Proper solutions of the $1/H$-flow and the Green kernel of the $p$-Laplacian

TL;DR

This paper establishes existence, growth estimates, and decay properties for the inverse mean curvature flow and the Green kernel of the p-Laplacian, advancing understanding of geometric flows and potential theory on manifolds.

Contribution

It provides new existence results, optimal growth estimates, and decay bounds for the Green kernel, connecting inverse mean curvature flow with p-capacitary potentials.

Findings

Existence of inverse mean curvature flow from a point under certain conditions

Optimal growth estimates for the flow on manifolds

Decay estimates for the Green kernel of the p-Laplacian

Abstract

We show existence and optimal growth estimate for the weak inverse mean curvature flow issuing from a point, on manifolds with certain curvature and isoperimetric conditions. These theorems imply analogous ones for the flow issuing from relatively compact sets. Some of the results are obtained by proving new decay estimates for the Green kernel of the $p$-Laplacian which fix a gap in the literature. Additionally, we address the convergence of renormalized $p$-capacitary potentials to the inverse mean curvature flow with outer obstacle.