Convex Functions are $p$-Subharmonic Functions, $p >1$ On $\mathbb{R}^n$ with Applications
Shihshu Walter Wei
备注：
12 pages, 1 table, to appear in Lecture Notes of Seminario Interdisciplinare di Matematica Vol 16 (2023). arXiv admin note: text overlap with arXiv:2104.05127
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 $q > p -1 > 0\, .$
We extend Prekopa's Theorem and the Brunn-Minkowski Theorem from convexity to $F$-subharmonicity. We apply this to the interpolation problem of convex functions and convex sets introducing a new notion of "harmonic interpolation" that we view as a generalization of Minkowski-addition.
If $u : Ω\subset \mathbb{R}^d \to {\rm X}$ is a harmonic map valued in a metric space ${\rm X}$ and ${\sf E} : {\rm X} \to \mathbb{R}$ is a convex function, in the sense that it generates an ${\rm EVI}_0$-gradient flow, we prove that the pullback ${\sf E} \circ u : Ω\to \mathbb{R}$ is subharmonic. This property was known in the smooth Riemannian manifold setting or with curvature restrictions on ${\rm X}$, while we prove it here in full generality. In addition, we establish generalized maximum principles, in the sense that the $L^q$ norm of ${\sf E} \circ u$ on $\partial Ω$ controls the $L^p$ …
A subequation on an open subset $X\subset \mathbb R^n$ is a subset $F$ of the space of $2$-jets on $X$ with certain properties. A smooth function is said to be $F$-subharmonic if all of its $2$-jets lie in $F$, and using the viscosity technique one can extend the notion of $F$-subharmonicity to any upper-semicontinuous function. Let $\mathcal P$ denote the subequation consisting of those $2$-jets whose Hessian part is semipositive. We introduce a notion of product subequation $F\#\mathcal P$ on $X\times \mathbb R^{m}$ and prove, under suitable hypotheses, that if $F$ is convex and $f(x,y)$ is …
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