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.

Schiffer's conjecture

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

• Pompeiu, Dimitrie (1929), "Sur certains systèmes d'équations linéaires et sur une propriété intégrale des fonctions de plusieurs variables", Comptes Rendus de l'Académie des Sciences, Série I, 188: 1138–1139
• Ciatti, Paolo (2008), Topics in mathematical analysis, Series on analysis, applications and computation, vol. 3, World Scientific, ISBN 978-981-281-105-9

External links

• Pompeiu problem at Department of Geometry, Bolyai Institute, University of Szeged, Hungary
• Pompeiu problem at SpringerLink encyclopaedia of mathematics
• The Pompeiu problem,
• Schiffer's conjecture,