THE CHORD GAUSS CURVATURE FLOW AND ITS ${L_{p}}$ CHORD MINKOWSKI PROBLEM

THE CHORD GAUSS CURVATURE FLOW AND ITS ${L_{p}}$ CHORD MINKOWSKI PROBLEM

THE CHORD GAUSS CURVATURE FLOW AND ITS ${L_{p}}$ CHORD MINKOWSKI PROBLEM

THE CHORD GAUSS CURVATURE FLOW AND ITS ${L_{p}}$ CHORD MINKOWSKI PROBLEM

THE CHORD GAUSS CURVATURE FLOW AND ITS ${L_{p}}$ CHORD MINKOWSKI PROBLEM

THE CHORD GAUSS CURVATURE FLOW AND ITS Lp{L_{p}} CHORD MINKOWSKI PROBLEM

THE CHORD GAUSS CURVATURE FLOW AND ITS Lp {L_{p}} CHORD MINKOWSKI PROBLEM

THE CHORD GAUSS CURVATURE FLOW AND ITS ${L_{p}}$ CHORD MINKOWSKI PROBLEM

THE CHORD GAUSS CURVATURE FLOW AND ITS ${L_{p}}$ CHORD MINKOWSKI PROBLEM

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doi:10.1007/s10473-025-0113-y

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (1): 161-179.doi: 10.1007/s10473-025-0113-y

Jinrong Hu^1,2, Yong Huang³, Jian Lu⁴, Sinan Wang⁵

1. School of Mathematics, Hunan University, Changsha, 410082, Hunan Province, China;
2. Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstrasse 8-10, 1040 Wien, Austria;
3. School of Mathematics, Hunan University, Changsha 410082, China;
4. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China;
5. School of Mathematics, Hunan University, Changsha 410082, China

收稿日期:2024-09-01 修回日期:2024-10-13 发布日期:2025-02-06
作者简介:Jinrong Hu, E-mail,: Hu_jinrong097@163.com; Yong Huang, E-mail,: huangyong@hnu.edu.cn; Jian Lu, E-mail,: lj-tshu04@163.com; Sinan Wang, E-mail,: wangsinan@hnu.edu.cn
基金资助:
National Natural Science Foundation of China (12171144, 12231006, 12122106).

Jinrong Hu^1,2, Yong Huang³, Jian Lu⁴, Sinan Wang⁵

1. School of Mathematics, Hunan University, Changsha, 410082, Hunan Province, China;
2. Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstrasse 8-10, 1040 Wien, Austria;
3. School of Mathematics, Hunan University, Changsha 410082, China;
4. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China;
5. School of Mathematics, Hunan University, Changsha 410082, China

Received:2024-09-01 Revised:2024-10-13 Published:2025-02-06
About author:Jinrong Hu, E-mail,: Hu_jinrong097@163.com; Yong Huang, E-mail,: huangyong@hnu.edu.cn; Jian Lu, E-mail,: lj-tshu04@163.com; Sinan Wang, E-mail,: wangsinan@hnu.edu.cn
Supported by:
National Natural Science Foundation of China (12171144, 12231006, 12122106).

摘要： In this paper, the $L_{p}$ chord Minkowski problem is concerned. Based on the results shown in [20], we obtain a new existence result of solutions to this problem in terms of smooth measures by using a nonlocal Gauss curvature flow for $p>-n$ with $p\neq 0$ .

关键词: $L_{p}$ chord Minkowski problem, new {M}onge-{A}mpère equation, geometric flow

Abstract: In this paper, the $L_{p}$ chord Minkowski problem is concerned. Based on the results shown in [20], we obtain a new existence result of solutions to this problem in terms of smooth measures by using a nonlocal Gauss curvature flow for $p>-n$ with $p\neq 0$ .

Key words: $L_{p}$ chord Minkowski problem, new {M}onge-{A}mpère equation, geometric flow

中图分类号:

35K55

Jinrong Hu, Yong Huang, Jian Lu, Sinan Wang. THE CHORD GAUSS CURVATURE FLOW AND ITS ${L_{p}}$ CHORD MINKOWSKI PROBLEM[J]. 数学物理学报(英文版), 2025, 45(1): 161-179.

Jinrong Hu, Yong Huang, Jian Lu, Sinan Wang. THE CHORD GAUSS CURVATURE FLOW AND ITS ${L_{p}}$ CHORD MINKOWSKI PROBLEM[J]. Acta mathematica scientia,Series B, 2025, 45(1): 161-179.

[1] Aleksandrov A D. On the surface area measure of convex bodies. Mat Sb (NS), 1939, 6: 167-174
[2] Aleksandrov A D. On the theory of mixed volumes. III. Extensions of two theorems of Minkowski on convex polyhedra to arbitrary convex bodies. Mat Sb, 1938, 3: 27-46
[3] Bianchi G, Böröczky K J, Colesanti A, Yang D. The

$L_p$ -Minkowski problem for

$-n＜p＜1$ . Adv Math, 2019, 341: 493-535
[4] Böröczky K J, Fodor F. The

$L_p$ dual Minkowski problem for

$p>1$ and

$q>0$ . J Differential Equations, 2019, 266(12): 7980-8033
[5] Böröczky K J, Lutwak E, Yang D, Zhang G. The logarithmic Minkowski problem. J Amer Math Soc, 2013, 26(3): 831-852
[6] Böröczky K J, Lutwak E, Yang D, Zhang G. The dual Minkowski problem for symmetric convex bodies. Adv Math, 2019, 356: Art 106805
[7] Böröczky K J, Lutwak E, Yang D, et al. The Gauss image problem. Comm Pure Appl Math, 2020, 73(7): 1406-1452
[8] Böröczky K J, Trinh H T. The planar

