Convergence of Deterministic Growth Models

We prove the uniform in space and time convergence of the scaled heights of large classes of deterministic growth models that are monotone and equivariant under translations by constants. The limits are characterized as the unique (viscosity solutions) of first- or second-order partial differential equations depending on whether the growth models are scaled hyperbolically or parabolically. One of the novelties is that for many relevant models, the parabolic scaling limit yields new equations with gradient discontinuities consistent with Finsler metrics, such as the crystalline infinity Laplacian. The results greatly simplify and extend a recent work by the first author to more general surface growth models, and are possibly the first such complete results about deterministic growth. The proofs are based on the methodology developed by Barles and the second author to prove convergence of approximation schemes.