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Augmented Lagrangian methods for degenerate Hamilton–Jacobi equations

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Abstract

We suggest a new approach to solve a class of degenerate Hamilton–Jacobi equations without any assumptions on the emptiness of the Aubry set. It is based on the characterization of the maximal subsolution by means of the Fenchel–Rockafellar duality. This approach enables us to use augmented Lagrangian methods as alternatives to the commonly used methods for numerical approximation of the solution, based on finite difference approximation or on optimal control interpretation of the solution.

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Notes

  1. ALG2 is an abbreviation of algorithm 2 in [20] to calculate efficiently saddle points of functionals of the form \(L_r\). It is essentially based on relaxation of Uzawa’s algorithm.

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Acknowledgements

The authors are grateful to the anonymous referees for carefully reading this paper and for the interesting remarks and suggestions. Research of Van Thanh NGUYEN is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.01-2018.309.

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Authors and Affiliations

  1. Faculté des Sciences et Techniques, Institut de recherche XLIM-DMI, UMR-CNRS 6172, Université de Limoges, Limoges, France

    Hamza Ennaji & Noureddine Igbida

  2. Department of Mathematics and Statistics, Quy Nhon University, Quy Nhon, Vietnam

    Van Thanh Nguyen

Authors
  1. Hamza Ennaji
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  2. Noureddine Igbida
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  3. Van Thanh Nguyen
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Correspondence to Noureddine Igbida.

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Communicated by Y. Giga.

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Ennaji, H., Igbida, N. & Nguyen, V.T. Augmented Lagrangian methods for degenerate Hamilton–Jacobi equations. Calc. Var. 60, 238 (2021). https://doi.org/10.1007/s00526-021-02092-5

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