Geodesic
Periodic solutions of the Hamilton–Jacobi equation with a periodic non-homogeneous term and Aubry–Mather theory
Sbornik. Mathematics, Tome 190 (1999) no. 10, pp. 1487-1504 
A. N. Sobolevskii. Periodic solutions of the Hamilton–Jacobi equation with a periodic non-homogeneous term and Aubry–Mather theory. Sbornik. Mathematics, Tome 190 (1999) no. 10, pp. 1487-1504. http://geodesic.mathdoc.fr/item/SM_1999_190_10_a4/
@article{SM_1999_190_10_a4,
     author = {A. N. Sobolevskii},
     title = {Periodic solutions of the {Hamilton{\textendash}Jacobi} equation with a~periodic non-homogeneous term and {Aubry{\textendash}Mather} theory},
     journal = {Sbornik. Mathematics},
     pages = {1487--1504},
     year = {1999},
     volume = {190},
     number = {10},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1999_190_10_a4/}
}
TY  - JOUR
AU  - A. N. Sobolevskii
TI  - Periodic solutions of the Hamilton–Jacobi equation with a periodic non-homogeneous term and Aubry–Mather theory
JO  - Sbornik. Mathematics
PY  - 1999
SP  - 1487
EP  - 1504
VL  - 190
IS  - 10
UR  - http://geodesic.mathdoc.fr/item/SM_1999_190_10_a4/
LA  - en
ID  - SM_1999_190_10_a4
ER  - 
%0 Journal Article
%A A. N. Sobolevskii
%T Periodic solutions of the Hamilton–Jacobi equation with a periodic non-homogeneous term and Aubry–Mather theory
%J Sbornik. Mathematics
%D 1999
%P 1487-1504
%V 190
%N 10
%U http://geodesic.mathdoc.fr/item/SM_1999_190_10_a4/
%G en
%F SM_1999_190_10_a4

Voir la notice de l'article provenant de la source Math-Net.Ru

It is proved that the one-dimensional Hamilton–Jacobi equation with a periodic non-homogeneous term admits a family of generalized solutions, each of which can be represented as the sum of a linear and a periodic function; a condition for the uniqueness of such a solution is given in terms of Aubry–Mather theory.

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

[2] Aubry S., “The twist map, the extended Frenkel–Kontorova model and the devil's staircase”, Physica D, 7 (1983), 240–258 | DOI | MR

[3] Aubry S., Le Daeron P. Y., “The discrete Frenkel–Kontorova model and its extensions. I: Exact results for the ground states”, Physica D, 8 (1983), 381–422 | DOI | MR

[4] Mather J. N., “Existence of quasi-periodic orbits for twist homeomorphisms of the annulus”, Topology, 21 (1982), 457–467 | DOI | MR | Zbl

[5] Sobolevskii A. N., “O periodicheskikh resheniyakh uravneniya Gamiltona–Yakobi s periodicheskoi siloi”, UMN, 53:6 (1998), 265–266 | MR | Zbl

[6] Jauslin H. R., Kreiss H. O., Moser J., On the forced Burgers equation with periodic boundary conditions, Preprint, 1997 | MR

[7] E W., “Aubry–Mather theory and periodic solutions of the forced Burgers equation”, Comm. Pure Appl. Math., 52:7 (1999), 811–828 | 3.0.CO;2-D class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[8] Cesari L., Optimization – theory and applications. Problems with ordinary differential equations, Springer-Verlag, New York, 1983 | MR

[9] Lions P.-L., Generalized solutions of Hamilton–Jacobi equations, Pitman, Boston, 1982 | MR

[10] Mather J. N., Forni G., “Action minimizing orbits in Hamiltonian systems”, Lecture Notes in Math., 1589, 1994, 92–186 | MR | Zbl

[11] Birkhoff G. D., “Quelques théorèmes sur les mouvements des systèmes dynamiques”, Bull. Soc. Math. France, 40 (1912) | Zbl