Convex Functions are $p$-Subharmonic Functions, $p >1$ On $\mathbb{R}^n$ with Applications

TL;DR

This paper establishes that convex functions on Euclidean space and Riemannian manifolds are inherently $p$-subharmonic for all $p > 1$, with implications for function growth and constancy on noncompact manifolds.

Contribution

It proves that convex functions are $p$-subharmonic for all $p > 1$ on $R^n$ and Riemannian manifolds, extending the understanding of convexity in geometric analysis.

Findings

Convex functions on $R^n$ are $p$-subharmonic for all $p > 1$.

Convex functions on Riemannian manifolds are $p$-subharmonic for all $p > 1$.

Nonnegative convex functions with certain growth conditions are constant on complete noncompact manifolds.

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 $\mathbb{R}^n$ is a $p$-subharmonic function, for every $p > 1$, and a $C^2$ convex function on a Riemannian manifold is a $p$-subharmonic function $f$, for every $p > 1\, .$ We also show that a $C^2$ convex function which is a submersion on a Riemannian manifold is a $p$-subharmonic function, for every $p \ge 1\, .$ This result is sharp. As further applications, via function growth estimates in $p$-harmonic geometry, we prove that every $p$-balanced nonnegative $C^2$ convex function on a complete noncompact Riemannian manifold is constant for $p > 1$. In particular, every $L^q$, nonnegative, convex function of class $C^2$ on a complete noncompact Riemannian manifold is constant for $q > p -1 > 0\, .$