Generalized $1$-harmonic Equation and The Inverse Mean Curvature Flow

TL;DR

This paper introduces generalized 1-harmonic equations, explores their analytic and geometric properties, and applies these findings to derive gradient bounds, Liouville theorems, and nonexistence results for inverse mean curvature flow and related functions.

Contribution

It develops a new framework for generalized 1-harmonic equations, linking analytic solutions with geometric properties, and applies this to inverse mean curvature flow and p-subharmonic functions.

Findings

Established a link between an analytic quantity and domain geometry.

Derived gradient bounds and nonexistence results for inverse mean curvature flow.

Proved Liouville theorems for p-subharmonic functions with constant p-tension.

Abstract

We introduce and study generalized $1$-harmonic equations (1.1). Using some ideas and techniques in studying $1$-harmonic functions from [W1] (2007), and in studying nonhomogeneous $1$-harmonic functions on a cocompact set from [W2, (9.1)] (2008), we find an analytic quantity $w$ in the generalized $1$-harmonic equations (1.1) on a domain in a Riemannian $n$-manifold that affects the behavior of weak solutions of (1.1), and establish its link with the geometry of the domain. We obtain, as applications, some gradient bounds and nonexistence results for the inverse mean curvature flow, Liouville theorems for $p$-subharmonic functions of constant $p$-tension field, $p \ge n$, and nonexistence results for solutions of the initial value problem of inverse mean curvature flow.