黎曼流形上k-海森堡方程的斜边值问题
Oblique Boundary Value Problems for k-Hessian Equations on Riemannian Manifolds
DOI: 10.12677/pm.2026.167169, PDF,    科研立项经费支持
作者: 李程明, 韩 菲：新疆师范大学数学科学学院，新疆 乌鲁木齐
关键词: k-海森堡；先验估计；斜边界值；脐边界；费米坐标；k-Hessian； A Priori Estimate； Oblique Boundary Value； Umbilical Boundary； Fermi Coordinate
摘要: 本文研究了具有脐边界的黎曼流形上k-海森堡方程的斜边界值问题。利用比较原理和最大值原理，我们得到了解的c0，c1和c2估计。特别地，我们证明了二阶导数可以由边界上双法向方向的导数控制，进而通过控制边界上双法向方向的导数，可实现对二阶导数的控制。
Abstract: This paper investigates oblique boundary value problems for k-Hessian equations on Riemannian manifolds with umbilical boundary. By virtue of the comparison principle and the maximum principle, we establish c0, c1 and c2 a priori estimates for solutions. In particular, we prove that the second-order derivatives can be dominated by the derivative in the double normal direction on the boundary; consequently, control over the second-order derivatives is achieved via bounding the double normal derivative at the boundary.
文章引用：李程明, 韩菲. 黎曼流形上k-海森堡方程的斜边值问题[J]. 理论数学, 2026, 16(7): 15-26. https://doi.org/10.12677/pm.2026.167169

参考文献

[1] Caffarelli, L.A., Nirenberg, L. and Spruck, J. (1985) The Dirichlet Problem for Nonlinear Second Order Elliptic Equations, III: Functions of the Eigenvalues of the Hessian. Acta Mathematica, 155, 261-301. [Google Scholar] [CrossRef] 
[2] Lions, P.L., Trudinger, N.S. and Urbas, J.I.E. (1986) The Neumann Problem for Equations of Monge‐Ampère Type. Communications on Pure and Applied Mathematics, 39, 539-563. [Google Scholar] [CrossRef] 
[3] Ma, X.N. and Qiu, G.H. (2019) The Neumann Problem for Hessian Equations. Communications in Mathematical Physics, 366, 1-28. [Google Scholar] [CrossRef] 
[4] Trudinger, N.S. (2007) Recent Developments in Elliptic Partial Differential Equations of Monge-Ampère Type. Proceedings of the International Congress of Mathematicians, Madrid, 22-30 August 2006, 291-301. [Google Scholar] [CrossRef] 
[5] Guan, B. (1999) The Dirichlet Problem for Hessian Equations on Riemannian Manifolds. Calculus of Variations and Partial Differential Equations, 8, 45-69. [Google Scholar] [CrossRef] 
[6] Guan, B. (2014) Second-order Estimates and Regularity for Fully Nonlinear Elliptic Equations on Riemannian Manifolds. Duke Mathematical Journal, 163, 1491-1524. [Google Scholar] [CrossRef] 
[7] Guan, B. and Jiao, H.M. (2015) Second Order Estimates for Hessian Type Fully Nonlinear Elliptic Equations on Riemannian Manifolds. Calculus of Variations and Partial Differential Equations, 54, 2693-2712. [Google Scholar] [CrossRef] 
[8] Guan, B. and Xiang, N. (2018) On Estimates for Fully Nonlinear Elliptic Equations with Neumann Boundary Conditions on Riemannian Manifolds.
https://www.researchgate.net/publication/329362377_On_estimates_for_fully_nonlinear_elliptic_equations_with_Neumann_boundary_conditions_on_Riemannian_Manifolds
[9] He, Y., Tu, Q. and Zhang, J.N. (2020) The Neumann Boundary Problem for k-Hessian Equations on Compact Manifolds. Scientia Sinica Mathematica, 52, 155-176. [Google Scholar] [CrossRef] 
[10] Gårding, L. (1959) An Inequality for Hyperbolic Polynomials. Indiana University Mathematics Journal, 8, 957-965. [Google Scholar] [CrossRef] 
[11] Chou, K.S. and Wang, X.J. (2001) A Variational Theory of the Hessian Equation. Communications on Pure and Applied Mathematics, 54, 1029-1064. [Google Scholar] [CrossRef] 

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