$\mathbb{R}$2上对偶Minkowski问题的可解性

数学物理学报 ›› 2019, Vol. 39 ›› Issue (6): 1314-1322.

$\mathbb{R}$²上对偶Minkowski问题的可解性

魏娜()

中南财经政法大学统计与数学学院武汉 430073

收稿日期:2019-05-24 出版日期:2019-12-26 发布日期:2019-12-28
作者简介:魏娜, E-mail: weina@zuel.edu.cn
基金资助:
中央高校基本科研业务费专项资金(2722019PY053);湖北省自然科学基金(2019CFB570)

The Solvability of Dual Minkowski Problem in $\mathbb{R}$²

Na Wei()

School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073

Received:2019-05-24 Online:2019-12-26 Published:2019-12-28
Supported by:
the Fundamental Research Funds for the Central Universities(2722019PY053);the Natural Science Foundation of Hubei Province(2019CFB570)

摘要/Abstract

摘要：

该文研究Sobolev空间W^1，4 ($\mathbb{S}$)中一类约束变分问题存在极小可达元.在$\int_{\rm{\mathbb{S}}} g (\theta )d\theta > 0$条件下，该极小可达元是相应Euler-Lagrange方程

${u^{\prime \prime }} + u = \frac{{g(\theta )}}{{u\left( {{u^2} + {u^{\prime 2}}} \right)}}, \theta \in {\rm{\mathbb{S}}}$

的严格正解.基于此，该文在$\mathbb{R}$²上证明了文献[Huang-Lutwak-Yang-Zhang.Acta Math，2016，216（2）：325-338]提出的对偶Minkowski问题的可解性.

关键词: 对偶Minkowski问题, 非线性方程, 变分方法

Abstract:

In this paper, we study the existence of minimum of a constrained variational problem in the Sobolev space W^{1, 4}($\mathbb{S}$). If ∫$_\mathbb{S}$g(θ)dθ>0, the minimum is a positive solution to the related Euler-Lagrange equation

${u^{\prime \prime }} + u = \frac{{g(\theta )}}{{u\left( {{u^2} + {u^{\prime 2}}} \right)}}, \theta \in {\rm{\mathbb{S}}}$

Based on this, we prove the solvability of the dual Minkowski problem in $\mathbb{R}$² posed by Huang-Lutwak-Yang-Zhang[Acta Math, 2016, 216(2):325-338].

Key words: Dual Minkowski problem, Nonlinear equation, Variational method

中图分类号:

O176

魏娜. $\mathbb{R}$²上对偶Minkowski问题的可解性[J]. 数学物理学报, 2019, 39(6): 1314-1322.

Na Wei. The Solvability of Dual Minkowski Problem in $\mathbb{R}$²[J]. Acta mathematica scientia,Series A, 2019, 39(6): 1314-1322.

参考文献 24

1	Alexandrov A D. Selected Works Part Ⅰ//Naidu P S V. Classics of Soviet Mathematics (Translated from the Russian). Amsterdam:Gordon and Breach Publishers, 1996
2	Aleksandrov A D . On the theory of mixed volumes Ⅰ:Extension of certain concepts in the theory of convex bodies. Mat Sbornik N S, 1973, 2: 947- 972
3	Böröczky K J , Lutwak E , Yang D , Zhang G . The logarithmic Minkowski problem. J Amer Math Soc, 2013, 26 (3): 831- 852
4	Böröczky K J , Lutwak E , Yang D , Zhang G . Affine images of isotropic measures. J Differential Geom, 2015, 99 (3): 407- 442 doi: 10.4310/jdg/1424880981
5	Caffarelli L A . A localization property of viscosity solutions to the Monge-Ampére equation and their strict convexity. Ann of Math, 1990, 131: 129- 134 doi: 10.2307/1971509
6	Caffarelli L A . Interior W^{2, p} estimates for solutions of the Monge-Ampére equation. Ann of Math, 1990, 131: 135- 150 doi: 10.2307/1971510
7	Chen S B , Li Q R . On the planar dual Minkowski problem. Adv Math, 2018, 333: 87- 117 doi: 10.1016/j.aim.2018.05.010
8	Chen W . L_p Minkowski problem with not necessarily positive data. Adv Math, 2006, 201 (1): 77- 89 doi: 10.1016/j.aim.2004.11.007
9	Cheng S Y , Yau S T . On the regularity of the solution of the n-dimensional Minkowski problem. Comm Pure Appl Math, 1976, 29 (5): 495- 516 doi: 10.1002/cpa.3160290504
10	Chou K S , Wang X J . The L_p-Minkowski problem and the Minkowski problem in centroaffine geometry. Adv Math, 2006, 205 (1): 33- 83 doi: 10.1016/j.aim.2005.07.004
11	Dou J B , Zhu M J . The two dimensional L_p Minkowski problem and nonlinear equations with negative exponents. Adv Math, 2012, 230 (3): 1209- 1221 doi: 10.1016/j.aim.2012.02.027
12	Huang Y , Lutwak E , Yang D , Zhang G . Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems. Acta Math, 2016, 216 (2): 325- 388 doi: 10.1007/s11511-016-0140-6
13	Jiang M . Remarks on the 2-dimensional L_p-Minkowski problem. Adv Nonlinear Stud, 2010, 10 (2): 297- 313
14	Jiang Y , Wu Y . On the 2-dimensional dual Minkowski problem. J Differential Equation, 2017, 263 (6): 3230- 3243 doi: 10.1016/j.jde.2017.04.033
15	Li Q, Sheng W, Wang X J. Flow by Gauss curvature to the Alekesandrov and dual Minkowski problems. J Eur Math Soc, 2017, arXiv:1712.07774v1
16	Lutwak E . The Brunn-Minkowski-Firey theory Ⅰ:Mixed volumes and the Minkowski problem. J Differential Geom, 1993, 38 (1): 131- 150 doi: 10.4310/jdg/1214454097
17	Minkowski H . Allgemeine Lehrsätze über die convexen Polyeder. Ges Abh, 1911, 2: 103- 121
18	Minkowski H . Volumen und Oberfläche. Math Ann, 1903, 57 (4): 447- 495 doi: 10.1007/BF01445180
19	Nirenberg L . The Weyl and Minkowski problems in differential geometry in the large. Comm Pure Appl Math, 1953, 6: 337- 394 doi: 10.1002/cpa.3160060303
20	Pogorelov A V. The Minkowski Multidimensional Problem. New York:John Wiley and Sons, 1978
21	Rabinowitz P H. Minimax Methods in Critical Point Theory with Applications to Differential Equations. Providence, RI:Amer Math Soc, 1986
22	Trudinger N S, Wang X J. The Monge-Ampére equation and its geometric applications//Ji L Z, et al. Handbook of Geometric Analysis, No 1. Somerville, MA:International Press, 2008:467-524
23	Zhao Y M . The dual Minkowski problem for negative indices. Calc Var Partial Differential Equations, 2017, 56 (2): 18 doi: 10.1007/s00526-017-1124-x
24	Zhu G X . The logarithmic Minkowski problem for polytopes. Adv Math, 2014, 262: 909- 931 doi: 10.1016/j.aim.2014.06.004

