Bulletin of the Korean Mathematical Society (대한수학회보)

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THE NEUMANN PROBLEM FOR A CLASS OF COMPLEX HESSIAN QUOTIENT EQUATIONS

  • Yuying Qian (Faculty of Mathematics and Statistics Hubei Key Laboratory of Applied Mathematics Hubei University) ;
  • Qiang Tu (Faculty of Mathematics and Statistics Hubei Key Laboratory of Applied Mathematics Hubei University) ;
  • Chenyue Xue (Faculty of Mathematics and Statistics Hubei Key Laboratory of Applied Mathematics Hubei University)
  • Received : 2023.09.26
  • Accepted : 2024.01.12
  • Published : 2024.07.31
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Abstract

In this paper, we study the Neumann problem for the complex Hessian quotient equation ${\frac{{\sigma}_k({\tau}{\Delta}uI+{\partial}{\bar{\partial}u)}}{{\sigma}_l({\tau}{\Delta}uI+{\partial}{\bar{\partial}u)}}}={\psi}$ with 0 ≤ 𝑙 < k ≤ n. We prove a priori estimate and global C1 estimates, in particular, we use the double normal second derivatives on the boundary to establish the global C2 estimates and prove the existence and the uniqueness for the Neumann problem of the above complex Hessian quotient equation.

Keywords

Acknowledgement

This research was supported by funds from the National Natural Science Foundation of China No. 12101206.

