This paper analyzes a regularization scheme of the Monge–Ampère 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–Ampère 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 \(u_\varepsilon \).

本文通过均匀椭圆形 Hamilton-Jacobi-Bellman 方程分析了 Monge-Ampère 方程的正则化方案。主要工具是来自粘度解理论的\(L^\infty \)范数的稳定性估计，其独立于正则化参数\(\varepsilon \)。对于任何非负\(L^n\)右侧和连续的狄利克雷数据，它们允许解\(u_\varepsilon \)统一收敛到正则化问题，从而得到 Monge-Ampère 方程的 Alexandrov 解u。主要应用保证u或\(u_\varepsilon \)的连续可微有限元近似在\(L^\infty \)范数中的后验误差界限。