Inverse Problem for Elliptic PDE on Compact Domain 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.
 A: Note that in general $f$ is a function of $n$ variables whereas $u|_{\partial\Omega}$, $\frac{\partial u}{\partial \nu}|_{\partial\Omega}$ are functions of $n-1$ variables and formally the problem of finding $f$ from $u|_{\partial\Omega}$, $\frac{\partial u}{\partial \nu}|_{\partial\Omega}$ is underdetermined and one can not expect to solve it uniquely.
A general empiric rule is the that in order to be able to determine the unknown coefficient it must depend on the same number of (real) variables as the input data. In particular, in the article that you cite the unknown coefficient depends on one variable only and one can expect that the inverse problem admits a unique solution.
If you are allowed to impose different boundary values $u|_{\partial\Omega} =u_0$ and measure $\Lambda u_0 = \frac{\partial u}{\partial \nu}|_{\partial\Omega}$ then this operator $\Lambda$ (known as Dirichlet-to-Neumann or Poincaré-Steklov map) determines $f$ uniquely, see R.G. Novikov (Funct. An. Appl. 22, 1988, §3, Corollary 2).  Note that the integral (Schwartz) kernel of $\Lambda$  depends on $2(n-1)$ real arguments and $2(n-1)\geq n$ for $n\geq 2$, so that this problem is non underdetermined for $n \geq 2$.  
