${C^{1, 1}}$ REGULARITY FOR SOLUTIONS TO THE DEGENERATE DUAL ORLICZ-MINKOWSKI PROBLEM

doi:10.1007/s10473-025-0203-x

摘要/Abstract

摘要： In this paper, $C^{1, 1}$ regularity for solutions to the degenerate dual Orlicz-Minkowski problem is considered. The dual Orlicz-Minkowski problem is a generalization of the $L_p$ dual Minkowski problem in convex geometry. The proof is adapted from Guan-Li [17] and Chen-Tu-Wu-Xiang [11].

关键词: $C^{1,1}$ regularity, Monge-Ampère type equation, dual Orlicz-Minkowski problem

Abstract: In this paper, $C^{1, 1}$ regularity for solutions to the degenerate dual Orlicz-Minkowski problem is considered. The dual Orlicz-Minkowski problem is a generalization of the $L_p$ dual Minkowski problem in convex geometry. The proof is adapted from Guan-Li [17] and Chen-Tu-Wu-Xiang [11].

Key words: $C^{1,1}$ regularity, Monge-Ampère type equation, dual Orlicz-Minkowski problem

中图分类号:

35J70

Di Wu. ${C^{1, 1}}$ REGULARITY FOR SOLUTIONS TO THE DEGENERATE DUAL ORLICZ-MINKOWSKI PROBLEM[J]. 数学物理学报(英文版), 2025, 45(2): 327-337.

Di Wu. ${C^{1, 1}}$ REGULARITY FOR SOLUTIONS TO THE DEGENERATE DUAL ORLICZ-MINKOWSKI PROBLEM[J]. Acta mathematica scientia,Series B, 2025, 45(2): 327-337.

参考文献

[1] Aleksandrov A. Existence and uniqueness of a convex surface with a given integral curvature. Doklady Acad Nauk Kasah SSSR, 1942, 36: 131-134
[2] Bianchi G, Böröczky K, Colesanti A. The Orlicz version of the

$L_p$ Minkowski problem for

$-n<p<0$ . Adv Appl Math, 2019,111: 101937
[3] Böröczky K.The logarithmic Minkowski conjecture and the

$L_p$ -Minkowski problem. arXiv: 2210.00194
[4] Böröczky K, Fodor F. The

$L_p$ dual Minkowski problem for

$p>1$ and

$q>0$ . J Differential Equations, 2019, 266: 7980-8033
[5] Böröczky K, Henk M, Pollehn H. Subspace concentration of dual curvature measures of symmetric convex bodies. J Differential Geom, 2018, 109: 411-429
[6] Böröczky K, Lutwak E, Yang D, Zhang G. The logarithmic Minkowski problem. J Amer Math Soc, 2013, 26: 831-852
[7] Chen C, Huang Y, Zhao Y. Smooth solutions to the

$L_p$ dual Minkowski problem. Math Ann, 2019, 373: 953-976
[8] Chen H, Chen S, Li Q. Variations of a class of MA type functionals and their applications. Anal & PDE, 2021, 14(3): 689-716
[9] Chen L, Liu Y, Lu J, Xiang N. Existence of smooth even solutions to the dual Orlicz-Minkowski problem. J Geom Anal, 2022, 32(2): 1-25
[10] Chen L, Tu Q, Wu D, Xiang N. Anisotropic Gauss curvature flows and their associated dual Orlicz-Minkowski problems. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 2022, 152(1): 148-162
[11] Chen L, Tu Q, Wu D, Xiang N.

$C^{1, 1}$ regularity for solutions to the degenerate

$L_p$ dual Minkowski problem. Calc Var Partial Differential Equations, 2021, 60(3): Art 115
[12] Chen S, Li Q, Zhu G. The logarithmic Minkowski problem for non-symmetric measures. Trans Amer Math Soc, 2019, 371: 2623-2641
[13] Chou K, Wang X. The

$L_p$ -Minkowski problem and the Minkowski problem in centroaffine geometry. Adv Math, 2006, 205: 33-83
[14] Ding S, Li G.A class of anisotropic inverse Gauss curvature flows and dual Orlicz Minkowski type problem. arXiv: 2209.04601
[15] Gardner R, Hug D, Weil W, et al. General volumes in the Orlicz-Brunn-Minkowski theory and a related Minkowski problem I. Calc Var Partial Differential Equations, 2019, 58: Art 12
[16] Gardner R, Hug D, Xing S, Ye D. General volumes in the Orlicz-Brunn-Minkowski theory and a related Minkowski problem II. Calc Var Partial Differential Equations, 2020, 59: Art 15
[17] Guan P, Li Y.

