Convex Functions are p-Subharmonic Functions, p >1 On Rn with Applications
Abstract
In this paper we discuss convexity, its average principle, an extrinsic average variational method in the Calculus of Variations, an average method in Partial Differential Equations, a link of convexity to p-subharmonicity, subsolutions to the p-Laplace equation, uniqueness, existence, isometric immersions in multiple settings. In particular, we show that a convex function on Rn is a p-subharmonic function, for every p > 1, and a C2 convex function on a Riemannian manifold is a p-subharmonic function f, for every p > 1\, . We also show that a C2 convex function which is a submersion on a Riemannian manifold is a p-subharmonic function, for every p 1\, . This result is sharp. As further applications, via function growth estimates in p-harmonic geometry, we prove that every p-balanced nonnegative C2 convex function on a complete noncompact Riemannian manifold is constant for p > 1. In particular, every Lq, nonnegative, convex function of class C2 on a complete noncompact Riemannian manifold is constant for q > p -1 > 0\, .