The Neumann problem for a class of mixed complex Hessian equations

Chuanqiang Chen; Li Chen; Xinqun Mei; Ni Xiang

doi:10.3934/dcds.2022049

Article Contents

2022, Volume 42, Issue 9: 4203-4218. Doi: 10.3934/dcds.2022049

This issue Previous Article Normal form coordinates for the Benjamin-Ono equation having expansions in terms of pseudo-differential operators Next Article Mean dimension theory in symbolic dynamics for finitely generated amenable groups

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
^*Corresponding author: Li Chen

Received: February 29, 2020

Revised: February 28, 2022

Early access: April 2022

Published: August 2022

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

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Abstract

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 $.

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Mathematics Subject Classification: Primary: 35J60; Secondary: 35B45.

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[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.