# A linear first order PDE with boundary condition

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 $$(\star)$$**

• What's wrong with the old good method of characteristics? It shows, by the way, that the system is unsolvable more often than not... – fedja May 12 at 9:32
• @fedja I mean, of course, you can assume that the $\partial \Omega$ is non-characteristic. – Ice sea May 12 at 10:11
• @fedja However, how to get the estimate of the solution in the norm of Sobolev spaces. Does any book contains such materials. – Ice sea May 12 at 10:14