Viscosity solutions of the augmented K-Hessian equations
DOI:
https://doi.org/10.56764/hpu2.jos.2022.1.1.3-9Abstract
In this paper, the Dirichlet problem for the augmented k – Hessian equations in the bounded domain with nonsmooth data will be investigated. We introduce the concept of (w,k) – convex function, show that all viscosity subsolutions and supersolutions of the considering Dirichlet problem are (w,k) – convex. Furthermore, we prove some sufficient conditions for the existence and uniqueness of the viscosity solutions of the Dirichlet problem
References
M. G. Crandall, H. Ishii, and P. L. Lions, “User’s guide to viscosity solutions of second order partial differential equations,” Bull. Am. Math. Soc., vol. 27, no. 1, pp. 1–67, 1992, doi: 10.1090/S0273-0979-1992-00266-5.
A. Colesanti and P. Salani, “Hessian equations in non-smooth domains,” Nonlinear Anal. Theory, Methods Appl., vol. 38, no. 6, pp. 803–812, 1999, doi: 10.1016/S0362-546X(98)00121-7.
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224, no. 1. Berlin, Heidelberg: Springer Berlin Heidelberg, 2001. doi: 10.1007/978-3-642-61798-0.
H. Ishii and P. L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, vol. 83, no. 1. 1990. doi: 10.1016/0022-0396(90)90068-Z.
F. Jiang, N. S. Trudinger, and X. P. Yang, “On the Dirichlet problem for a class of augmented Hessian equations,” J. Differ. Equ., vol. 258, no. 5, pp. 1548–1576, 2015, doi: 10.1016/j.jde.2014.11.005.
F. Jiang and N. S. Trudinger, “Oblique boundary value problems for augmented Hessian equations II,” Nonlinear Anal. Theory, Methods Appl., vol. 154, pp. 148–173, 2017, doi: 10.1016/j.na.2016.08.007.
H. T. Ngoan and T. T. K. Chung, “Elliptic Solutions to Nonsymmetric Monge-Ampère Type Equations II. A Priori Estimates and the Dirichlet Problem,” Acta Math. Vietnamica, vol. 44, no. 3, pp. 723–749, 2019, doi: 10.1007/s40306-018-0270-3.
J. Rauch and B. A. Taylor, “The Dirichlet problem for the multidimensional Monge-Ampère equation,” Rocky Mt. J. Math., vol. 7, no. 2, pp. 345–364, 1977, doi: 10.1216/RMJ-1977-7-2-345
