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A note on the Monge–Ampère type equations with general source terms


Authors: Weifeng Qiu and Lan Tang
Journal: Math. Comp. 89 (2020), 2675-2706
MSC (2010): Primary 65N30, 65L12
DOI: https://doi.org/10.1090/mcom/3554
Published electronically: June 19, 2020
MathSciNet review: 4136543
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Abstract: In this paper we consider numerical approximation to the generalised solutions to the Monge–Ampère type equations with general source terms. We first give some important propositions for the border of generalised solutions. Then, for both the classical and weak Dirichlet boundary conditions, we present well-posed numerical methods for the generalised solutions with general source terms. Finally, we prove that the numerical solutions converge to the generalised solution.


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Additional Information

Weifeng Qiu
Affiliation: Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Hong Kong, People’s Republic of China
MR Author ID: 845089
Email: weifeqiu@cityu.edu.hk

Lan Tang
Affiliation: School of Mathematics and Statistics, Central China Normal University, Wuhan, Hubei 430079, People’s Republic of China
Email: lantang@mail.ccnu.edu.cn

Keywords: Subdifferential, Monge–Ampère equation, Oliker–Prussner method, generalised solution, convex domain, convex function
Received by editor(s): April 11, 2019
Received by editor(s) in revised form: November 12, 2019, and February 11, 2020
Published electronically: June 19, 2020
Additional Notes: The first author was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 11302219).
The second author was partially supported by NNSFC grant of China (No. 11831009) and the Fundamental Research Funds for the Central Universities (No. CCNU19TS032).
The second author is the corresponding author.
Article copyright: © Copyright 2020 American Mathematical Society