Geometric functionals for the \(p\)-Laplace operator on the graph (Q2692700)

From the authors' abstract:: Let \(G = (V, E)\) be a connected finite graph. Assume that \(G\) satisfies the \(\mathrm{CD}_p^\psi\) condition for \(p > 1\) and some \(C^1\), concave function \(\psi : (0, + \infty) \to \mathbf{R}\) introduced by \textit{F. Münch} [J. Math. Pures Appl. 120, 130-164 (2018; Zbl 1400.05219)]. Based on the gradient estimate for positive solutions to the \(p\)-Laplace parabolic equation on \(G\), the authos establish evolving formulas for the Fisher information, Shannon entropy and Perelman's \(\mathcal{W}_p\)-functional along the \(p\)-Laplace parabolic equation on \(G\). The monotonicity of a \(p\)-parabolic frequency on \(G\) without any curvature condition is provided. This 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, the authors get backward uniqueness for the \(p\)-Laplace operator on the graph.