Consider the elliptic PDE $\Delta u + f u = 0$ on some compact domain $\Omega \subset \mathbb{R}^n$; here $f$ is some function on $\Omega$ (and $\Delta$ is the Laplacian). My question is the following: is it possible to reconstruct $f$ everywhere on $\Omega$ just from properties of $u$ at the boundary $\partial \Omega$? If so, what such properties are sufficient to recover $f$? For instance, certainly I'd expect I'd need to know $u|_{\partial \Omega}$ and perhaps some normal derivative $\partial_n u|_{\partial \Omega}$. Is any other information sufficient?
In the special case of $n = 2$ and where $\Omega$ is the quarter-plane $x> 0$, $y>0$, this paper claims that knowledge of $u(x,0)$, $u(0,y)$, and $\partial_x u(0,y)$ is sufficient to recover $f = f(x)$. I'm interested in whether this result generalizes to arbitrary $\Omega$ in arbitrary dimensions and for $f$ a function of all coordinates.