Geodesic
On the structure of the solution of the Hamilton-Jacobi equation with piecewise linear input data
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 2 (2008), pp. 144-147 
N. N. Subbotina; L. G. Shagalova. On the structure of the solution of the Hamilton-Jacobi equation with piecewise linear input data. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 2 (2008), pp. 144-147. http://geodesic.mathdoc.fr/item/VUU_2008_2_a47/
@article{VUU_2008_2_a47,
     author = {N. N. Subbotina and L. G. Shagalova},
     title = {On the structure of the solution of the {Hamilton-Jacobi} equation with piecewise linear input data},
     journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
     pages = {144--147},
     year = {2008},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VUU_2008_2_a47/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

The Cauchy problem is considered for the Hamilton Jacobi equation with Hamiltonian depending on the impulse variable only. Estimations have been obtained for the minimax (and/or viscosity) solution to this problem in the case of piecewise linearity of the Hamiltonian or the border function. The proposed estimations provide explicit formulas for the minimax solution, if «minimaxes» and «maximins» contained in them coincide.

[1] Subbotin A. I., Minimaksnye neravenstva i uravneniya Gamiltona–Yakobi, Nauka, M., 1991 | MR

[2] Subbotin A. I., Generalized Solutions of First Order PDEs: The Dynamical Optimization Perspective, Birkhäuser, Boston, 1995 | MR

[3] Crandall M. G., Lions P. L., “Viscosity solutions of Hamilton Jacobi equations”, Trans. Amer. Math. Soc., 277:1 (1983), 1–42 | DOI | MR | Zbl

[4] Hopf E., “Generalized solutions of nonlinear equations of first order”, J. Math. Mech., 14 (1965), 951–973 | MR | Zbl

[5] Bardi M., Evans L. C., “On Hopf 's formulas for solutions of Hamilton–Jacobi equations”, Nonlinear Analysys, Theory, Methods, Appl., 8:11 (1984), 1373–1381 | DOI | MR | Zbl

[6] Barron E. N., Jensen R., Liu W., “Hopf-Lax-type formula for $u_t+H(u,Du)=0$”, J. Differ. Equations, 126:1 (1996), 48–61 | DOI | MR | Zbl

[7] Pshenichnyi B. N., Sagaidak M. I., “O differentsialnykh igrakh s fiksirovannym vremenem”, Kibernetika, 1970, no. 2, 54–63 | MR

[8] Subbotin A. I., Shagalova L. G., “Kusochno-lineinoe reshenie zadachi Koshi dlya uravneniya Gamiltona–Yakobi”, Doklady RAN, 325:5 (1992), 932–936 | MR | Zbl

[9] Demyanov V. F., Rubinov A. M., Osnovy negladkogo analiza i kvazidiffepentsialnoe ischislenie, Nauka, M., 1990 | MR