I want to solve the following first order PDE $$ (\star)\quad\begin{cases} \nabla u\cdot \nabla\xi=f \quad\text{in}\,\Omega, \\ u\mid_{\partial \Omega}=0 \end{cases} $$ where $\xi\in C^2(\overline{\Omega})$ and $|\nabla \xi|$ does not vanish everywhere.
If we ignore the boundary condition, this problem is not difficult. By setting $w=e^{-\lambda \xi} u$, the original equation is equivalent to $$ \lambda w+\frac{1}{|\nabla \xi|^2}\nabla w\cdot \nabla \xi=\frac{e^{-\lambda \xi} }{|\nabla \xi|^2}f $$
For sufficient large $\lambda$, we can consider the the following elliptic equation $$ -\epsilon\Delta w+\lambda w+\nabla w\cdot \Xi=F \quad \text{in}\ \Omega, $$ where $\epsilon>0$ is a parameter and $\Xi=\frac{1}{|\nabla \xi|^2}\nabla \xi$. By extending the coefficients $\Xi$ and $F$ to a large domain $U$ and equip it with a necessary boundary condition. i.e. $$ -\epsilon\Delta w+\lambda w+\nabla w\cdot \Xi=F \quad \text{in}\ U. \quad w\mid_{\partial U}=0 $$ Such elliptic equation has a unique solution $w^\epsilon$ for each $\epsilon>0$. In addition, by multiplying both hands with $-\Delta w^\epsilon$ and integration by parts, we can show that $$ \|w^{\epsilon}\|_{1,U}\lesssim\|F\|_{1,U} $$ By passing limit $\epsilon \to 0$, one can find at least one solution. However, this technique does not work for the case with boundary condition.
**i) For which $F$ can we ensure that the equation $(\star)$ has a solution $u\in H_0^1(\Omega)$. Of course, we can assume $\partial\Omega$ is non-characteristic with respect to $\nabla \xi$.
ii) Is there some papers/book for PDE for the equation $(\star)$**
