Geodesic
A~boundary value problem for a~quasilinear equation of first order with arbitrary dependence of the direction of the characteristics at the boundary on the unknown function
Izvestiya. Mathematics , Tome 11 (1977) no. 2, pp. 397-416.

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A boundary value problem for the equation $$ \frac d{dx_k}a_k(x,u)+b(x,u)+cu=0 $$ is posed and investigated in a domain $\Omega\subset\mathbf R^n$ with boundary $S$. Let $a_\nu$ be the normal component on $S$ of the vector $\vec a=(a_1,\dots,a_n)$. In contrast to previous papers, an arbitrary dependence of $a_\nu(x,u)$ on $u$ is permitted. Bibliography: 7 titles. 
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É. B. Bykhovskii. A~boundary value problem for a~quasilinear equation of first order with arbitrary dependence of the direction of the characteristics at the boundary on the unknown function. Izvestiya. Mathematics , Tome 11 (1977) no. 2, pp. 397-416. http://geodesic.mathdoc.fr/item/IM2_1977_11_2_a10/

[1] Bykhovskii E. B., “Kraevaya i nachalno-kraevaya zadachi “v tselom” dlya kvazilineinogo zakona sokhraneniya”, Dokl. AN SSSR, 215:1 (1974), 17–20

[2] Bykhovskii E. B., “Globalnaya kraevaya zadacha dlya kvazilineinogo differentsialnogo uravneniya pervogo poryadka”, Izvestiya AN SSSR. Seriya matem., 38 (1974), 1408–1456

[3] Bykhovskii E. B., “Povedenie reshenii kraevoi zadachi dlya ellipticheskogo uravneniya s malym parametrom pri starshikh proizvodnykh na mnogoobrazii tochek vstrechi kharakteristik vyrozhdennogo uravneniya”, Vestnik Leningr. un-ta, 1975, no. 13, 7–13

[4] Bykhovskii E. B., “Otsenki proizvodnykh resheniya ellipticheskoi kraevoi zadachi s malym parametrom pri starshikh proizvodnykh”, Vestnik Leningr. un-ta, 1977, no. 1

[5] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1973 | MR

[6] Volpert A. I., “Prostranstva $BV$ i kvazilineinye uravneniya”, Matem. sb., 73(115):2 (1967), 255–302 | MR

[7] Vishik M. I., Lyusternik L. A., “Regulyarnoe vyrozhdenie i pogranichnyi sloi dlya lineinykh differentsialnykh uravnenii s malym parametrom”, Uspekhi matem. nauk, 12:5(77) (1957), 3–120 | MR