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We prove a strong form of the comparison principle for the ellipticMonge-Ampere equation, with a Dirichlet boundary condition interpreted in theviscosity sense. This comparison principle is valid when the equation admits aLipschitz continuous weak solution. The result is tight, as demonstrated byexamples in which the strong comparison principle fails in the absence ofLipschitz continuity. This form of comparison principle closes a significantgap in the convergence analysis of many existing numerical methods for theMonge-Ampere equation. An important corollary is that any consistent, monotone,stable approximation of the Dirichlet problem for the Monge-Ampere equationwill converge to the viscosity solution.

A strong comparison principle for the generalized Dirichlet problem for Monge-Ampere