[1]	张申贵. 带类p(x)-拉普拉斯算子的双非局部问题的无穷多解[J]. 数学物理学报, 2018, 38(3): 514-526.
[2]	廖家锋, 李红英. 带Sobolev临界指数的超线性Kirchhoff型方程正解的存在性与多解性[J]. 数学物理学报, 2017, 37(6): 1119-1124.
[3]	陈丽珍, 李安然, 李刚. 带有次线性项和超线性项的Klein-Gordon-Maxwell系统多重解的存在性[J]. 数学物理学报, 2017, 37(4): 663-670.
[4]	刘健, 赵增勤. 基于变分方法的奇异脉冲方程弱解的存在性[J]. 数学物理学报, 2016, 36(3): 521-530.
[5]	叶一蔚, 唐春雷. 带有超线性项或次线性项的Schrödinger-Poisson系统解的存在性和多重性[J]. 数学物理学报, 2015, 35(4): 668-682.
[6]	邓义华, 罗李平, 周立君. 一类带负位势函数的超线性p-Laplace方程基态解的存在性[J]. 数学物理学报, 2015, 35(3): 578-586.
[7]	赵金虎, 刘白羽, 徐尔. 一类完全非线性椭圆型方程组解的对称性[J]. 数学物理学报, 2015, 35(2): 312-323.
[8]	裘良华. 体积元保持变换下的一类特殊预给定数量曲率问题[J]. 数学物理学报, 2011, 31(5): 1317-1322.
[9]	葛斌, 薛小平, 周庆梅. 一类具有非光滑位势p(x)-Kirchhoff 型问题多解性[J]. 数学物理学报, 2011, 31(5): 1431-1438.
[10]	彭艳芳, 张正杰. 具临界指数及奇异性的双调和方程解的存在性[J]. 数学物理学报, 2009, 29(6): 1492-1499.
[11]	刘祥清; 黄毅生; 邓志颖. 双调和方程特征值问题[J]. 数学物理学报, 2009, 29(1): 57-69.
[12]	邓志颖; 黄毅生. 一类非线性椭圆方程正解的存在性[J]. 数学物理学报, 2008, 28(5): 971-976.
[13]	林丽珊;李永青. 带权的 p-Laplacian 非线性特征值问题解的存在性[J]. 数学物理学报, 2007, 27(6): 996-1005.
[14]	刘祥清; 黄毅生. 一类带权函数的拟线性椭圆方程[J]. 数学物理学报, 2007, 27(2): 277-287.
[15]	孙金丽;孙经先. 拟增算子的多个不动点[J]. 数学物理学报, 2006, 26(5): 695-699.

$\mathbb{R}$²上对偶Minkowski问题的可解性

The Solvability of Dual Minkowski Problem in $\mathbb{R}$²

RichHTML

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献 24

相关文章 15

Metrics

本文评价

推荐阅读 3