[proofplan] We construct a local $C^2$ solution to the fully nonlinear first-order PDE $F(\nabla u, u, x) = 0$ using the method of characteristics. The proof proceeds in six steps: extend the initial gradient field along $\Gamma$ using the implicit function theorem and the non-characteristic condition; solve the characteristic ODE system with Picard--Lindelof; show the flow map is a local diffeomorphism via the inverse function theorem (the non-characteristic condition ensures the Jacobian is non-degenerate); define $u$ by composing the ODE solution with the inverse flow; verify PDE satisfaction using conservation of $F$ along characteristics; and confirm the boundary condition and uniqueness. [/proofplan] [step:Extend the initial gradient field along $\Gamma$ via the implicit function theorem] The tangential consistency condition fixes the components of $\tilde{a}$ tangent to $\Gamma$: they equal $\nabla_\Gamma g(\tilde{c})$. The normal component is determined by the requirement $F(\tilde{a}, g(\tilde{c}), \tilde{c}) = 0$. The non-characteristic condition $\nabla_a F(\tilde{a}, g(\tilde{c}), \tilde{c}) \cdot \nu(\tilde{c}) \neq 0$ means the implicit function theorem can be applied to solve uniquely for the normal component of $\tilde{a}$ as a smooth function of the base point. This yields a neighbourhood $\Gamma' \subseteq \Gamma$ of $\tilde{c}$ and a unique $C^1$ admissible gradient field \begin{align*} a_0: \Gamma' &\to \mathbb{R}^n \end{align*} with $a_0(\tilde{c}) = \tilde{a}$ and $F(a_0(c_0), g(c_0), c_0) = 0$ for all $c_0 \in \Gamma'$. [/step] [step:Solve the characteristic ODE system] For each $c_0 \in \Gamma'$, the Picard--Lindelof theorem applied to the characteristic system (from the general characteristic ODE framework) with initial data \begin{align*} X(0) = c_0, \quad U(0) = g(c_0), \quad G(0) = a_0(c_0), \end{align*} yields a unique $C^1$ solution $(G(\cdot, c_0), U(\cdot, c_0), X(\cdot, c_0))$ on a common interval $(-\delta, \delta)$. The solution depends smoothly on the initial data $c_0$ by the smooth dependence theorem for ODEs. [/step] [step:Prove the flow map is a local diffeomorphism via the inverse function theorem] Define the flow map $\Psi: (-\delta, \delta) \times \Gamma' \to \mathbb{R}^n$ by $\Psi(s, c_0) = X(s, c_0)$. The Jacobian of $\Psi$ at $(0, \tilde{c})$ has columns: - $\partial_s\Psi|_{s=0} = \dot{X}(0) = \nabla_a F(\tilde{a}, g(\tilde{c}), \tilde{c})$, which has nonzero normal component by the non-characteristic condition; - $\partial_{c_0}\Psi|_{s=0}$, which consists of the $n-1$ tangent vectors to $\Gamma$ at $\tilde{c}$. These $n$ vectors are linearly independent: the first has non-zero projection onto the normal direction $\nu(\tilde{c})$, while the remaining $n-1$ span the tangent space $T_{\tilde{c}}\Gamma$. Hence $\det(D\Psi)|_{(0,\tilde{c})} \neq 0$, and the [inverse function theorem](/page/Inverse%20Function%20Theorem) provides a neighbourhood $V$ of $\tilde{c}$ on which $\Psi$ is a $C^1$ diffeomorphism. [/step] [step:Define the solution $u$ via the inverse flow] For $x \in V \cap \overline{\Omega}$, write $(s(x), c_0(x)) = \Psi^{-1}(x)$ and define \begin{align*} u(x) := U(s(x), c_0(x)). \end{align*} Since $\Psi^{-1}$ is $C^1$ and $U$ is $C^2$ in its arguments (by regularity of the characteristic ODE), $u \in C^2(V \cap \Omega)$. [/step] [step:Verify PDE satisfaction by showing $F$ is constant along characteristics] Using the characteristic ODEs, compute: \begin{align*} \frac{d}{ds} F(G(s), U(s), X(s)) &= \sum_i \frac{\partial F}{\partial a_i} \dot{G}_i + \frac{\partial F}{\partial b} \dot{U} + \sum_j \frac{\partial F}{\partial c_j} \dot{X}_j = 0, \end{align*} as each of the three ODE equations is designed to cancel a corresponding group of terms in this total derivative. Since $F(G(0), U(0), X(0)) = F(a_0(c_0), g(c_0), c_0) = 0$ by admissibility of the initial data, we conclude $F(G(s), U(s), X(s)) = 0$ for all $s \in (-\delta, \delta)$. Re-expressing in terms of $u$ gives $F(\nabla u(x), u(x), x) = 0$ at every $x \in V \cap \Omega$. [/step] [step:Confirm the boundary condition and uniqueness] At $s = 0$: $u(c_0) = U(0, c_0) = g(c_0)$ for every $c_0 \in \Gamma' \cap V$, so the boundary condition $u = g$ on $V \cap \Gamma$ is satisfied. Uniqueness of $u$ on $V \cap \Omega$ follows from the uniqueness of the ODE solution (Picard--Lindelof) and the injectivity of $\Psi$ on $V$. [/step]