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Convergence of Deterministic Growth Models

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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. 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.

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  • 15 September 2022

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Acknowledgements

The first author was partially supported by National Science Foundation Grant DMS-1855484. The second author was partially supported by the National Science Foundation Grant DMS-1900599, the Office for Naval Research Grant N000141712095 and the Air Force Office for Scientific Research Grant FA9550-18-1-0494. The authors would also like to thank Ery Arias-Castro, Peter Morfe and Lexing Ying for helpful suggestions.

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Authors and Affiliations

  1. Department of Statistics, Stanford University, 390 Jane Stanford Way, Stanford, 94305, CA, USA

    Sourav Chatterjee

  2. Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, 60637, IL, USA

    Panagiotis E. Souganidis

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  1. Sourav Chatterjee
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  2. Panagiotis E. Souganidis
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Correspondence to Panagiotis E. Souganidis.

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Communicated by P.-L. Lions.

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Chatterjee, S., Souganidis, P.E. Convergence of Deterministic Growth Models. Arch Rational Mech Anal 245, 863–898 (2022). https://doi.org/10.1007/s00205-022-01798-w

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