  We consider the numerical construction of minimal Lagrangian graphs, which is
related to recent applications in materials science, molecular engineering, and
theoretical physics. It is known that this problem can be formulated as an
eigenvalue problem for a fully nonlinear elliptic partial differential
equation. We introduce and implement a two-step generalized finite difference
method, which we prove converges to the solution of the eigenvalue problem.
Numerical experiments validate this approach in a range of challenging
settings. We further discuss the generalization of this new framework to
Monge-Ampere type equations arising in optimal transport. This approach holds
great promise for applications where the data does not naturally satisfy the
mass balance condition, and for the design of numerical methods with improved
stability properties.
