-
Previous Article
Asymptotics for the concentrated field between closely located hard inclusions in all dimensions
- CPAA Home
- This Issue
-
Next Article
An optimal osmotic control problem for a concrete dam system
A class of the non-degenerate complex quotient equations on compact Kähler manifolds
a. | School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, Anhui, China |
b. | School of Mathematics and Statistics, Fuyang Normal University, Fuyang 236037, Anhui, China |
In this paper, we are concerned with the equations that are in the form of the linear combinations of the elementary symmetric functions of a Hermitian matrix on compact K$ \ddot{a} $hler manifolds. Under the assumption of the cone condition, we obtain a priori estimates for the class of complex quotient equations. Then using the method of continuity, we prove an existence result.
References:
[1] |
Z. Blocki,
Weak solutions to the complex Hessian equation, Ann. Inst. Fourier (Grenoble), 55 (2005), 1735-1756.
|
[2] |
P. Cherrier,
Equations de Monge-Amp$\grave{e}$re sur les vari$\acute{e}$t$\acute{e}$s Hermitienne compactes, Bull. Sc. Math., 2 (1987), 343-385.
|
[3] |
J. Chu and N. McCleerey, Fully non-linear degenerate elliptic equations in complex geometry, preprint, arXiv: 2010.03431v1. |
[4] |
S. Dinew and S. Kolodziej,
Liouville and Calabi-Yau type theorems for complex Hessian equations, Amer. J. Math., 139 (2017), 403-415.
doi: 10.1353/ajm.2017.0009. |
[5] |
H. Fang, M. Lai and X. Ma,
On a class of fully nonlinear flows in Kähler geometry, J. Reine Angew. Math., 653 (2011), 189-220.
doi: 10.1515/CRELLE.2011.027. |
[6] |
B. Guan and Q. Li,
Complex Monge-Amp$\grave{e}$re equations and totally real submanifolds, Adv. Math., 225 (2010), 1185-1223.
doi: 10.1016/j.aim.2010.03.019. |
[7] |
P. Guan and X. Zhang, A class of curvature type equations. Pure and Applied Math Quarterly, preprint, arXiv: 1909.03645. |
[8] |
A. Hanani,
Equations du type de Monge-Amp$\grave{e}$re sur les vari$\acute{e}$t$\acute{e}$s hermitiennes compactes, J. Funct. Anal., 137 (1996), 49-75.
doi: 10.1006/jfan.1996.0040. |
[9] |
Z. L. Hou, X. N. Ma and D. M. Wu,
A second order estimate for complex Hessian equations on a compact Kähler manifold, Math. Res. Lett., 17 (2010), 547-561.
doi: 10.4310/MRL.2010.v17.n3.a12. |
[10] |
N. V. Krylov,
On the general notion of fully nonlinear second order elliptic equation, Trans. Amer. Math. Soc., 3 (1995), 857-895.
doi: 10.2307/2154876. |
[11] |
M. Lin and N. S. Trudinger.,
On some inequalities for elementary symmetric functions, Bull. Austral. Math. Soc., 50 (1994), 317-326.
doi: 10.1017/S0004972700013770. |
[12] |
C. Li, C. Ren and Z. Wang,
Curvature estimates for convex solutions of some fully nonlinear Hessian-type equations, Calc. Var. Partial Differ. Equ., 58 (2019), 1-32.
doi: 10.1007/s00526-019-1623-z. |
[13] |
D. Phong and T. Dat, Fully non-linear parabolic equations on compact Hermitian manifolds, preprint, arXiv: 1711.10697v2. |
[14] |
J. Spruck, Geometric aspects of the theory of fully nonlinear elliptic equations, Clay Mathematics Proceedings, 2 (2005), 283-309. |
[15] |
J. Song and B. Weinkove,
On the convergence and singularities of the J -flow with applications to the Mabuchi energy, Commun. Pure Appl. Math., 61 (2008), 210-229.
