Pompeiu problem [SEP] named after
Pompeiu problem
In mathematics, the Pompeiu problem is a conjecture in integral geometry, named for Dimitrie Pompeiu, who posed the problem in 1929, 
as follows. Suppose "f" is a nonzero continuous function defined on a Euclidean space, and "K" is a simply connected Lipschitz domain, so that the integral of "f" vanishes on every congruent copy of "K". Then the domain is a ball.
A special case is Schiffer's conjecture.
External links.
- Pompeiu problem
From a purely academic point of view free boundaries belong to a larger class of problems usually referred to as overdetermined problems, or as David Kinderlehrer and Guido Stampacchia addressed it in their book: The problem of matching Cauchy data. 
Other related FBP that can be mentioned are Pompeiu problem, Schiffer’s conjectures. See the external links below.
References.
- The Matching Problem in PDE,
- The Pompeiu problem,
- Schiffer's conjecture,