Skip to main content
SpringerLink
Log in

A Near-Optimal Rate of Periodic Homogenization for Convex Hamilton–Jacobi Equations

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We consider a Hamilton–Jacobi equation where the Hamiltonian is periodic in space and coercive and convex in momentum. Combining the representation formula from optimal control theory and a theorem of Alexander, originally proved in the context of first-passage percolation, we find a rate of homogenization which is within a log-factor of optimal and holds in all dimensions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alexander, K.S.: Approximation of subadditive functions and convergence rates in limiting-shape results. Ann. Probab. 25(1), 30–55, 1997. ISSN: 0091-1798, 2168-894X. https://doi.org/10.1214/aop/1024404277

  2. Alexander, K.S.: Lower bounds on the connectivity function in all directions for Bernoulli per-colation in two and three dimensions. Ann. Probab. 18(4), 1547–1562, 1990. ISSN: 0091-1798

  3. Capuzzo-Dolcetta, I., Ishii, H.: On the rate of convergence in homogenization of Hamilton–Jacobi equations. Indiana Univ. Math. J. 50(3), 1113–1129, 2001. ISSN: 0022-2518

  4. Mitake, H., Tran, H.V., Yu, Y.: Rate of convergence in periodic homogenization of Hamilton–Jacobi equations: the convex setting. Arch. Ration. Mech. Anal. 233(2), 901–934, 2019. ISSN: 1432-0673. https://doi.org/10.1007/s00205-019-01371-y

  5. Papanicolaou, G., Lions, P.-L., Varadhan, S.R.S.: Homogenization of Hamilton–Jacobi equations, 1987

  6. Tran, H.: Hamilton–Jacobi Equations: Theory and Applications. American Mathematical Society, Providence, Rhode Island, 2021. ISBN: 978-1-4704-6555-1

Download references

Acknowledgements

I would like to thank my advisor, Charles Smart, for many helpful discussions and comments on earlier drafts of this paper.

Author information

Authors and Affiliations

  1. University of Chicago Division of the Physical Sciences, Chicago, IL, USA

    William Cooperman

Authors
  1. William Cooperman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to William Cooperman.

Additional information

Communicated by F. Lin.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cooperman, W. A Near-Optimal Rate of Periodic Homogenization for Convex Hamilton–Jacobi Equations. Arch Rational Mech Anal 245, 809–817 (2022). https://doi.org/10.1007/s00205-022-01797-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-022-01797-x

Search

Navigation