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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     author = {\'E. B. Bykhovskii},
     title = {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},
     journal = {Izvestiya. Mathematics },
     pages = {397--416},
     publisher = {mathdoc},
     volume = {11},
     number = {2},
     year = {1977},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1977_11_2_a10/}
}
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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/