Siconolfi, Antonio; Zabad, Sahar Scalar reduction techniques for weakly coupled Hamilton-Jacobi systems. (English) Zbl 1415.35094 NoDEA, Nonlinear Differ. Equ. Appl. 25, No. 6, Paper No. 50, 20 p. (2018). Summary: We study a class of weakly coupled systems of Hamilton-Jacobi equations at the critical level. We associate to it a family of scalar discounted equation. Using control-theoretic techniques we construct an algorithm which allows obtaining a critical solution to the system as limit of a monotonic sequence of subsolutions. We moreover get a characterization of isolated points of the Aubry set and establish semiconcavity properties for critical subsolutions. Cited in 1 Document MSC: 35F21 Hamilton-Jacobi equations 49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games 37J50 Action-minimizing orbits and measures (MSC2010) 35D40 Viscosity solutions to PDEs 35Q93 PDEs in connection with control and optimization Keywords:Aubry set; control-theoretic techniques; monotonic sequence of subsolutions; semiconcavity properties PDFBibTeX XMLCite \textit{A. Siconolfi} and \textit{S. Zabad}, NoDEA, Nonlinear Differ. Equ. Appl. 25, No. 6, Paper No. 50, 20 p. (2018; Zbl 1415.35094) Full Text: DOI arXiv References: [1] Bardi, M., CapuzzoDolcetta, I.: Optimal Control Theory and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston (1997). https://doi.org/10.1007/978-0-8176-4755-1 · doi:10.1007/978-0-8176-4755-1 [2] Barles, G.: Solutions de viscosité des équations de Hamilton-Jacobi, Mathématiques & Applications (Berlin) 17. Springer, Paris (1994) · Zbl 0819.35002 [3] Camilli, F., Ley, O., Loreti, P., Nguyen, V.D.: Large time behavior of weakly coupled systems of first-order Hamilton-Jacobi equations. NoDEA Nonlinear Differ. Equ. Appl. 19(6), 719-749 (2012). https://doi.org/10.1007/s00030-011-0149-7 · Zbl 1254.49016 · doi:10.1007/s00030-011-0149-7 [4] Davini, A., Fathi, A., Iturriaga, R., Zavidovique, M.: Convergence of the solutions of the discounted equation. Inventiones Mathematicae 206(1), 29-55 (2016). https://doi.org/10.1007/s00222-016-0648-6 · Zbl 1362.35094 · doi:10.1007/s00222-016-0648-6 [5] Davini, A., Siconolfi, A., Zavidovique, M.: Random Lax-Oleinik Semigroups for Hamilton-Jacobi Systems, arXiv:1608.01836 · Zbl 1415.35088 [6] Davini, A., Zavidovique, M.: Aubry sets for weakly coupled systems of Hamilton-Jacobi equations. SIAM J. Math. Anal. 46(5), 3361-3389 (2014). https://doi.org/10.1137/120899960 · Zbl 1327.35063 · doi:10.1137/120899960 [7] Fathi, A., Siconolfi, A.: PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians. Calc. Var. Partial Differ. Equ. 22(2), 185-228 (2005). https://doi.org/10.1007/s00526-004-0271-z · Zbl 1065.35092 · doi:10.1007/s00526-004-0271-z [8] Ibrahim, H., Siconolfi, A., Zabad, S.: Cycle Chacterization of the Aubry Set for Weakly Coupled Hamilton-Jacobi Systems, arXiv: 1604.08012 · Zbl 1415.35091 [9] Mitake, H., Siconolfi, A., Tran, H.V., Yamada, N.: A Lagrangian approach to weakly coupled Hamilton-Jacobi systems. SIAM J. Math. Anal. 48(2), 821-846 (2016). https://doi.org/10.1137/15M1010841 · Zbl 1343.35065 · doi:10.1137/15M1010841 [10] Mitake, H., Tran, H.V.: Remarks on the large time behavior of viscosity solutions of quasi-monotone weakly coupled systems of Hamilton-Jacobi equations. Asymptot. Anal. 77(1-2), 43-70 (2012). https://doi.org/10.3233/ASY-2011-1071 · Zbl 1241.35020 · doi:10.3233/ASY-2011-1071 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.