Given the curve $\gamma(t):=(\cos(t),\sin(t))$ for $t\in(0,\pi)$, the differential operator $L(u):=uu_{xy}+0.5u_yu_{yy}+u^2$ and some prescribed Cauchy data
$u|_\Gamma=z|_\Gamma$ and $\partial_n u|_\Gamma=\partial_n z|_\Gamma$
($\Gamma:=\gamma((0,\pi))\subset\mathbb{R}^2$, $n$ denotes the outer normal to $\Gamma$) where $z(x,y):=\frac{1}{y}$ if $y>0$, it should be found out which parts of $\Gamma$ are characteristic to $L$.
My first idea was to make use of a theorem that involves the principal part $L_p(z;u)=uz_1z_2+0.5u_yz_2^2$ of this quasilinear differential operator: This result states that $\Gamma$ is characteristic to $L$ in $x\in\Gamma$ iff $L_p(n;u)$ vanishes if it is evaluated at $x$. Unfortunately, I do not see how to deal with the occuring $u_y(\gamma(t))$ when trying to solve this equation. Moreover, it is not clear to me how to use the condition about the normal derivative. Does someone have an idea?