper extends gradient estimates to an inhomogeneous normalized p-Laplacian evolution, showing that initial pointwise bounds are preserved over time, with implications for elliptic equations and De Giorgi's conjecture.

Contribution

It establishes that initial gradient bounds for an inhomogeneous normalized p-Laplacian evolution are maintained throughout the evolution, generalizing Modica's estimate.

Findings

Gradient estimates are preserved over time for solutions.

A general pointwise gradient bound for elliptic equations is obtained.

Connections to De Giorgi's conjecture are discussed.

Abstract

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