References

  1. L. Caffarelli, J. J. Kohn, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. II. Complex Monge-Ampere, and uniformly elliptic, equations, Comm. Pure Appl. Math. 38 (1985), no. 2, 209-252. https://doi.org/10.1002/cpa.3160380206
  2. L. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampere equation, Comm. Pure Appl. Math. 37 (1984), no. 3, 369-402. https://doi.org/10.1002/cpa.3160370306
  3. L. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985), no. 3-4, 261-301. https://doi.org/10.1007/BF02392544
  4. C. Chen, W. Dong, and F. Han, Interior Hessian estimates for a class of Hessian type equations, Calc. Var. Partial Differential Equations 62 (2023), no. 2, Paper No. 52, 15 pp. https://doi.org/10.1007/s00526-022-02385-3
  5. C. Chen and W. Wei, The Neumann problem of complex Hessian quotient equations, Int. Math. Res. Not. IMRN 2021 (2021), no. 23, 17652-17672. https://doi.org/10.1093/imrn/rnaa081
  6. C. Chen and D. Zhang, The Neumann problem of Hessian quotient equations, Bull. Math. Sci. 11 (2021), no. 1, Paper No. 2050018, 26 pp. https://doi.org/10.1142/S1664360720500186
  7. P. Cherrier and A. Hanani, Le probleme de Dirichlet pour les equations de Monge-Ampere en metrique hermitienne, Bull. Sci. Math. 123 (1999), no. 7, 577-597. https://doi.org/10.1016/S0007-4497(99)00115-3
  8. T. C. Collins and S. Picard, The Dirichlet problem for the k-Hessian equation on a complex manifold, Amer. J. Math. 144 (2022), no. 6, 1641-1680. https://doi.org/10.1353/ajm.2022.0040
  9. S. Dinew and S. Ko lodziej, Liouville and Calabi-Yau type theorems for complex Hessian equations, Amer. J. Math. 139 (2017), no. 2, 403-415. https://doi.org/10.1353/ajm.2017.0009
  10. W. Dong and W. Wei, The Neumann problem for a type of fully nonlinear complex equations, J. Differential Equations 306 (2022), 525-546. https://doi.org/10.1016/j.jde.2021.10.040
  11. H. Fang, M. Lai, and X. Ma, On a class of fully nonlinear flows in Kahler geometry, J. Reine Angew. Math. 653 (2011), 189-220. https://doi.org/10.1515/CRELLE.2011.027
  12. K. Feng, H. B. Ge, and T. Zheng, The Dirichlet problem of fully nonlinear equations on Hermitian manifolds, Beijing: Peking University, 2020. arXiv:1905.02412v4
  13. D. Gu and N. C. Nguyen, The Dirichlet problem for a complex Hessian equation on compact Hermitian manifolds with boundary, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18 (2018), no. 4, 1189-1248.
  14. B. Guan and Q. Li, Complex Monge-Ampere equations and totally real submanifolds, Adv. Math. 225 (2010), no. 3, 1185-1223. https://doi.org/10.1016/j.aim.2010.03.019
  15. B. Guan and Q. Li, A Monge-Ampere type fully nonlinear equation on Hermitian manifolds, Discrete Contin. Dyn. Syst. Ser. B 17 (2012), no. 6, 1991-1999. https://doi.org/10.3934/dcdsb.2012.17.1991
  16. B. Guan and Q. Li, The Dirichlet problem for a complex Monge-Ampere type equation on Hermitian manifolds, Adv. Math. 246 (2013), 351-367. https://doi.org/10.1016/j.aim.2013.07.006
  17. B. Guan and W. Sun, On a class of fully nonlinear elliptic equations on Hermitian manifolds, Calc. Var. Partial Differential Equations 54 (2015), no. 1, 901-916. https://doi.org/10.1007/s00526-014-0810-1
  18. Z. Hou, X. Ma, and D. Wu, A second order estimate for complex Hessian equations on a compact Kahler manifold, Math. Res. Lett. 17 (2010), no. 3, 547-561. https://doi.org/10.4310/MRL.2010.v17.n3.a12
  19. N. M. Ivochkina, Solution of the Dirichlet problem for certain equations of Monge-Ampere type, Mat. Sb. (N.S.) 128(170) (1985), no. 3, 403-415, 447.
  20. H. Jiao and T. Wang, The Dirichlet problem for a class of Hessian type equations, J. Math. Anal. Appl. 453 (2017), no. 1, 509-528. https://doi.org/10.1016/j.jmaa.2017.04.015
  21. S.-Y. Li, On the Neumann problems for complex Monge-Ampere equations, Indiana Univ. Math. J. 43 (1994), no. 4, 1099-1122. https://doi.org/10.1512/iumj.1994.43.43048
  22. P.-L. Lions, N. S. Trudinger, and J. I. E. Urbas, The Neumann problem for equations of Monge-Ampere type, Comm. Pure Appl. Math. 39 (1986), no. 4, 539-563. https://doi.org/10.1002/cpa.3160390405
  23. X. Ma and G. Qiu, The Neumann problem for Hessian equations, Comm. Math. Phys. 366 (2019), no. 1, 1-28. https://doi.org/10.1007/s00220-019-03339-1
  24. X. Ma, G. Qiu, and J. J. Xu, Gradient estimates on Hessian equations for Neumann problem, Sci. Sin. Math. 46 (2016), 1-10. https://doi.org/10.1360/N012014-00278
  25. G. Qiu, Neumann problems for Hessian equations and geometric applications, [Ph D Thesis]. Hefei: University of Sciences and Technology of China, 2016.
  26. G. Qiu and C. Xia, Classical Neumann problems for Hessian equations and Alexandrov-Fenchel's inequalities, Int. Math. Res. Not. IMRN 2019 (2019), no. 20, 6285-6303. https://doi.org/10.1093/imrn/rnx296
  27. J. Song and B. Weinkove, On the convergence and singularities of the J-flow with applications to the Mabuchi energy, Comm. Pure Appl. Math. 61 (2008), no. 2, 210-229. https://doi.org/10.1002/cpa.20182
  28. W. Sun, On a class of fully nonlinear elliptic equations on closed Hermitian manifolds II: L∞ estimate, Comm. Pure Appl. Math. 70 (2017), no. 1, 172-199. https://doi.org/10.1002/cpa.21652
  29. G. Szekelyhidi, Fully non-linear elliptic equations on compact Hermitian manifolds, J. Differential Geom. 109 (2018), no. 2, 337-378. https://doi.org/10.4310/jdg/1527040875
  30. V. Tosatti and B. Weinkove, Estimates for the complex Monge-Ampere equation on Hermitian and balanced manifolds, Asian J. Math. 14 (2010), no. 1, 19-40. https://doi.org/10.4310/AJM.2010.v14.n1.a3
  31. V. Tosatti and B. Weinkove, The complex Monge-Ampere equation on compact Hermitian manifolds, J. Amer. Math. Soc. 23 (2010), no. 4, 1187-1195. https://doi.org/10.1090/S0894-0347-2010-00673-X
  32. N. S. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math. 175 (1995), no. 2, 151-164. https://doi.org/10.1007/BF02393303
  33. S.-T. Yau, On the Ricci curvature of a compact Kahler manifold and the complex Monge-Ampere equation. I, Comm. Pure Appl. Math. 31 (1978), no. 3, 339-411. https://doi.org/10.1002/cpa.3160310304
  34. D. Zhang, Hessian equations on closed Hermitian manifolds, Pacific J. Math. 291 (2017), no. 2, 485-510. https://doi.org/10.2140/pjm.2017.291.485