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

TL;DR

This paper establishes a strong comparison principle for the elliptic Monge-Ampere equation with Dirichlet boundary conditions in the viscosity sense, ensuring convergence of numerical methods under Lipschitz continuity.

Contribution

It proves a tight strong comparison principle for the Monge-Ampere equation that guarantees convergence of numerical schemes assuming Lipschitz solutions.

Findings

Strong comparison principle holds for Lipschitz solutions

Counterexamples show failure without Lipschitz continuity

Ensures convergence of consistent, monotone, stable numerical methods

Abstract

We prove a strong form of the comparison principle for the elliptic Monge-Ampere 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-Ampere equation. An important corollary is that any consistent, monotone, stable approximation of the Dirichlet problem for the Monge-Ampere equation will converge to the viscosity solution.