# Numerical Solution of the Simple Monge-Amp\`ere Equation with Non-convex Dirichlet Data on Non-convex Domains

## TL;DR

This paper provides numerical evidence that the semi-Lagrangian method for the Monge-Ampère equation converges even on non-convex domains and with non-convex boundary data, extending known convergence results.

## Contribution

It demonstrates convergence of the semi-Lagrangian method without convexity assumptions, supported by numerical experiments and analysis.

## Key findings

- Numerical solutions converge on non-convex domains.
- Multi-valued functions can approximate solutions satisfying boundary conditions.
- Convergence observed without strict convexity assumptions.

## Abstract

The existence of a unique numerical solution of the semi-Lagrangian method for the simple Monge-Amp\`ere equation is known independently of the convexity of the domain or Dirichlet boundary data -- when the Monge-Amp\`ere equation is posed as Bellman problem. However, the convergence to the viscosity solution has only been proved on strictly convex domains. In this paper we provide numerical evidence that convergence of numerical solutions is observed more generally without convexity assumptions. We illustrate how in the limit multi-valued functions may be approximated to satisfy the Dirichlet conditions on the boundary as well as local convexity in the interior of the domain.