I'm looking for a reference about existence of first order PDE. Maybe this problem is quite standard, but I haven't found anything about it.

Consider the coordinates $(x,y)\in\mathbb{R}^{n}\times\mathbb{R}^{m}$. Given a vectorial function $F:\mathbb{R}^{n+m}\to \mathbb{R}^{n+m}$ and a function $g(x) \in C^1 $, I'm looking for a function $u$ which solves (locally) $$ \begin{cases} \nabla u\cdot F=0\\ u(x,0)=g(x), \end{cases} $$ considering that $$ F\cdot e_{y_j}\not=0, \quad \text{for every } j=1\dots, m. $$

The first thing that I want to note is that the boundary condition is on a set lower dimension. I think that this is better for finding solutions: we can impose a more stronger boundary condition, e.g., $u(x,y_1,\dots,y_{n-1},0)=g(x)$.

My question is about the minimal regularity assumptions on $F$ to find a solution.

I know that if $F\in C^1$, then we have (local) existence and uniqueness for our problem. This theorem can be found in the Evans's book for example.

If we only suppose that $F$ is continuous, the existence of solutions for the problem is guaranteed? I'm pretty sure that one lose uniqueness of solutions.

My guess is that one can find solutions. The strategy used to solve this problem is to find the characteristic curves by solving an ODE system and to applying the inverse function theorem.

Let us consider $(s,\sigma)\in \mathbb{R}^{n+m-1}\times\mathbb{R}$ and solves $$ \partial_\sigma x(\sigma,s)=F(x),\\ x(s,0)=(s,0), $$ which admits solution if $F$ is continuous by Peano's existences theorem. One has that the function $x=x(s,\sigma)$ satisfies $|J x (0)|\not=0$ and $x(0)=0$, by the orthogonality assumption and $J x$ has the same regularity of $F$, that is $x \in C^1$. So, we can apply the inverse function theorem to find that there the inverse map $x^{-1}$ of class $C^1$. Then, the function $g\circ x^{-1}$ is a $C^1$ solution to our problem.

Any suggestions or references are welcome!