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Min–max formulas and other properties of certain classes of nonconvex effective Hamiltonians

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This paper is the first attempt to systematically study properties of the effective Hamiltonian \(\overline{H}\) arising in the periodic homogenization of some coercive but nonconvex Hamilton–Jacobi equations. Firstly, we introduce a new and robust decomposition method to obtain min–max formulas for a class of nonconvex \(\overline{H}\). Secondly, we analytically and numerically investigate other related interesting phenomena, such as “quasi-convexification” and breakdown of symmetry, of \(\overline{H}\) from other typical nonconvex Hamiltonians. Finally, in the appendix, we show that our new method and those a priori formulas from the periodic setting can be used to obtain stochastic homogenization for the same class of nonconvex Hamilton–Jacobi equations. Some conjectures and problems are also proposed.

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Authors and Affiliations

  1. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Jianliang Qian

  2. Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI, 48824, USA

    Jianliang Qian

  3. Department of Mathematics, University of Wisconsin Madison, Van Vleck Hall, 480 Lincoln Drive, Madison, WI, 53706, USA

    Hung V. Tran

  4. Department of Mathematics, University of California, Irvine, 410G Rowland Hall, Irvine, CA, 92697, USA

    Yifeng Yu

Authors
  1. Jianliang Qian
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  2. Hung V. Tran
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  3. Yifeng Yu
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Correspondence to Hung V. Tran.

Additional information

Communicated by Y. Giga.

The work of JQ is partially supported by NSF Grants 1522249 and 1614566, the work of HT is partially supported by NSF Grant DMS-1615944, the work of YY is partially supported by NSF CAREER Award #1151919.

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Qian, J., Tran, H.V. & Yu, Y. Min–max formulas and other properties of certain classes of nonconvex effective Hamiltonians. Math. Ann. 372, 91–123 (2018). https://doi.org/10.1007/s00208-017-1601-8

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  • DOI: https://doi.org/10.1007/s00208-017-1601-8

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