Math Conjectures

Pompeiu–Schiffer conjecture

OPENMajorConjectureProposed 1929 · Standard version

Canonical statement

Let n≥2n\ge2 and let Ω⊂Rn\Omega\subset\mathbb R^n be a bounded connected domain with connected C∞C^\infty boundary and outward unit normal ν\nu. If there are λ>0\lambda>0 and a nonconstant u∈C∞(Ω‾)u\in C^\infty(\overline\Omega) such that
−Δu=λuin Ω,∂νu=0on ∂Ω,u∣∂Ω=c -\Delta u=\lambda u\quad\text{in }\Omega,\qquad \partial_\nu u=0\quad\text{on }\partial\Omega,\qquad u|_{\partial\Omega}=c
for one constant cc, then Ω\Omega is a Euclidean ball.
View source LaTeX
Let \(n\ge2\) and let \(\Omega\subset\mathbb R^n\) be a bounded connected domain with connected \(C^\infty\) boundary and outward unit normal \(\nu\). If there are \(\lambda>0\) and a nonconstant \(u\in C^\infty(\overline\Omega)\) such that \[ -\Delta u=\lambda u\quad\text{in }\Omega,\qquad \partial_\nu u=0\quad\text{on }\partial\Omega,\qquad u|_{\partial\Omega}=c \] for one constant \(c\), then \(\Omega\) is a Euclidean ball.
ANAL-FA-009:[BrownSchreiberTaylor1973Pompeiu][Williams1976Pompeiu][Zalcman1992PompeiuSurvey][Dai2025SchifferWithdrawn]
Analysis and functional analysisPartial differential equationsGeometry and topology

Notes

The problem goes back to Pompeiu, who asked in 1929 whether a bounded domain other than a ball can fail the Pompeiu property, that is, admit a nonzero continuous function on Rn\mathbb R^n integrating to zero over every rigid motion of the domain. Schiffer later recast the question as an overdetermined eigenvalue problem: if a smooth bounded domain Ω\Omega admits a nonconstant Neumann eigenfunction that is constant on ∂Ω\partial\Omega, must Ω\Omega be a ball? Under the smoothness hypotheses of this record the two formulations merge, failure of the Pompeiu property corresponding to the existence of such an eigenfunction.

Brown, Schreiber and Taylor connected the problem to spectral synthesis and Fourier analysis [BrownSchreiberTaylor1973Pompeiu], Williams obtained a partial solution [Williams1976Pompeiu], and Fourier-analytic and boundary-regularity results now settle many special classes of domains; Zalcman's bibliographic survey collects much of this work [Zalcman1992PompeiuSurvey]. A 2025 claim of a full proof by Dai was withdrawn by its author because its constant-breadth argument was insufficient [Dai2025SchifferWithdrawn].

The conjecture thus remains open: no argument yet covers every smooth domain in the statement.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.