2023/09/08 by Shihshu Walter Wei, Wei, Shihshu Walter
Computer Science · Mathematics · #53C40 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2309.04463
openalex publication_date 2023/09/08 · openalex created_date 2023/09/12 · openalex updated_date 2026/07/28
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 .