> [!theorem] > > Let $\Omega \subset \real^d$ be an [[Open Set|open]] domain, $A: \Omega \to \real^d$ be a $C^1$-[[Vector Field|vector field]] on $U$, $b \in C^1(\Omega, \real)$, > $ > L: C^1(\Omega) \to C(\Omega) \quad u \mapsto \angles{A, u} + bu > $ > be a [[Linear Partial Differential Operator|linear partial differential operator]], and $S \subset \Omega$ be a [[Hypersurface|hypersurface]] of class $C^1$ non-characteristic for $L$. > > Then there exists a [[Neighbourhood|neighbourhood]] $U \in \cn^o(S)$ such that for any $\phi \in C^1(S, \real)$, there exists a unique $u \in C^1(U)$ such that $Lu|_{S} = \phi$. > > *Proof*. Let $x \in S$. By the [[Existence Theorem for Differential Equations]], there exists a neighbourhood $U_x \in \cn^o(x)$, $\eps > 0$, and a unique $C^1$ [[Flow|local flow]] $\alpha: I_0 \times U_x \to \Omega$ such that for each $y \in U_x$, $\alpha(0, y) = y$ and $\alpha_y' = A(y)$. Since $S$ is non-characteristic for $L$, there exists $I_1 \subset I_0$ and $V_x \in \cn^o(x)$ such that $\alpha(t, y) \not\in S$ for all $y \in U_x$ and $t \in I_0 \setminus \bracs{0}$, and $\alpha: I_1 \times (V_x \cap S) \to \alpha(I_1 \times (V_x \cap S)) = W_x$ is a $C^1$-diffeomorphism. > > Let $u \in C^1(U)$, then $Lu|_{V_x \cap S} = \phi$ on if and only if for each $y \in V_x \cap S$, > $ > \frac{d(u \circ \alpha_y)}{dt} = Du \cdot \frac{d\alpha_y}{dt} = \angles{A, u} = \phi \circ \alpha_y - bu \circ \alpha_y \quad u \circ \alpha_y(0) = \phi(y) > $ > By the existence theorem for differential equations, there exists $I_2 \subset I_1$, $V_x' \subset V_x$, and a $C^1$ map $\beta: I_2 \times W_x \to \real$ such that > $ > \frac{\partial \beta}{\partial t} = \phi \circ \alpha_y(t) - bu \circ \alpha_y(t) > $ > for all $y \in V_x$. In which case, $u_x = \beta \circ \alpha^{-1}: W_x \to \real$ is a unique local solution. By uniqueness of local solution and the [[Gluing Lemma|gluing lemma]], there exists a unique $C^1$ map $u: \bigcup_{x \in S}W_x \to S$ such that $Lu|_S = \phi$.