带平均曲率算子的离散混合边值问题凸解的存在性

数学物理学报 ›› 2022, Vol. 42 ›› Issue (2): 379-386.

带平均曲率算子的离散混合边值问题凸解的存在性

段磊(),陈天兰*()

西北师范大学数学与统计学院兰州 730070

收稿日期:2021-06-23 出版日期:2022-04-26 发布日期:2022-04-18
通讯作者: 陈天兰 E-mail:gsxsdl@163.com;chentianlan511@126.com
作者简介:段磊, E-mail: gsxsdl@163.com
基金资助:
国家自然科学基金(11801453);国家自然科学基金(11901464);甘肃省青年科技基金(20JR10RA100)

Existence of Convex Solutions for a Discrete Mixed Boundary Value Problem with the Mean Curvature Operator

Lei Duan(),Tianlan Chen*()

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070

Received:2021-06-23 Online:2022-04-26 Published:2022-04-18
Contact: Tianlan Chen E-mail:gsxsdl@163.com;chentianlan511@126.com
Supported by:
the NSFC(11801453);the NSFC(11901464);the Youth Science and Technology Fund of Gansu Province(20JR10RA100)

摘要/Abstract

摘要：

运用锥上的不动点定理讨论了Minkowski空间平均曲率算子的离散混合边值问题 $\Delta[\phi(\Delta v(t-1))]=f(t, -v(t)), {\quad} t\in[2, T-1]_{{\Bbb Z}},$ $\Delta v(1)=0, {\quad} v(T)=0$ 非平凡凸解的存在性, 其中 $\phi(s)=\frac{s}{\sqrt{1-s^{2}}}, s\in(-1, 1),$ $[2, T-1]_{{\Bbb Z}}:=\{2, 3, \cdots, T-2,$ $T-1\},$ $T\geqslant4$ 是正整数, 非线性项 $f(t, u)$ 非负连续, 在 $u=1$ 处允许具有奇异性.

关键词: 平均曲率算子, 离散混合边值问题, 非平凡凸解, 锥, 不动点定理

Abstract:

In this paper, by using the fixed point theorem in cones, we discuss the existence of $\Delta[\phi(\Delta v(t-1))]=f(t, -v(t)), {\quad} t\in[2, T-1]_{{\Bbb Z}},$ $\Delta v(1)=0, {\quad} v(T)=0$ nontrivial convex solutions for a discrete mixed boundary value problem of mean curvature operator in Minkowski space, where $\phi(s)=\frac{s}{\sqrt{1-s^{2}}}, s\in(-1, 1),$ $[2, T-1]_{{\Bbb Z}}:=\{2, 3, \cdots, T-2,$ $T-1\},$ $T\geqslant4$ and $T\in{\Bbb N}^{\ast}$ , the nonlinear term $f(t, u)$ is nonnegative and continuous, and singularity is allowed at $u=1$ .

Key words: Mean curvature operator, Discrete mixed boundary value problem, Nontrivial convex solutions, Cone, Fixed point theorem

中图分类号:

O175.7

段磊,陈天兰. 带平均曲率算子的离散混合边值问题凸解的存在性[J]. 数学物理学报, 2022, 42(2): 379-386.

Lei Duan,Tianlan Chen. Existence of Convex Solutions for a Discrete Mixed Boundary Value Problem with the Mean Curvature Operator[J]. Acta mathematica scientia,Series A, 2022, 42(2): 379-386.

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