Ni Xiang , Yuni Xiong . THE L_P DUAL MINKOWSKI TYPE PROBLEM FOR MIXED HESSIAN QUOTIENT TYPE EQUATIONS WITH $p\geq q$[J]. Acta mathematica scientia, Series B, 2026 , 46(2) : 697 -713 . DOI: 10.1007/s10473-026-0209-z

[1] Alexandrov A.Zur Theorie der gemischten Volumina von konvexen Körpern II, Neue Ungleichungen zwischen den gemischten Volumina und ihre Anwendungen(in Russian). Mat Sb, 1937, 2: 1205-1238
[2] Berg C. Corps convexes et potentials sphériques. Det Kgl Danske Videnskab Selskab Math fys Medd,1969, 37: 1-64
[3] Bian B, Guan P. A microscopic convexity principle for nonlinear partial differential equations. Invent Math, 2009, 177: 307-335
[4] Böröczky K, Fodor F. The $L_p$ dual Minkowski problem for $p>1$ and $q>0$. J Diff Equ,2019, 266: 7980-8033
[5] Böröczky K, Lutwak E, Yang D, Zhang G. The logarithmic Minkowski problem. J Amer Math Soc,2013, 26: 831-852
[6] Böröczky K, Henk M, Pollehn H. Subspace concentration of dual curvature measures of symmetric convex bodies. J Differ Geom,2018, 109: 411-429
[7] Caffarelli L, Nirenberg L, Spruck J. The Dirichlet problem for nonlinear second order elliptic equations III, Functions of the eigenvalues of the Hessian. Acta Math, 1985, 155: 261-301
[8] Chen C, Huang Y, Zhao Y. Smooth solutions to the $L_p$ dual Minkowski problem. Math Ann, 2019, 373: 953-976
[9] Chen C, Xu L. The $L_p$ Minkowski type problem for a class of mixed Hessian quotient equations. Adv Math, 2022, 411: Art 108794
[10] Chen C, Dong W, Han F. Interior Hessian estimates for a class of Hessian type equations. Calc Var Partial Differ Equ, 2023, 62: Article 52
[11] Chen H, Chen S, Li Q. Variations of a class of Monge-Ampère type functionals and their applications. Anal PDE,2021, 14: 689-716
[12] Chen H, Li Q. The $L_p$ dual Minkowski problem and related parabolic flows. J Funct Anal, 2021, 281: Art 109139
[13] Cheng S, Yau S. On the regularity of the solution of the $n$-dimensional Minkowski problem. Comm Pure Appl Math, 1976, 29: 495-516
[14] Christoffel E. über die Bestimmung der Gestalt einer krummen Oberfläche durch lokale Messungen auf derselben. J Reine Angew Math,1865, 64: 193-209
[15] Chou K, Wang X. The $L_p$-Minkowski problem and the Minkowski problem in centroaffine geometry. Adv Math, 2006, 205: 33-83
[16] Dong W, Wei W.The Neumann problem for a type of fully nonlinear complex equation. J Differ Equ, 306: 525-546
[17] Firey W. The determination of convex bodies from their mean radius of curvature functions. Mathematika, 1976, 14: 1-14
[18] Firey W. Christoffel problems for general convex bodies. Mathematika, 1968, 15: 7-21
[19] Guan P, Ma X. Christoffel-Minkowski problem I: convexity of solutions of a Hessian Equation. Invent Math, 2003, 151: 553-577
[20] Guan P, Lin C.On equation $\det(u_{ij}+\delta_{ij}u)=u^pf$ on $\mathbb{S}^n$. Beijing: Tsing-Hua University, 2000
[21] Guan P, Lin C, Ma X. The Christoffel-Minkowski problem II: Weingarten curvature equations. Chin Ann Math, 2006, 27: 595-614
[22] Guan P, Ma X, Zhou F. The Christoffel-Minkowski problem III: Existence and convexity of admissible solutions. Comm Pure Appl Math, 2010, 59: 1352-1376
[23] Guan P, Xia C. $L^p$ Christoffel-Minkowski problem: The case $1< p < k+1$. Calc Var Partial Diff Equ, 2018, 57: 69
[24] Hu C, Ma X, Shen C. On Christoffel-Minkowski problem of Firey's p-sum. Calc Var Partial Diff Equ, 2004, 21: 137-155
[25] 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
[26] Huang Y, Zhao Y. On the $L_p$ dual Minkowski problem. Adv Math, 2018, 332: 57-84
[27] Ivaki M.Deforming a hypersurface by principal radii of curvature and support function. Calc Var Partial Diff Equ, 2019, 58: 2133-2165
[28] Lei Y, Xu L, Zhang P.The $L_p$ dual Minkowski type problem for a class of mixed Hessian equations. Private communication in 2024
[29] Lewy H. On differential geometry in the large. Trans Amer Math Soc, 1938, 43: 258-270
[30] Lieberman G.Second Order Parabolic Differential Equations. Singapore: World Scientific, 1996
[31] Lutwak E. The Brunn-Minkowski-Firey theory I, Mixed volumes and the Minkowski problem. J Diff Geom, 1993, 38: 131-150
[32] Lutwak E, Oliker V. On the regularity of solution to a generalization of the Minkowski problem. J Diff Geom, 1995, 40: 227-246
[33] Lutwak E, Yang D, Zhang G. $L_p$ dual curvature measures. Adv Math, 2018, 329: 85-132
[34] Minkowski H.Allgemeine Lehrsätzeüber die konvexen Polyeder. Nachr Ges Wiss Göttingen, 1897, 198-219
[35] Nirenberg L. The Weyl and Minkowski problems in differential geometry in the large. Comm Pure Appl Math, 1953, 6: 337-394
[36] Pogorelov A.The Minkowski Multidimensional Problem. Wiley: New York, 1978
[37] Sheng W, Yi C.A class of anisotropic expanding curvature flow. Disc Conti Dynam Systems-A, 2020, 40: 2017-2035
[38] Spruck J. Geometric aspects of the theory of fully nonlinear elliptic equations. Clay Math Proc, 2005, 2: 283-309
[39] Zhao Y. Existence of solutions to the even dual Minkowski problem. J Diff Geom, 2018, 110: 543-572