A STRONG COMPARISON PRINCIPLE FOR THE GENERALIZED DIRICHLET PROBLEM FOR MONGE--AMPERE

被引：0
作者：
Hamfeldt, Brittany froese [1 ]
机构：
[1] New Jersey Inst Technol, Dept Math Sci, Univ Hts, Newark, NJ 07102 USA
来源：
SIAM JOURNAL ON MATHEMATICAL ANALYSIS | 2025年 / 57卷 / 01期
关键词：
Monge--Ampere equation; comparison principle; numerical analysis; VISCOSITY SOLUTIONS; DIFFERENCE-SCHEMES; EQUATIONS; EIGENVALUES; MONOTONE;
D O I：
10.1137/23M1577006
中图分类号：
O29 [应用数学];
学科分类号：
070104 ;
摘要：
We prove a strong form of the comparison principle for the elliptic Monge--Ampe`\re equation, with a Dirichlet boundary condition interpreted in the viscosity sense. This comparison principle is valid when the equation admits a Lipschitz continuous weak solution. The result is tight, as demonstrated by examples in which the strong comparison principle fails in the absence of Lipschitz continuity. This form of comparison principle closes a significant gap in the convergence analysis of many existing numerical methods for the Monge--Ampe`\re equation. An important corollary is that any consistent, monotone, stable approximation of the Dirichlet problem for the Monge--Ampe`\re equation will converge to the viscosity solution.
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页码：1 / 13
页数：13
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共 32 条
  • [1] Bakelman I. J., Generalized elliptic solutions of the Dirichlet problem for n-dimensional Monge-Ampere equations, Nonlinear Functional Analysis and its Applications, Proc. Sympos. Pure Math, 45, pp. 73-102, (1986)
  • [2] Bardi M., Mannucci P., Comparison principles and Dirichlet problem for fully nonlinear degenerate equations of Monge-Ampere type, Forum Math, 25, pp. 1291-1330, (2013)
  • [3] Barles G., Souganidis P. E., Convergence of approximation schemes for fully nonlinear second order equations, Asymptot. Anal, 4, pp. 271-283, (1991)
  • [4] Benamou J.-D., Collino F., Mirebeau J.-M., Monotone and consistent discretization of the Monge-Ampere operator, Math. Comp, 85, pp. 2743-2775, (2016)
  • [5] Caffarelli L., Nirenberg L., Spruck J., The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian, Acta Math, 155, pp. 261-301, (1985)
  • [6] Caffarelli L. A., Some Regularity Properties of Solutions of Monge Ampere Equation, (1991)
  • [7] Chen Y.-Y., Wan J., Lin J., Monotone mixed finite difference scheme for Monge-Amp'ere equations, J. Sci. Comput, 76, pp. 1839-1867, (2018)
  • [8] Crandall M. G., Ishii H., Lions P.-L., User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27, pp. 1-67, (1992)
  • [9] Feng X., Jensen M., Convergent semi-Lagrangian methods for the Monge-Ampere equation on unstructured grids, SIAM J. Numer. Anal, 55, pp. 691-712, (2017)
  • [10] Feng X., Lewis T., A narrow-stencil finite difference method for approximating viscosity solutions of Hamilton-Jacobi-Bellman equations, SIAM J. Numer. Anal, 59, pp. 886-924, (2021)
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