# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2022049
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## The Neumann problem for a class of mixed complex Hessian equations

 1 School of Mathematics and Statistics, Ningbo University, Ningbo 315211, China 2 Faculty of Mathematics and Statistics, Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan 430062, China 3 School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

*Corresponding author: Li Chen

Received  March 2020 Revised  March 2022 Early access April 2022

Fund Project: Research of the first author was supported by ZJNSF No. LXR22A010001, NSFC No.11771396 and No.12171260, and research of the second and the fourth authors was supported by funds from Hubei Provincial Department of Education Key Projects D20171004, D20181003 and NSFC No.11971157

In this paper, we consider the Neumann problem of a class of mixed complex Hessian equations $\sigma_k(\partial \bar{\partial} u) = \sum\limits _{l = 0}^{k-1} \alpha_l(z) \sigma_l (\partial \bar{\partial} u)$ with $2 \leq k \leq n$, and establish the global $C^1$ estimates and reduce the global second derivative estimate to the estimate of double normal second derivatives on the boundary. In particular, we can prove the global $C^2$ estimates and the existence theorems when $k = n$.

Citation: Chuanqiang Chen, Li Chen, Xinqun Mei, Ni Xiang. The Neumann problem for a class of mixed complex Hessian equations. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022049
##### References:
 [1] L. Caffarelli, J. Kohn, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. Ⅱ. Complex Monge-Ampère and uniformly elliptic equations, Comm. Pure Applied Math., 38 (1985), 209-252.  doi: 10.1002/cpa.3160380206. [2] L. Caffarelli, L. Nirenberg and J. Spruck, Dirichlet problem for nonlinear second order elliptic equations Ⅰ, Monge-Ampère equations, Comm. Pure Appl. Math., 37 (1984), 369-402.  doi: 10.1002/cpa.3160370306. [3] L. Caffarelli, L. Nirenberg and J. Spruck, Dirichlet problem for nonlinear second order elliptic equations Ⅲ, Functions of the eigenvalues of the Hessian, Acta Math., 155 (1985), 261-301.  doi: 10.1007/BF02392544. [4] C. Q. Chen, L. Chen, X. Q. Mei and N. Xiang, The classical neumann problem for a class of mixed Hessian equations, Stud. Appl. Math., 148 (2022), 5-26. [5] C. Q. Chen and W. Wei, The Neumann problem of complex Hessian quotient equations, Int. Math. Res. Not., 2021 (2021), 17652-17672.  doi: 10.1093/imrn/rnaa081. [6] C. Q. Chen and D. K. Zhang, The Neumann problem of Hessian quotient equations, Bull. Math. Sci., 11 (2021), Paper No. 2050018, 26 pp. doi: 10.1142/S1664360720500186. [7] K. S. Chou and X. J. Wang, A variation theory of the Hessian equation, Comm. Pure Appl. Math., 54 (2001), 1029-1064.  doi: 10.1002/cpa.1016. [8] J. X. Fu and S. T. Yau, A Monge-Ampère type equation motivated by string theorey, Comm. Anal. Geom., 15 (2007), 29-75.  doi: 10.4310/CAG.2007.v15.n1.a2. [9] J. X. Fu and S. T. Yau, The theory of superstring with flux on non-K$\ddot{a}$hler manifolds and the complex Monge-Ampère equation, J. Diff. Geom., 78 (2008), 369-428. [10] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin-New York, 1977. [11] P. F. Guan and X. W. Zhang, A class of curvature type equations, Pure Appl. Math. Q., 17 (2021), 865-907.  