doi: 10.1002/cpa.20182. |
[16] |
W. Sun,
On a class of fully nonlinear elliptic equations on closed Hermitian manifolds, J. Geom. Anal., 26 (2016), 2459-2473.
doi: 10.1007/s12220-015-9634-2. |
[17] |
W. Sun,
On a class of fully nonlinear elliptic equations on closed Hermitian manifolds II: $L^{\infty}$estimate, Commun. Pure Appl. Math., 70 (2017), 172-199.
doi: 10.1002/cpa.21652. |
[18] |
G. Sz$\acute{e}$kelyhidi,
Fully non-linear elliptic equations on compact Hermitian manifolds, J. Differ. Geom., 109 (2018), 337-378.
doi: 10.4310/jdg/1527040875. |
[19] |
V. Tosatti and B. Weinkove,
Estimates for the complex Monge-Amp$\grave{e}$re equation on Hermitian and balanced manifolds, Asian J. Math., 14 (2010), 19-40.
doi: 10.4310/AJM.2010.v14.n1.a3. |
[20] |
V. Tosatti and B. Weinkove,
The complex Monge-Amp$\grave{e}$re equation on compact Hermitian manifolds, J. Amer. Math. Soc., 23 (2010), 1187-1195.
doi: 10.1090/S0894-0347-2010-00673-X. |
[21] |
V. Tosatti, Y. Wang, B. Weinkove and X. Yang,
$C^{2, \alpha}$ estimates for nonlinear elliptic equations in complex and almost complex geometry, Calc. Var. Partial Differ. Equ., 54 (2015), 431-453.
doi: 10.1007/s00526-014-0791-0. |
[22] |
S. Yau,
On the Ricci curvature of a compact Kähler manifold and the complex Monge-Amp$\grave{e}$re equation, I, Commun. Pure Appl. Math., 31 (1978), 339-411.
doi: 10.1002/cpa.3160310304. |
[23] |
X. Zhang and X. Zhang,
Regularity estimates of solutions to complex Monge-Amp$\grave{e}$re equations on Hermitian manifolds, J. Funct. Anal., 260 (2011), 2004-2026.
doi: 10.1016/j.jfa.2010.12.024. |
[24] |
D. Zhang,
Hessian equations on closed Hermitian manifolds, Pacific J. Math., 291 (2017), 485-510.
doi: 10.2140/pjm.2017.291.485. |
show all references
References:
[1] |
Z. Blocki,
Weak solutions to the complex Hessian equation, Ann. Inst. Fourier (Grenoble), 55 (2005), 1735-1756.
|
[2] |
P. Cherrier,
Equations de Monge-Amp$\grave{e}$re sur les vari$\acute{e}$t$\acute{e}$s Hermitienne compactes, Bull. Sc. Math., 2 (1987), 343-385.
|
[3] |
J. Chu and N. McCleerey, Fully non-linear degenerate elliptic equations in complex geometry, preprint, arXiv: 2010.03431v1. |
[4] |
S. Dinew and S. Kolodziej,
Liouville and Calabi-Yau type theorems for complex Hessian equations, Amer. J. Math., 139 (2017), 403-415.
doi: 10.1353/ajm.2017.0009. |
[5] |
H. Fang, M. Lai and X. Ma,
On a class of fully nonlinear flows in Kähler geometry, J. Reine Angew. Math., 653 (2011), 189-220.
doi: 10.1515/CRELLE.2011.027. |
[6] |
B. Guan and Q. Li,
Complex Monge-Amp$\grave{e}$re equations and totally real submanifolds, Adv. Math., 225 (2010), 1185-1223.
doi: 10.1016/j.aim.2010.03.019. |
[7] |
P. Guan and X. Zhang, A class of curvature type equations. Pure and Applied Math Quarterly, preprint, arXiv: 1909.03645. |
[8] |
A. Hanani,
Equations du type de Monge-Amp$\grave{e}$re sur les vari$\acute{e}$t$\acute{e}$s hermitiennes compactes, J. Funct. Anal., 137 (1996), 49-75.
