The Minimum Principle for Convex Subequations
The Minimum Principle for Convex Subequations
复制标题
凸子方程的极小值原理
DOI：
10.1007/s12220-021-00782-2
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
Nyström, David Witt
中科院分区：
文献类型：
--
作者：
Ross, Julius;Nyström, David Witt
A subequation, in the sense of Harvey–Lawson, on an open subsetis a subsetFof the space of 2-jets onXwith certain properties. A smooth function is said to beF-subharmonic if all of its 2-jets lie inF, and using the viscosity technique one can extend the notion ofF-subharmonicity to any upper-semicontinuous function. Letdenote the subequation consisting of those 2-jets whose Hessian part is semipositive. We introduce a notion of product subequationonand prove, under suitable hypotheses, that ifFis convex andf(x,y) is-subharmonic then the marginal function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} g(x):= \inf _y f(x,y) \end{aligned}$$\end{document}isF-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.
A subequation, in the sense of Harvey–Lawson, on an open subsetis a subsetFof the space of 2-jets onXwith certain properties. A smooth function is said to beF-subharmonic if all of its 2-jets lie inF, and using the viscosity technique one can extend the notion ofF-subharmonicity to any upper-semicontinuous function. Letdenote the subequation consisting of those 2-jets whose Hessian part is semipositive. We introduce a notion of product subequationonand prove, under suitable hypotheses, that ifFis convex andf(x,y) is-subharmonic then the marginal function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} g(x):= \inf _y f(x,y) \end{aligned}$$\end{document}isF-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.
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