First order nonlinear PDEs

Now we discuss about general solution of first order non-linear PDE given by

$$F(x, y, u, u_x, u_y) =0,~ ~\Omega,$$ (1.13)

subjected to the boundary condition

$$u=g \hspace{0.5cm} \Gamma$$ (1.14)

, where $\Gamma \subseteq \partial \Omega,$ $g: \Gamma \rightarrow \mathbb{R}$ given. Assume $F$ and $g$ are smooth functions. Here $F$ is not linear in $u_x$ and $u_y$. We also assume that $F$ has continuous second order partial derivatives over a suitable domain of $(x,y,u,u_x,u_y)$ space with $F_{u_{x}}^{2} + F_{u_{y}}^{2} \neq 0$.