doi: 10.1006/jfan.1996.0040. |
[9] |
Z. L. Hou, X. N. Ma and D. M. Wu,
A second order estimate for complex Hessian equations on a compact Kähler manifold, Math. Res. Lett., 17 (2010), 547-561.
doi: 10.4310/MRL.2010.v17.n3.a12. |
[10] |
N. V. Krylov,
On the general notion of fully nonlinear second order elliptic equation, Trans. Amer. Math. Soc., 3 (1995), 857-895.
doi: 10.2307/2154876. |
[11] |
M. Lin and N. S. Trudinger.,
On some inequalities for elementary symmetric functions, Bull. Austral. Math. Soc., 50 (1994), 317-326.
doi: 10.1017/S0004972700013770. |
[12] |
C. Li, C. Ren and Z. Wang,
Curvature estimates for convex solutions of some fully nonlinear Hessian-type equations, Calc. Var. Partial Differ. Equ., 58 (2019), 1-32.
doi: 10.1007/s00526-019-1623-z. |
[13] |
D. Phong and T. Dat, Fully non-linear parabolic equations on compact Hermitian manifolds, preprint, arXiv: 1711.10697v2. |
[14] |
J. Spruck, Geometric aspects of the theory of fully nonlinear elliptic equations, Clay Mathematics Proceedings, 2 (2005), 283-309. |
[15] |
J. Song and B. Weinkove,
On the convergence and singularities of the J -flow with applications to the Mabuchi energy, Commun. Pure Appl. Math., 61 (2008), 210-229.
doi: 10.1002/cpa.20182. |
[16] |
W. Sun,
On a class of fully nonlinear elliptic equations on closed Hermitian manifolds, J. Geom. Anal., 26 (2016), 2459-2473.
doi: 10.1007/s12220-015-9634-2. |
[17] |
W. Sun,
On a class of fully nonlinear elliptic equations on closed Hermitian manifolds II: $L^{\infty}$estimate, Commun. Pure Appl. Math., 70 (2017), 172-199.
doi: 10.1002/cpa.21652. |
[18] |
G. Sz$\acute{e}$kelyhidi,
Fully non-linear elliptic equations on compact Hermitian manifolds, J. Differ. Geom., 109 (2018), 337-378.
doi: 10.4310/jdg/1527040875. |
[19] |
V. Tosatti and B. Weinkove,
Estimates for the complex Monge-Amp$\grave{e}$re equation on Hermitian and balanced manifolds, Asian J. Math., 14 (2010), 19-40.
doi: 10.4310/AJM.2010.v14.n1.a3. |
[20] |
V. Tosatti and B. Weinkove,
The complex Monge-Amp$\grave{e}$re equation on compact Hermitian manifolds, J. Amer. Math. Soc., 23 (2010), 1187-1195.
doi: 10.1090/S0894-0347-2010-00673-X. |
[21] |
V. Tosatti, Y. Wang, B. Weinkove and X. Yang,
$C^{2, \alpha}$ estimates for nonlinear elliptic equations in complex and almost complex geometry, Calc. Var. Partial Differ. Equ., 54 (2015), 431-453.
doi: 10.1007/s00526-014-0791-0. |
[22] |
S. Yau,
On the Ricci curvature of a compact Kähler manifold and the complex Monge-Amp$\grave{e}$re equation, I, Commun. Pure Appl. Math., 31 (1978), 339-411.
doi: 10.1002/cpa.3160310304. |
[23] |
X. Zhang and X. Zhang,
Regularity estimates of solutions to complex Monge-Amp$\grave{e}$re equations on Hermitian manifolds, J. Funct. Anal., 260 (2011), 2004-2026.
doi: 10.1016/j.jfa.2010.12.024. |
[24] |
D. Zhang,
Hessian equations on closed Hermitian manifolds, Pacific J. Math., 291 (2017), 485-510.
