Geometric functionals for the p-Laplace operator on the graph

Abstract

Let G=(V,E) be a connected finite graph. In the first part of this paper we assume that G satisfies the CDpψ condition for p>1 and some C1, concave function ψ:(0,+∞)→R. Based on the gradient estimate for positive solutions to the p-Laplace parabolic equation on G, we establish evolving formulas for the Fisher information, Shannon entropy and Perelman's Wp-functional along the p-Laplace parabolic equation on G. In the second part of this paper, we prove monotonicity of a p-parabolic frequency on G without any curvature condition, which generalizes Colding-Minicozzi II's monotone formula, both from Laplace operator to the p-Laplace operator, and also from the manifold to the graph setting. From the monotonicity, we get backward uniqueness for the p-Laplace operator on the graph.