The motivation is that I am trying to understand the equation $$\tag{1}\label{eq:1}u_{nn} - \Delta u = -(\kappa u_n + u_{tt})$$ on the boundary $\partial\Omega$ of some bounded open set $\Omega\subset\mathbb R^2$ ($\kappa$ is the signed curvature of the boundary). Basically, if $\partial \Omega$ is piecewise straight, then $\kappa=0$ and the tangential $t$ and normal $n$ simply form a local coordinate system by rotating the standard coordinates and hence \eqref{eq:1} is clear since the Laplacian $\Delta u := D^2u(x_1,x_1) + D^2u(x_2,x_2) = D^2u(t,t) + D^2u(n,n)$ is invariant under such rotation.
In the case of arbitrary, say piecewise $C^2$, boundary I am a bit confused. What does $u_{tt}$ and $u_{nn}$ even stand for, it cannot simply be $D^2u(t,t)$ and $D^2u(n,n)$ since then there would be no curvature term in \eqref{eq:1}.
My intuition for $u_{tt}$: The tangential derivative $u_t=\frac{\partial}{\partial t} u=\nabla u\cdot t$ can be written as $\frac{d}{ds}u(\gamma(s))$ for some arc-length parametrisation $\gamma$ of the boundary. Hence $u_{tt} = \frac{d^2}{ds^2}u(\gamma(s)) = D^2u(\gamma'(s), \gamma'(s)) + \nabla u\cdot \gamma''(s)= D^2u(t(s), t(s)) - \kappa u_n$ since $\gamma'(s)= t(s), \gamma''(s) = -\kappa n(s)$.
For $u_{nn}$: The normal derivative $\frac{\partial}{\partial n} u_n$ as a directional derivative is the limit when approaching the boundary normally, but $u_n$ is only defined on the boundary. So does this mean that for any extension $n^*$ of the normal field $n$, that$\frac{\partial}{\partial n} u_n = \frac{\partial}{\partial n} u_{n^*}$ is equal? And how to see this? Is there a better way of interpreting $u_{nn}$ in the first place?
I guess I am making this way to complicated - thanks for your help!
I don't think it is wise to think of these derivatives on the boundary in terms of derivatives w.r.t. the Euclidean coordinates. A more natural way is to define suitable normal coordinates, so that a neighborhood of a boundary point looks like a neighborhood $U$ of the origin in the half-plane ${\Bbb R}_- \times {\Bbb R}$. We write $(x_\alpha)_{\alpha=1,2}$ for the coordinates in $\Omega\supset \Omega$.
Let $\gamma(s)$, $s\in ]-\delta,\delta[$ be a piece of the arc-length parametrized boundary with $p=\gamma(0) \in \partial \Omega$. Let $t(s)=\gamma'(s)$ be the corresponding unit tangent vector and $n(s)$ an outward unit normal at $\gamma(s)$. We then have $t'(s)=-k n(s)$ and consequently $n'(s)=k t(s)$.
An explicit way to construct normal coordinates is now to pose $\xi(r,s):=\gamma(s) + r n(s)$ in a neighborhood of the origin $(r,s)=(0,0)$. The map verifies for $r=0$ (Fact 2): $$ \partial_s \xi_{|r=0} = t(s), \ \ \partial_r \xi_{|r=0} = n(s), \ \ \partial^2_s \xi_{|r=0} = -k n(s), \ \ \partial^2_r \xi_{|r=0} = 0. $$ In particular, we see that the jacobian matrix at a boundary point is the orthogonal matrix $J=J_{|r=0}=\begin{pmatrix} n(s) & t(s) \end{pmatrix} $. Note that from $JJ^T=I$ we get (Fact 1) for $\alpha,\beta=1,2$: $n_\alpha n_\beta+t_\alpha t_\beta = \delta_{\alpha,\beta}$ (Kronecker-delta). As $J$ is invertible, $\xi$ defines a local diffeomorphism $$ \xi : U \cap ({\Bbb R}_- \times {\Bbb R}) \to V \cap \Omega$$ from a neighborhood $U$ of $(0,0)$ to a neighborhood $V$ of $p\in \Omega$.
Under this coordinate transformation we get the following derivatives (using the Einstein convention of summing over repeated indices and natural abbreviations): $$ \partial_r = \frac{\partial}{\partial r} = \frac{\partial x_\alpha}{\partial r} \frac{\partial }{\partial x_\alpha} = n_\alpha(s) \partial_\alpha $$
$$ \partial_s = \frac{\partial}{\partial s} = \frac{\partial x_\alpha}{\partial s} \frac{\partial }{\partial x_\alpha} = (1+kr) t_\alpha(s) \partial_\alpha $$
$$ \partial_r^2 = n_\alpha n \partial_\beta \partial_\alpha \partial_\beta $$
$$ \partial_s^2 = (1+kr) (-k n_\alpha) \partial_\alpha + (1+kr)^2 t_\alpha t_\beta \partial_\alpha \partial_\beta $$
Finally, restricting to the boundary, i.e. $r=0$ and using Fact 1 for the equality in the middle: $$ \left(\partial_r^2 + \partial_s^2 \right)_{|r=0} = -k n_\alpha \partial_\alpha + ( n_\alpha n_\beta + t_\alpha t_\beta) \partial_\alpha \partial_\beta = -k \partial_r + \partial_\alpha \partial_\alpha = -k \partial_r + \Delta , $$ as we wanted. A priori the above is defined only in the interior of $U$ (or $\Omega$). In the $(r,s)$ coordinates, however, $C^2$-smoothness with suitable boundary behavior is easy to define (one advantage of the half-plane) and on the boundary you may identify the $r,s$ derivatives with the $n,t$ derivatives which you mention. Regarding generality, any smooth map $\xi$ verifying Fact 2 will do.