We have the PDE $$a\cdot Du + bu = f, \hspace{2mm} u = g \hspace{2mm} \text{on} \hspace{2mm} S$$
with $a, b, f, g \in C^1$ and $S$ is a $(d-1)$ dimensional $C^1$-hypersurface.
Call $S$ compatible with the equation if there is no such $x \in S$ such that $\nu (x) \cdot a(x) = 0$, where $\nu$ is the normal to $S$.
The notes then say:
we need compatibility as otherwise $a(x_0)$ is tangential to $S$, which implies $a(x_0) \cdot Du(x_0) = a(x_0) \cdot Dg(x_0)$
I am not sure why $a(x_0)$ being tangential to $S$ implies this.
- Surely if $x_0 \in S$ and $u=g$ on $S$, then this is always true by the initial data?
Help with this is appreciated, thank you