共形几何中的一类退化的完全非线性方程的边值问题——献给李工宝教授 70 寿辰

摘要/Abstract

摘要：

该文得到了带边流形上的一类源于共形几何的、退化的完全非线性方程的解的先验估计. 并进一步使用连续性方法得到了这类方程解的存在性.关键词：共形几何; 退化方程; 先验估计.

Abstract:

This paper considers the a priori estimates for a class of degenerate fully nonlinear equations arising in conformal geometry on manifolds with boundary. Based on these a priori estimates, we obtain an existence result using the continuity method.

Key words: conformal geometry, degenerate equations, a priori estimates

中图分类号:

O186.12

贺妍, 张元正. 共形几何中的一类退化的完全非线性方程的边值问题——献给李工宝教授 70 寿辰[J]. 数学物理学报, 2025, 45(6): 1825-1838.

Yan He, Yuanzheng Zhang. Boundary Value Problems for Some Degenerate Fully Nonlinear Elliptic Equations Arising in Conformal Geometry[J]. Acta mathematica scientia,Series A, 2025, 45(6): 1825-1838.

参考文献 59

[1]	Aubin T. Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. J Math Pures Appl, 1976, 55(3): 269-296
[2]	Blocki Z. Regularity of the degenerate Monge-Ampère equation on compact Kähler manifolds. Math Z, 2003, 244(1): 153-161 doi: 10.1007/s00209-002-0483-x
[3]	Chang A, Gursky M, Yang P. An equation of Monge-Ampère type in conformal geometry and four-manifolds of positive Ricci curvature. Ann of Math, 2002, 155(3): 709-787 doi: 10.2307/3062131
[4]	Chang A, Gursky M, Yang P. An a priori estimate for a fully nonlinear equation on four-manifolds. J Anal Math, 2002, 87: 151-186 doi: 10.1007/BF02868472
[5]	Chen L, He Y. A class of fully nonlinear equations on Riemannian manifolds with negative curvature. Calc Var Partial Differential Equations, 2024, 63(6): Article 158
[6]	Chen S. Boundary value problems for some fully nonlinear elliptic equations. Calc Var Partial Differential Equations, 2007, 30(1): 1-15 doi: 10.1007/s00526-006-0072-7
[7]	Chen S. Conformal deformation on manifolds with boundary. Geom Funct Anal, 2009, 19(4): 1029-1064 doi: 10.1007/s00039-009-0028-0
[8]	Chen X J, Tu Q, Xiang N. Dirichlet problem for degenerate Hessian quotient type curvature equations. Calc Var Partial Differential Equations, 2025, 64(3): Article 99
[9]	Chen X Z, Wei W. The $\sigma_{2} $-curvature equation on a compact manifold with boundary. arXiv 2307.13942, 2023
[10]	Chen X Z, Wei W. Estimates for modified $ \sigma_{2} $ curvature equation on compact manifolds with boundary. arXiv 2405.139134, 2024
[11]	Chu B Z, Li Y Y, Li Z Y. Liouville theorems for conformally invariant fully nonlinear equations. I. arXiv 2311.07542, 2023
[12]	Hessian equations with elementary symmetric functions. Comm Partial Differential Equations, 2006, 31(7/9): 1005-1025 doi: 10.1080/03605300500481491
[13]	Duncan J A, Nguyen L. Local pointwise second derivative estimates for strong solutions to the $ k $-Yamabe equation on Euclidean domains. Calc Var Partial Differential Equations, 2021, 60(5): Article 177
[14]	Ge Y X, Lin C S, Wang G F. On the $ \sigma_2 $-scalar curvature. J Differential Geom, 2010, 84(1): 45-86
[15]	Ge Y X, Wang G F. On a fully nonlinear Yamabe problem. Ann Sci Ecole Norm Sup, 2006, 39(4): 569-598
[16]	Ge Y X, Wang G F. On a conformal quotient equation. II. Comm Anal Geom, 2013, 21(1): 1-38
[17]	Ge Y X, Wang G F. On a conformal quotient equation. Int Math Res Not IMRN, 2007, 2007(6): Article 19
[18]	Guan P F. $ C^2 $ a priori estimate for degenerate Monge-Ampère equations. Duke Math J, 1997, 86: 323-346