$L_p$ -Minkowski problem for

$0＜p＜1$ . Adv Appl Math, 2017, 87: 58-81
[9] Chen C, Huang Y, Zhao Y. Smooth solutions to the

$L_p$ dual Minkowski problem. Math Ann, 2019, 373(3/4): 953-976
[10] Chen S, Li Q, Zhu G. On the

$L_p$ Monge-Ampère equation. J Differential Equations, 2017, 263(8): 4997-5011
[11] 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
[12] Chou K S, Wang X J. A logarithmic Gauss curvature flow and the Minkowski problem. Ann Inst H Poincaré Anal Non Linéaire, 2000, 17(6): 733-751
[13] 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
[14] Fenchel W, Jessen B. Mengenfunktionen und konvexe Körper. Danske Vid Selsk Mat-Fys Medd, 1938, 16: 1-31
[15] Firey W J.

$p$ -means of convex bodies. Math Scand, 1962, 10: 17-24
[16] Firey W J. Shapes of worn stones. Mathematika, 1974, 21: 1-11
[17] Gardner R J. Geometric Tomography. New York: Cambridge University Press, 2006
[18] Guan P. Topics in Geometric Fully Nonlinear Equations. Lecture notes of the workshop Monge-Ampére equations. Zhejiang University, Hangzhou, 2002
[19] Guo L, Xi D, Zhao Y. The

$L_p$ chord Minkowski problem in a critical interval. Math Ann, 2024, 389(3): 3123-3162
[20] Hu J, Huang Y, Lu J. Boundary regularity of Riesz potential, smooth solution to the chord log-Minkowski problem. arXiv:2304.14220
[21] 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
[22] Huang Y, Lutwak E, Yang D, Zhang G. The

$L_p$ -Aleksandrov problem for

$L_p$ -integral curvature. J Differential Geom, 2018, 110(1): 1-29
[23] Huang Y, Zhao Y. On the

$L_p$ dual Minkowski problem. Adv Math, 2018, 332: 57-84
[24] Hug D, Lutwak E, Yang D, Zhang G. On the

$L_p$ Minkowski problem for polytopes. Discrete Comput Geom, 2005, 33(4): 699-715
[25] Jerison D. Prescribing harmonic measure on convex domains. Invent Math, 1991, 105(2): 375-400
[26] Jiang Y, Wu Y. On the 2-dimensional dual Minkowski problem. J Differential Equations, 2017, 263(6): 3230-3243
[27] Li Q R, Sheng W, Wang X J. Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems. J Eur Math Soc (JEMS), 2020, 22(3): 893-923
[28] Li Y. The

$L_p$ chord Minkowski problem for negative

$p$ . J Geom Anal, 2024, 34(3): Art 82
[29] Lu J, Wang X J. Rotationally symmetric solutions to the

$L_p$ -Minkowski problem. J Differential Equations, 2013, 254(3): 983-1005
[30] Lutwak E. Dual mixed volumes. Pacific J Math, 1975, 58(2): 531-538
[31] Lutwak E. The Brunn-Minkowski-Firey theory I: Mixed volumes and the Minkowski problem. J Differential Geom, 1993, 38(1): 131-150
[32] Lutwak E. The Brunn-Minkowski-Firey theory II: Affine and geominimal surface areas. Adv Math, 1996, 118(2): 244-294
[33] Lutwak E, Xi D, Yang D, Zhang G. Chord measures in integral geometry and their Minkowski problems. Comm Pure Appl Math, 2024, 77(7): 3277-3330
[34] Lutwak E, Yang D, Zhang G. On the

$L_p$ -Minkowski problem. Trans Amer Math Soc, 2004, 356(11): 4359-4370
[35] Krylov N V. Nonlinear Elliptic and Parabolic Equations of the Second Order. Translated from the Russian by P. L. Buzytsky [P. L. Buzytskiĭ]. Mathematics and Its Applications (Soviet Series). Dordrecht: D. Reidel Publishing Co, 1987
[36] Minkowski H. Allgemeine Lehrsätze über die convexen Polyeder. Nachr Ges Wiss Göttingen, 1897, 1897: 198-220
[37] Minkowski H. Volumen und Oberfläche. Math Ann, 1903, 57(4): 447-495
[38] Nirenberg L. The Weyl and Minkowski problems in differential geometry in the large. Comm Pure Appl Math, 1953, 6: 337-394
[39] Ren D. Topics in Integral Geometry. Singapore: World Scientific, 1994
[40] Santaló L A. Integral Geometry and Geometric Probability. Cambridge: Cambridge University Press, 2004
[41] Schneider R. Convex Bodies: the Brunn-Minkowski Theory. Cambridge: Cambridge University Press, 2014
[42] Urbas J I E. An expansion of convex hypersurfaces. J Differential Geom, 1991, 33(1): 91-125
[43] Xi D, Yang D, Zhang G, Zhao Y. The

$L_p$ chord Minkowski problem. Adv Nonlinear Stud, 2023, 23(1): Art 20220041
[44] Zhao Y. The dual Minkowski problem for negative indices. Calc Var Partial Differential Equations, 2017, 56(2): Art 18
[45] Zhao Y. Existence of solutions to the even dual Minkowski problem. J Differential Geom, 2018, 110(3): 543-572
[46] Zhu G. The logarithmic Minkowski problem for polytopes. Adv Math, 2014, 262: 909-931
[47] Zhu G. The

$L_p$ Minkowski problem for polytopes for

$0＜p＜1$ . J Funct Anal, 2015, 269(4): 1070-1094
[48] Zhu G. The centro-affine Minkowski problem for polytopes. J Differential Geom, 2015, 101(1): 159-174

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