doi: 10.2140/pjm.2017.291.485. |
[1] |
Dmitry Jakobson, Alexander Strohmaier, Steve Zelditch. On the spectrum of geometric operators on Kähler manifolds. Journal of Modern Dynamics, 2008, 2 (4) : 701-718. doi: 10.3934/jmd.2008.2.701 |
[2] |
Carlos Kenig, Tobias Lamm, Daniel Pollack, Gigliola Staffilani, Tatiana Toro. The Cauchy problem for Schrödinger flows into Kähler manifolds. Discrete and Continuous Dynamical Systems, 2010, 27 (2) : 389-439. doi: 10.3934/dcds.2010.27.389 |
[3] |
Gang Tian. Finite-time singularity of Kähler-Ricci flow. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1137-1150. doi: 10.3934/dcds.2010.28.1137 |
[4] |
Wei Sun. On uniform estimate of complex elliptic equations on closed Hermitian manifolds. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1553-1570. doi: 10.3934/cpaa.2017074 |
[5] |
Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2619-2633. doi: 10.3934/dcds.2020377 |
[6] |
Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete and Continuous Dynamical Systems, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233 |
[7] |
Maciej J. Capiński, Piotr Zgliczyński. Cone conditions and covering relations for topologically normally hyperbolic invariant manifolds. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 641-670. doi: 10.3934/dcds.2011.30.641 |
[8] |
M. L. M. Carvalho, Edcarlos D. Silva, C. Goulart. Choquard equations via nonlinear rayleigh quotient for concave-convex nonlinearities. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3445-3479. doi: 10.3934/cpaa.2021113 |
[9] |
Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2505-2518. doi: 10.3934/cpaa.2020272 |
[10] |
Xianmin Geng, Shengli Zhou, Jiashan Tang, Cong Yang. A sufficient condition for classified networks to possess complex network features. Networks and Heterogeneous Media, 2012, 7 (1) : 59-69. doi: 10.3934/nhm.2012.7.59 |
[11] |
Joel Coacalle, Andrew Raich. Compactness of the complex Green operator on non-pseudoconvex CR manifolds. Communications on Pure and Applied Analysis, 2021, 20 (6) : 2139-2154. doi: 10.3934/cpaa.2021061 |
[12] |
Zaihong Jiang, Li Li, Wenbo Lu. Existence of axisymmetric and homogeneous solutions of Navier-Stokes equations in cone regions. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4231-4258. doi: 10.3934/dcdss.2021126 |
[13] |
D.J. Georgiev, A. J. Roberts, D. V. Strunin. Nonlinear dynamics on centre manifolds describing turbulent floods: k-$\omega$ model. Conference Publications, 2007, 2007 (Special) : 419-428. doi: 10.3934/proc.2007.2007.419 |
[14] |
Yazhou Han. Integral equations on compact CR manifolds. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2187-2204. doi: 10.3934/dcds.2020358 |
[15] |
Tiancong Chen, Qing Han. Smooth local solutions to Weingarten equations and $\sigma_k$-equations. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 653-660. doi: 10.3934/dcds.2016.36.653 |
[16] |
Giuseppe Savaré. Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in $RCD (K, \infty)$ metric measure spaces. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1641-1661. doi: 10.3934/dcds.2014.34.1641 |
[17] |
Chuanqiang Chen, Li Chen, Xinqun Mei, Ni Xiang. The Neumann problem for a class of mixed complex Hessian equations. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022049 |
[18] |
Jean-François Coulombel, Frédéric Lagoutière. The Neumann numerical boundary condition for transport equations. Kinetic and Related Models, 2020, 13 (1) : 1-32. doi: 10.3934/krm.2020001 |
[19] |
Zhihua Huang, Xiaochun Liu. Existence theorem for a class of semilinear totally characteristic elliptic equations involving supercritical cone sobolev exponents. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3201-3216. doi: 10.3934/cpaa.2019144 |
[20] |
A. V. Rezounenko. Inertial manifolds with delay for retarded semilinear parabolic equations. Discrete and Continuous Dynamical Systems, 2000, 6 (4) : 829-840. doi: 10.3934/dcds.2000.6.829 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]