$C^{1,1}$ estimates for solutions of a problem of Alexandrov. Comm Pure Appl Math, 1997, 50: 789-811
[18] Guang Q, Li Q, Wang X.The

$L_p$ -Minkowski problem with super-critical exponents. arXiv: 2203.05099
[19] Haberl C, Lutwak E,Yang D, Zhang G. The even Orlicz Minkowski problem. Adv Math, 2010, 224: 2485-2510
[20] He Y, Li Q, Wang X. Multiple solutions of the

$L_p$ -Minkowski problem. Calc Var Partial Differential Equations, 2016, 55(5): Art 117
[21] Henk M, Pollehn H. Necessary subspace concentration conditions for the even dual Minkowski problem. Adv Math, 2018, 323: 114-141
[22] Huang Y, Jiang Y. Variational characterization for the planar dual Minkowski problem. J Funct Anal, 2019, 277: 2209-2236
[23] 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: 325-388
[24] Huang Y, Lutwak E, Yang D, Zhang G. The

$L_p$ -Aleksandrov problem for

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

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

$L_p$ Minkowski problem for polytopes. Discrete Comput Geom, 2005, 33: 699-715
[27] Jian H, Lu J. Existence of solutions to the Orlicz-Minkowski problem. Adv Math, 2019, 344: 262-288
[28] Jian H, Lu J, Wang X. A priori estimates and existence of solutions to the prescribed centroaffine curvature problem. J Funct Anal, 2018, 274: 826-862
[29] Jian H, Lu J, Zhu G. Mirror symmetric solutions to the centro-affine Minkowski problem. Calc Var Partial Differential Equations, 2016, 55: Art 41
[30] Li Q, Liu J, Lu J. Non-uniqueness of solutions to the

$L_p$ dual Minkowski problem. Inter Math Rese Noti, 2022, 12: 9114-9150
[31] Li Q, Sheng W, Wang X. Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems. J Eur Math Soc, 2020, 22: 893-923
[32] Liu Y, Lu J. A flow method for the dual Orlicz-Minkowski problem. Trans Amer Math Soc, 2020: 373: 5833-5853
[33] Lu J. Nonexistence of maximizers for the functional of the centroaffine Minkowski problem. Sci China Math, 2018, 61: 511-516
[34] Lu J. A remark on rotationally symmetric solutions to the centroaffine Minkowski problem. J Differential Equations, 2019, 266: 4394-4431
[35] Lu J, Wang X. Rotationally symmetric solutions to the

$L_p$ -Minkowski problem. J Differential Equations, 2013, 254: 983-1005
[36] Lutwak E. The Brunn-Minkowski-Firey theory I: Mixed volumes and the Minkowski problem. J Differential Geom, 1993, 38: 131-150
[37] Lutwak E, Yang D, Zhang G. On the

$L_p$ -Minkowski problem. Trans Amer Math Soc, 2004, 356: 4359-4370
[38] Lutwak E, Yang D, Zhang G.

$L_p$ dual curvature measures. Adv Math, 2018, 329: 85-132
[39] Minkowski H.Allgemeine Lehrsätze über die konvexen Polyeder// Ausgewählte Arbeiten zur Zahlentheorie und zur Geometrie. Leipzig: Teubner, 1897: 198-219
[40] Minkowski H. Volumen and Oberfläche. Math Ann, 1903: 57: 447-495
[41] Oliker V.Existence and uniqueness of convex hypersurfaces with prescribed Gaussian curvature in spaces of constant curvature// Sem Inst Mate Appl ``Giovanni Sansone", Univ Studi Firenze, 1983
[42] Pogorelov A.Extrinsic Geometry of Convex Surfaces. Providence, RI: Amer Math Soc, 1973
[43] Schneider R.Convex Bodies: the Brunn-Minkowski theory. Cambridge: Cambridge University Press, 2013
[44] Sheng W, Yi C.An anisotropic shrinking flow and

$L_p$ Minkowski problem. arXiv: 1905.04679
[45] Stancu A. The discrete planar

$L_0$ -Minkowski problem. Adv Math, 2002, 167: 160-174
[46] Xie F. The Orlicz Minkowski problem for general measures. Proc Amer Math Soc, 2022, 150(10): 4433-4445
[47] Xing S, Ye D. The general dual Orlicz-Minkowski problem. Indiana Univ Math J, 2020, 69: 621-655
[48] Zhao Y. The dual Minkowski problem for negative indices. Calc Var Partial Differential Equations, 2017, 56: Art 18
[49] Zhao Y. Existence of solutions to the even dual Minkowski problem. J Differential Geom, 2018, 110: 543-572
[50] Zhu B, Xing S, Ye D. The dual Orlicz-Minkowski problem. J Geom Anal, 2018, 28: 3829-3855
[51] Zhu G. The logarithmic Minkowski problem for polytopes. Adv Math, 2014, 262: 909-931
[52] Zhu G. The centro-affine Minkowski problem for polytopes. J Differential Geom, 2015, 101: 159-174