[19]	Guan P F, Li Y Y. The Weyl problem with nonnegative Gauss curvature. J Differential Geom, 1994, 39: 331-342
[20]	Guan P F, Li Y Y. $ C^1$, 1 estimates for solutions of a problem of Alexandrov. Commun Pure Appl Math, 1997, 50: 789-811 doi: 10.1002/(ISSN)1097-0312
[21]	Guan P F, Lin C S, Wang G F. Application of the method of moving planes to conformally invariant equations. Math Z, 2004, 247(1): 1-19 doi: 10.1007/s00209-003-0608-x
[22]	Guan P F, Trudinger N S, Wang X J. On the Dirichlet problem for degenerate Monge-Ampère equations. Acta Math, 1999, 182(1): 87-104 doi: 10.1007/BF02392824
[23]	Guan P F, Wang G F. A fully nonlinear conformal flow on locally conformally flat manifolds. J Reine Angew Math, 2003, 557: 219-238
[24]	Guan P F, Zhang X W. A class of curvature type equations. Pure Appl Math Q, 2021, 17(3): 865-907 doi: 10.4310/PAMQ.2021.v17.n3.a2
[25]	Gursky M, Viaclovsky J. Fully nonlinear equations on Riemannian manifolds with negative curvature. Indiana U Math J, 2003, 52: 399-419 doi: 10.1512/iumj.2003.52.2313
[26]	Gursky M J, Viaclovsky J A. Prescribing symmetric functions of the eigenvalues of the Ricci tensor. Ann of Math, 2007, 166: 475-531 doi: 10.4007/annals
[27]	He Y, Sheng W M. On existence of the prescribing $ k $-curvature problem on manifolds with boundary. Comm Anal Geom, 2011, 19(1): 53-77 doi: 10.4310/CAG.2011.v19.n1.a3
[28]	He Y, Sheng W M. Local estimates for some elliptic equations arising from conformal geometry. Int Math Res Not IMRN, 2013, 2013(2): 258-290 doi: 10.1093/imrn/rnr262
[29]	He Y, Tu Q, Xiang N. Existence for some degenerate Hessian type equations arising in conformal geometry. Pacific J Math, 2025, 335(1): 97-117 doi: 10.2140/pjm
[30]	Ivochkina N M, Trudinger N S, Wang X J. The Dirichlet problem for degenerate Hessian equations. Comm Partial Differential Equations, 2004, 29: 219-235 doi: 10.1081/PDE-120028851
[31]	Jiao H M, Jiao Y. The Pogorelov estimates for degenerate curvature equations. Int Math Res Not IMRN, 2024, 2024(18): 12504-12529 doi: 10.1093/imrn/rnae161
[32]	Jiao H M, Wang Z Z. The Dirichlet problem for degenerate curvature equations. J Funct Anal, 2022, 283(1): Art 109485
[33]	Jiao H M, Wang Z Z. Second order estimates for convex solutions of degenerate $ k $-Hessian equations. J Funct Anal, 2024, 286(3): Art 110248
[34]	Jiang F D, Trudinger N S. Oblique boundary value problems for augmented Hessian equations III. Comm Partial Differential Equations, 2019, 44(8): 708-748 doi: 10.1080/03605302.2019.1597113
[35]	Jiang F D, Trudinger N S. Oblique boundary value problems for augmented Hessian equations I. Bull Math Sci, 2018, 8(2): 353-411 doi: 10.1007/s13373-018-0124-2
[36]	Jiang F D, Trudinger N S. Oblique boundary value problems for augmented Hessian equations II. Nonlinear Anal, 2017, 154: 148-173 doi: 10.1016/j.na.2016.08.007
[37]	Jin Q N, Li A B, Li Y Y. Estimates and existence results for a fully nonlinear Yamabe problem on manifolds with boundary. Calc Var Partial Differential Equations, 2007, 28: 509-543 doi: 10.1007/s00526-006-0057-6
[38]	Krylov N V. Weak interior second order derivative estimates for degenerate nonlinear elliptic equations. Differ Integral Equ, 1994, 7: 133-156