doi: 10.4310/PAMQ.2021.v17.n3.a2. [12] R. Harvey and B. Lawson, Calibrated geometries, Acta Math., 148 (1982), 47-157.  doi: 10.1007/BF02392726. [13] 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. [14] G. Huisken and C. Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math., 183 (1999), 45-70.  doi: 10.1007/BF02392946. [15] N. Ivochkina, Solutions of the Dirichlet problem for certain equations of Monge-Ampère type (in Russian), Mat. Sb., 128 (1985), 403-415. [16] F. D. Jiang and N. Trudinger, Oblique boundary value problems for augmented Hessian equations Ⅰ, Bull. Math. Sci., 8 (2018), 353-411.  doi: 10.1007/s13373-018-0124-2. [17] F. D. Jiang and N. Trudinger, Oblique boundary value problems for augmented Hessian equations Ⅱ, Nonlinear Anal., 154 (2017), 148-173.  doi: 10.1016/j.na.2016.08.007. [18] F. D. Jiang and N. Trudinger, Oblique boundary value problems for augmented Hessian equation Ⅲ, Comm. Part. Diff. Equa., 44 (2019), 708-748.  doi: 10.1080/03605302.2019.1597113. [19] N. V. Krylov, On the general notion of fully nonlinear second order elliptic equation, Trans. Amer. Math. Soc., 347 (1995), 857-895.  doi: 10.1090/S0002-9947-1995-1284912-8. [20] S. Y. Li, On the Neumann problems for Complex Monge-Ampère equations, Indiana Univ. Math. J., 43 (1994), 1099-1122.  doi: 10.1512/iumj.1994.43.43048. [21] Y. Y. Li, Interior gradient estimates for solutions of certain fully nonlinear elliptic equations, J. Diff. Equa., 90 (1991), 172-185.  doi: 10.1016/0022-0396(91)90166-7. [22] G. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302. [23] G. Lieberman, Oblique Boundary Value Problems for Elliptic Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8679. [24] G. Lieberman and N. Trudinger, Nonlinear oblique boundary value problems for nonlinear elliptic equations, Trans. Amer. Math. Soc., 295 (1986), 509-546.  doi: 10.1090/S0002-9947-1986-0833695-6. [25] 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. [26] P. L. Lions, N. Trudinger and J. Urbas, The Neumann problem for equations of Monge-Ampère type, Comm. Pure Appl. Math., 39 (1986), 539-563.  doi: 10.1002/cpa.3160390405. [27] X. N. Ma and G. H. Qiu, The Neumann problem for hessian equations, Comm. Math. Phys., 366 (2019), 1-28.  doi: 10.1007/s00220-019-03339-1. [28] G. H. Qiu and C. Xia, Classical Neumann problems for hessian equations and Alexandrov-Fenchel's inequalities, Int. Math. Res. Not., 2019 (2019), 6285-6303.  doi: 10.1093/imrn/rnx296. [29] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University, 1993. doi: 10.1017/CBO9780511526282. [30] J. Spruck, Geometric aspects of the theory of fully nonlinear elliptic equations, Amer. Math. Soc., Providence, RI, 2 (2005), 283-309. [31] N. Trudinger, On degenerate fully nonlinear elliptic equations in balls, Bull. Aust. Math. Soc., 35 (1987), 299-307.  doi: 10.1017/S0004972700013253. [32] N. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math., 175 (1995), 151-164.  doi: 10.1007/BF02393303. [33] J. Urbas, Nonlinear oblique boundary value problems for Hessian equations in two dimensions, Ann. Inst. H. Poincarè-Anal. Non Lin., 12 (1995), 507-575.  doi: 10.1016/s0294-1449(16)30150-0. [34] J. Urbas, Nonlinear oblique boundary value problems for two-dimensional curvature equations, Adv. Diff. Equa., 1 (1996), 301-336.

