This paper analyzes a regularization scheme of the Monge--Amp\`ere equation by uniformly elliptic Hamilton--Jacobi--Bellman equations. The main tools are stability estimates in the $L^\infty$ norm from the theory of viscosity solutions which are independent of the regularization parameter $\varepsilon$. They allow for the uniform convergence of the solution $u_\varepsilon$ to the regularized problem towards the Alexandrov solution $u$ to the Monge--Amp\`ere equation for any nonnegative $L^n$ right-hand side and continuous Dirichlet data. The main application are guaranteed a posteriori error bounds in the $L^\infty$ norm for continuously differentiable finite element approximations of $u$ or $u_\varepsilon$.
翻译:本文分析了蒙-安普-安普-安普-埃雷方程式的正规化方案。 主要工具是来自不依赖正规化参数的粘度解决方案理论的“美元”标准值的“美元”标准值的“美元”稳定性估算值。 这使得“美元”解决方案与“亚历山德罗-安普-安普-安普-安普-安普-安普-安普-安普-安普-安普-安普-安普-安普-安普-安普-安普-安普-安普-安格”方案在任何非负值的汉密尔顿-汉密尔顿-贾科比-贝-贝尔曼方程式和连续的 Dirichlet数据中的统一性方案。 主要的应用程序保证了“美元”标准“美元”的事后误差,用于持续不同的有限要素近值$或“美元”的“美元”的“美元”标准值。