[39]	Krylov N V. Barriers for derivatives of solutions of nonlinear elliptic equations on a surface in Euclidean space. Comm Partial Differential Equations, 1994, 19: 1909-1944 doi: 10.1080/03605309408821077
[40]	Krylov N V. A theorem on the degenerate elliptic Bellman equations in bounded domains. Differ Integral Equ, 1995, 8: 961-980
[41]	Krylov N V. On the general notion of fully nonlinear second-order elliptic equations. Trans Am Math Soc, 1995, 347: 857-895 doi: 10.1090/tran/1995-347-03
[42]	Li A B, Li Y Y. On some conformally invariant fully nonlinear equations. Comm Pure Appl Math, 2003, 56: 1416-1464 doi: 10.1002/cpa.v56:10
[43]	Li A B, Li Y Y. On some conformally invariant fully nonlinear equations, part II: Liouville, Harnack and Yamabe. Acta Math, 2005, 195: 117-154 doi: 10.1007/BF02588052
[44]	Li A B, Li Y Y. A fully nonlinear version of the Yamabe problem on manifolds with boundary. J Eur Math Soc (JEMS), 2006, 8: 295-316 doi: 10.4171/jems
[45]	Li J Y, Sheng W M. Deforming metrics with negative curvature by a fully nonlinear flow. Calc Var Partial Differential Equations, 2005, 23: 33-50 doi: 10.1007/s00526-004-0287-4
[46]	Li Y Y. Local gradient estimates of solutions to some conformally invariant fully nonlinear equations. Comm Pure Appl Math, 2009, 62: 1293-1326 doi: 10.1002/cpa.v62:10
[47]	Li Y Y, Nguyen L. A compactness theorem for a fully nonlinear Yamabe problem under a lower Ricci curvature bound. J Funct Anal, 2014, 266(6): 3741-3771 doi: 10.1016/j.jfa.2013.08.004
[48]	Li Y Y, Nguyen L. A fully nonlinear version of the Yamabe problem on locally conformally flat manifolds with umbilic boundary. Adv Math, 2014, 251: 87-110 doi: 10.1016/j.aim.2013.10.011
[49]	Li Y Y, Nguyen L, Wang B. Comparison principles and Lipschitz regularity for some nonlinear degenerate elliptic equations. Calc Var Partial Differential Equations, 2018, 57: 1-29 doi: 10.1007/s00526-017-1276-8
[50]	Schoen R. Conformal deformation of a Riemannian metric to constant scalar curvature. J Differential Geom, 1984, 20: 479-495
[51]	Sheng W M, Trudinger N, Wang X J. The Yamabe problem for higher order curvatures. J Differential Geom, 2007, 77: 515-553
[52]	Sheng W M, Yuan L X. A class of Neumann problems arising in conformal geometry. Pacific J Math, 2014, 270(1): 211-235 doi: 10.2140/pjm
[53]	Trudinger N S. Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann Scuola Norm Sup Pisa, 1968, 22(2): 265-274
[54]	Trudinger N S, Wang X J. The intermediate case of the Yamabe problem for higher order curvatures. IMRN: Int Math Res Not, 2010, 8(13): 2437-2458
[55]	Trudinger N S, Wang X J. On Harnack inequalities and singularities of admissible metrics in the Yamabe problem. Calc Var Partial Differential Equations, 2009, 35(3): 317-338 doi: 10.1007/s00526-008-0207-0
[56]	Viaclovsky J. Conformal geometry, contact geometry and the calculus of variations. Duke Math J, 2000, 101: 283-316
[57]	Wang X J. A priori estimates and existence for a class of fully nonlinear elliptic equations in conformal geometry. Chin Ann Math Ser B, 2006, 27(2): 1-10 doi: 10.1007/s11401-005-0247-0
[58]	Wang X J. Some counterexamples to the regularity of Monge-Ampère equations. Proc Am Math Soc, 1995, 123: 841-845
[59]	Ya Yamabe H. On a deformation of Riemannian structures on compact manifolds. Osaka Math J, 1960, 12: 21-37