A closely related and highly significant formulation is Schiffer's conjecture, named after the mathematician
Menahem Max Schiffer. While the Pompeiu problem is rooted in integral geometry, Schiffer's conjecture is framed in the language of
partial differential equations. Schiffer's conjecture proposes that if a bounded,
simply connected domain \Omega with a sufficiently smooth boundary \partial \Omega admits a non-trivial solution u to the following overdetermined
boundary value problem: \begin{cases}\Delta u + \lambda u = 0 \quad &\text{ in }\Omega \\ \frac{\partial u}{\partial n} = 0, \quad &\text{ on } \partial \Omega \\ u = 1, \quad &\text{ on } \partial \Omega \end{cases} for some eigenvalue \lambda > 0 and some constant, then the domain \Omega must be a ball. Balls always admit solutions to such overdetermined value problem. On a ball, one can pick u to be a radially symmetric Neumann eigenfunction of the Laplacian, which will satisfy the first two equations above. Since moreover is radially symmetric, u is constant at \partial \Omega, so rescaling the Neumann eigenfunction, one can ensure the third equation as well.
Equivalence to the Pompeiu problem The deep connection between the two problems was brought to light by Williams (1976). By utilizing the
Fourier transform and techniques from
complex analysis, it was proven that Schiffer's conjecture is mathematically equivalent to the Pompeiu problem for the case of \Omega \subset \mathbb{R}^n smooth and contractible. ==References==