show all references

##### References:
 [1] L. Caffarelli, J. Kohn, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. Ⅱ. Complex Monge-Ampère and uniformly elliptic equations, Comm. Pure Applied Math., 38 (1985), 209-252.  doi: 10.1002/cpa.3160380206. [2] L. Caffarelli, L. Nirenberg and J. Spruck, Dirichlet problem for nonlinear second order elliptic equations Ⅰ, Monge-Ampère equations, Comm. Pure Appl. Math., 37 (1984), 369-402.  doi: 10.1002/cpa.3160370306. [3] L. Caffarelli, L. Nirenberg and J. Spruck, Dirichlet problem for nonlinear second order elliptic equations Ⅲ, Functions of the eigenvalues of the Hessian, Acta Math., 155 (1985), 261-301.  doi: 10.1007/BF02392544. [4] C. Q. Chen, L. Chen, X. Q. Mei and N. Xiang, The classical neumann problem for a class of mixed Hessian equations, Stud. Appl. Math., 148 (2022), 5-26. [5] C. Q. Chen and W. Wei, The Neumann problem of complex Hessian quotient equations, Int. Math. Res. Not., 2021 (2021), 17652-17672.  doi: 10.1093/imrn/rnaa081. [6] C. Q. Chen and D. K. Zhang, The Neumann problem of Hessian quotient equations, Bull. Math. Sci., 11 (2021), Paper No. 2050018, 26 pp. doi: 10.1142/S1664360720500186. [7] K. S. Chou and X. J. Wang, A variation theory of the Hessian equation, Comm. Pure Appl. Math., 54 (2001), 1029-1064.  doi: 10.1002/cpa.1016. [8] J. X. Fu and S. T. Yau, A Monge-Ampère type equation motivated by string theorey, Comm. Anal. Geom., 15 (2007), 29-75.  doi: 10.4310/CAG.2007.v15.n1.a2. [9] J. X. Fu and S. T. Yau, The theory of superstring with flux on non-K$\ddot{a}$hler manifolds and the complex Monge-Ampère equation, J. Diff. Geom., 78 (2008), 369-428. [10] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin-New York, 1977. [11] P. F. Guan and X. W. Zhang, A class of curvature type equations, Pure Appl. Math. Q., 17 (2021), 865-907.  doi: 10.4310/PAMQ.2021.v17.n3.a2. [12] R. Harvey and B. Lawson, Calibrated geometries, Acta Math., 148 (1982), 47-157.  doi: 10.1007/BF02392726. [13] 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. [14] G. Huisken and C. Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math., 183 (1999), 45-70.  doi: 10.1007/BF02392946. [15] N. Ivochkina, Solutions of the Dirichlet problem for certain equations of Monge-Ampère type (in Russian), Mat. Sb., 128 (1985), 403-415. [16] F. D. Jiang and N. Trudinger, Oblique boundary value problems for augmented Hessian equations Ⅰ, Bull. Math. Sci., 8 (2018), 353-411.  doi: 10.1007/s13373-018-0124-2. [17] F. D. Jiang and N. Trudinger, Oblique boundary value problems for augmented Hessian equations Ⅱ, Nonlinear Anal., 154 (2017), 148-173.  doi: 10.1016/j.na.2016.08.007. [18] F. D. Jiang and N. Trudinger, Oblique boundary value problems for augmented Hessian equation Ⅲ, Comm. Part. Diff. Equa., 44 (2019), 708-748.  doi: 10.1080/03605302.2019.1597113. [19] N. V. Krylov, On the general notion of fully nonlinear second order elliptic equation, Trans. Amer. Math. Soc., 347 (1995), 857-895.  doi: 10.1090/S0002-9947-1995-1284912-8. [20] S. Y. Li, On the Neumann problems for Complex Monge-Ampère equations, Indiana Univ. Math. J., 43 (1994), 1099-1122.  doi: 10.1512/iumj.1994.43.43048. [21] Y. Y. Li, Interior gradient estimates for solutions of certain fully nonlinear elliptic equations, J. Diff. Equa., 90 (1991), 172-185.  doi: 10.1016/0022-0396(91)90166-7. [22] G. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302. [23] G. Lieberman, Oblique Boundary Value Problems for Elliptic Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8679. [24] G. Lieberman and N. Trudinger, Nonlinear oblique boundary value problems for nonlinear elliptic equations, Trans. Amer. Math. Soc., 295 (1986), 509-546.  doi: 10.1090/S0002-9947-1986-0833695-6. [25] 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. [26] P. L. Lions, N. Trudinger and J. Urbas, The Neumann problem for equations of Monge-Ampère type, Comm. Pure Appl. Math., 39 (1986), 539-563.  doi: 10.1002/cpa.3160390405. [27] X. N. Ma and G. H. Qiu, The Neumann problem for hessian equations, Comm. Math. Phys., 366 (2019), 1-28.  doi: 10.1007/s00220-019-03339-1. [28] G. H. Qiu and C. Xia, Classical Neumann problems for hessian equations and Alexandrov-Fenchel's inequalities, Int. Math. Res. Not., 2019 (2019), 6285-6303.  