The Minimum Principle for Convex Subequations

2018/06/15 by Ross, Julius, Nyström, David Witt
#32U05 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1806.06033

Abstract

A subequation on an open subset X⊂ \mathbb Rⁿ 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× \mathbb R^m and prove, under suitable hypotheses, that if F is convex and f(x,y) is F#\mathcal P-subharmonic then the marginal function g(x):= inf_y f(x,y) is F-subharmonic. This generalises the classical statement that the marginal function of a convex function is again convex. We also prove a complex version of this result that generalises the Kiselman minimum principle for the marginal function of a plurisubharmonic function.

Related

• Convex Functions are p-Subharmonic Functions, p >1 On ℝⁿ with Applications
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