Geodesic
Generalized solutions of the Hamilton–Jacobi equations of eikonal type. I. Formulation of the problems; existence, uniqueness and stability theorems; some properties of the solutions
Sbornik. Mathematics, Tome 27 (1975) no. 3, pp. 406-446 
S. N. Kruzhkov. Generalized solutions of the Hamilton–Jacobi equations of eikonal type. I. Formulation of the problems; existence, uniqueness and stability theorems; some properties of the solutions. Sbornik. Mathematics, Tome 27 (1975) no. 3, pp. 406-446. http://geodesic.mathdoc.fr/item/SM_1975_27_3_a6/
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Voir la notice de l'article provenant de la source Math-Net.Ru

A nonlocal theory of boundary value problems in bounded and unbounded domains is constructed for nonlinear equations of first order of the type of the classical eikonal equation in geometric optics with Cauchy–Dirichlet boundary conditions and with additional conditions at infinity in the case of unbounded domains. Existence, uniqueness and stability theorems are proved; some representations of the solutions are indicated; and properties of the solutions generalizing the principles of Fermat and Huygens in geometric optics are established. Bibliography: 14 titles.

[1] M. Born, E. Volf, Osnovy optiki, izd-vo «Nauka», Moskva, 1973

[2] I. G. Petrovskii, Lektsii po teorii obyknovennykh differentsialnykh uravnenii, izd-vo «Nauka», Moskva, 1964 | MR

[3] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mischenko, Matematicheskaya teoriya optimalnykh protsessov, Fizmatgiz, Moskva, 1961

[4] S. N. Kruzhkov, “Zadacha Koshi v tselom dlya nelineinykh uravnenii i nekotorykh kvazilineinykh sistem pervogo poryadka so mnogimi peremennymi”, DAN SSSR, 155:4 (1964), 743–746

[5] S. N. Kruzhkov, “Obobschennye resheniya nelineinykh uravnenii pervogo poryadka i nekotorye zadachi dlya kvazilineinykh parabolicheskikh uravnenii”, Vestnik MGU, seriya matem. i mekh., 1964, no. 6, 65–74

[6] S. N. Kruzhkov, “O resheniyakh nelineinykh uravnenii pervogo poryadka”, DAN SSSR, 167:2 (1966), 286–289 | Zbl

[7] S. N. Kruzhkov, “Metod konechnykh raznostei dlya nelineinykh uravnenii pervogo poryadka so mnogimi nezavisimymi peremennymi”, Zh. vych. matem. i matem. fiziki, 6:5 (1966), 884–894 | Zbl

[8] S. N. Kruzhkov, “Obobschennye resheniya nelineinykh uravnenii pervogo poryadka so mnogimi nezavisimymi peremennymi, I”, Matem. sb., 70(112) (1966), 394–415 | Zbl

[9] S. N. Kruzhkov, “Obobschennye resheniya nelineinykh uravnenii pervogo poryadka so mnogimi nezavisimymi peremennymi, II”, Matem. sb., 72(114) (1967), 108–134 | Zbl

[10] S. N. Kruzhkov, Nelineinye uravneniya s chastnymi proizvodnymi (lektsii), ch. 2. Uravneniya pervogo poryadka, rotaprint MGU, Moskva, 1970

[11] S. L. Sobolev, Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, izd-vo LGU, Leningrad, 1950

[12] J. Serrin, “The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables”, Phil. Trans. Royal Soc. London, Ser. A, 264:1153 (1969), 413–496 | DOI | MR | Zbl

[13] K. Miranda, Uravneniya s chastnymi proizvodnymi ellipticheskogo tipa, IL, Moskva, 1957

[14] F. Frank, R. Mizes, Differentsialnye i integralnye uravneniya matematicheskoi fiziki, ONTI, Moskva, 1937