doi: 10.1093/imrn/rnx296. [29] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University, 1993. doi: 10.1017/CBO9780511526282. [30] J. Spruck, Geometric aspects of the theory of fully nonlinear elliptic equations, Amer. Math. Soc., Providence, RI, 2 (2005), 283-309. [31] N. Trudinger, On degenerate fully nonlinear elliptic equations in balls, Bull. Aust. Math. Soc., 35 (1987), 299-307.  doi: 10.1017/S0004972700013253. [32] N. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math., 175 (1995), 151-164.  doi: 10.1007/BF02393303. [33] J. Urbas, Nonlinear oblique boundary value problems for Hessian equations in two dimensions, Ann. Inst. H. Poincarè-Anal. Non Lin., 12 (1995), 507-575.  doi: 10.1016/s0294-1449(16)30150-0. [34] J. Urbas, Nonlinear oblique boundary value problems for two-dimensional curvature equations, Adv. Diff. Equa., 1 (1996), 301-336.
 [1] Wenxiong Chen, Congming Li. A priori estimate for the Nirenberg problem. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 225-233. doi: 10.3934/dcdss.2008.1.225 [2] Weisong Dong, Tingting Wang, Gejun Bao. A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1769-1780. doi: 10.3934/cpaa.2016013 [3] Xinqun Mei, Jundong Zhou. The interior gradient estimate of prescribed Hessian quotient curvature equation in the hyperbolic space. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1187-1198. doi: 10.3934/cpaa.2021012 [4] Julii A. Dubinskii. Complex Neumann type boundary problem and decomposition of Lebesgue spaces. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 201-210. doi: 10.3934/dcds.2004.10.201 [5] Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5217-5226. doi: 10.3934/dcdsb.2020340 [6] Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367 [7] Lucie Baudouin, Emmanuelle Crépeau, Julie Valein. Global Carleman estimate on a network for the wave equation and application to an inverse problem. Mathematical Control and Related Fields, 2011, 1 (3) : 307-330. doi: 10.3934/mcrf.2011.1.307 [8] Soumen Senapati, Manmohan Vashisth. Stability estimate for a partial data inverse problem for the convection-diffusion equation. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021060 [9] Yanni Guo, Genqi Xu, Yansha Guo. Stabilization of the wave equation with interior input delay and mixed Neumann-Dirichlet boundary. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2491-2507. doi: 10.3934/dcdsb.2016057 [10] Mamadou Sango. Homogenization of the Neumann problem for a quasilinear elliptic equation in a perforated domain. Networks and Heterogeneous Media, 2010, 5 (2) : 361-384. doi: 10.3934/nhm.2010.5.361 [11] 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 [12] Ziwei Zhou, Jiguang Bao. On the exterior problem for parabolic k-Hessian equations. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022106 [13] Atsushi Kawamoto. Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation. Inverse Problems and Imaging, 2018, 12 (2) : 315-330. doi: 10.3934/ipi.2018014 [14] John Sylvester. An estimate for the free Helmholtz equation that scales. Inverse Problems and Imaging, 2009, 3 (2) : 333-351. doi: 10.3934/ipi.2009.3.333 [15] 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 [16] Sándor Kelemen, Pavol Quittner. Boundedness and a priori estimates of solutions to elliptic systems with Dirichlet-Neumann boundary conditions. Communications on Pure and Applied Analysis, 2010, 9 (3) : 731-740. doi: 10.3934/cpaa.2010.9.731 [17] Dimitri Mugnai, Kanishka Perera, Edoardo Proietti Lippi. A priori estimates for the Fractional p-Laplacian with nonlocal Neumann boundary conditions and applications. Communications on Pure and Applied Analysis, 2022, 21 (1) : 275-292. doi: 10.3934/cpaa.2021177 [18] Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101 [19] Aymen Jbalia. On a logarithmic stability estimate for an inverse heat conduction problem. Mathematical Control and Related Fields, 2019, 9 (2) : 277-287. doi: 10.3934/mcrf.2019014 [20] Bo Guan, Heming Jiao. The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 701-714. doi: 10.3934/dcds.2016.36.701

2021 Impact Factor: 1.588