Maybe this is a canonical result but I'm facing difficulties to find a reference.
It is well known the following theorem about Linear Partial Differential Equations:
Theorem: Let $\Omega \subset \mathbb{R}^2$ be an open set and $\Gamma \subset \Omega$ a $\mathcal{C}^1$ curve. Let $\gamma (s) = (\alpha(s),\beta(s))$ be a $\mathcal{C}^1$ parametrization of $\Gamma$, defined in the interval $I \subset \mathbb{R}$. Suppose that $a,b,c$ $\in$ $\mathcal{C}^1 (\Omega , \mathbb{R})$ and $f\in\mathcal{C}^1 (I,\mathbb{R})$, such that $a$ and $b$ never simultaneously vanish in $\Omega$ and satisfy $$\text{det} \begin{bmatrix} \alpha'(s) & a(\alpha(s),\beta(s)) \\ \beta'(s) & b (\alpha(s),\beta(s)) \end{bmatrix} \neq 0, \ \forall \ s \in I. $$ Then, there is a solution of the Cauchy Problem \begin{align*} \left\{\begin{array}{l} a(x,y)u_x + b(x,y) u_y = c(x,y),\quad &\text{if}\ (x,y) \in \Omega, \\ u(\alpha(s),\beta(s)) = f(s)\ &\text{if} \ s \in I. \end{array}\right. \end{align*} defined in a neighborhood of $\Gamma$.
However, I wasn't able to find a 3-dimensional version of this theorem. All the books in which I looked for such a theorem only presented the 2-dimensional version. I think that a 3-dimensional version of this theorem would look like the following
Possible Theorem: Let $\Omega \subset \mathbb{R}^3$ be an open set and $\Sigma \subset \Omega$ be a $\mathcal{C}^1$ surface. Let $\sigma (u,v) = (\sigma_1(u,v),\sigma_2(u,v),\sigma_3(u,v))$ a $\mathcal{C}^1$ parametrization of $\Sigma$, defined in the square $I\times I \subset \mathbb{R}^2$. Suppose that $a,b,c,d$ $\in$ $\mathcal{C}^1 (\Omega , \mathbb{R})$ and $f\in\mathcal{C}^1 (I,\mathbb{R})$, such that $a$, $b$ and $c$ never simultaneously vanish in $\Omega$ and satisfy $$\text{det} \begin{bmatrix} \frac{\partial \sigma_1}{\partial u}(u,v) & \frac{\partial \sigma_1}{\partial v}(u,v) & a(\sigma(u,v))\\ \frac{\partial \sigma_2}{\partial u}(u,v) & \frac{\partial \sigma_2}{\partial v}(u,v) &b(\sigma(u,v))\\ \frac{\partial \sigma_3}{\partial u}(u,v) & \frac{\partial \sigma_3}{\partial v}(u,v) & c(\sigma(u,v)) \end{bmatrix} \neq 0, \ \forall \ (u,v) \in I\times I. $$ Then, there is a solution of the Cauchy Problem \begin{align*} \left\{\begin{array}{l} a(x,y,z)u_x + b(x,y,z) u_y + c(x,y,z)u_z = d(x,y,z) ,\quad& \text{if}\ (x,y,z) \in \Omega, \\ u(\sigma(u,v) ) = f(u,v)\ &\text{if} \ (u,v) \in I\times I. \\ \end{array}\right. \end{align*} defined in a neighborhood of $\Sigma$.
Does anyone know if this theorem is true and could indicate me a reference?