Pompeiu problem

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 ${\displaystyle \Omega }$ with a sufficiently smooth boundary ${\displaystyle \partial \Omega }$ admits a non-trivial solution ${\displaystyle u}$ to the following overdetermined boundary value problem:

${\displaystyle {\begin{cases}\Delta u+\lambda u=0\quad &{\text{ in }}\Omega \\{\frac {\partial u}{\partial n}}=0,\quad &{\text{ on }}\partial \Omega \\u=1,\quad &{\text{ on }}\partial \Omega \end{cases}}}$

for some eigenvalue ${\displaystyle \lambda >0}$ and some constant, then the domain ${\displaystyle \Omega }$ must be a ball. Balls always admit solutions to such overdetermined value problem. On a ball, one can pick ${\displaystyle u}$ to be a radially symmetric Neumann eigenfunction of the Laplacian, which will satisfy the first two equations above. Since moreover is radially symmetric, ${\displaystyle u}$ is constant at ${\displaystyle \partial \Omega }$, so rescaling the Neumann eigenfunction, one can ensure the third equation as well.

• 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

• 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,

This mathematical analysis–related article is a stub. You can help Wikipedia by adding missing information.

• v
• t
• e