  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.
