Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
1893119 | Journal of Geometry and Physics | 2011 | 9 Pages |
We introduce and study generalized 1-harmonic equations (1.1). Using some ideas and techniques in studying 1-harmonic functions from Wei (2007) [1], and in studying nonhomogeneous 1-harmonic functions on a cocompact set from Wei (2008) [2, (9.1)], we find an analytic quantity ww in the generalized 1-harmonic equations (1.1) on a domain in a Riemannian nn-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 pp-subharmonic functions of constant pp-tension field, p≥np≥n, and nonexistence results for solutions of the initial value problem of inverse mean curvature flow.