Convergence of deterministic growth models

TL;DR

This paper proves uniform convergence of scaled heights in a broad class of deterministic growth models, characterizing the limits as solutions to PDEs, thus extending previous work and simplifying proofs.

Contribution

It extends and simplifies convergence results for deterministic growth models, characterizing limits as viscosity solutions of PDEs under more general conditions.

Findings

Uniform convergence of scaled heights established

Limits characterized as viscosity solutions of PDEs

Results apply to more general surface growth models

Abstract

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. The results greatly simplify and extend a recent work by the first author to more general surface growth models. The proofs are based on the methodology developed by Barles and the second author to prove convergence of approximation schemes.