Schiffer and Pompeiu conjectures

The Schiffer conjecture asserts that if a bounded connected domain $\Omega\subset\mathbb{R}^n$ with sufficiently smooth boundary admits some $\lambda>0$ and a nonconstant function $u$ satisfying $\Delta u+\lambda u=0$ in $\Omega$ , together with $u=1$ and $\nabla u=0$ on $\partial\Omega$ , then $\Omega$ is a Euclidean ball. The Pompeiu conjecture asserts that a bounded domain $\Omega\subset\mathbb{R}^n$ fails the Pompeiu property if and only if it is a ball: namely, if a continuous function $f$ on $\mathbb{R}^n$ satisfies $\int_{g(\Omega)}f(x)\,dx=0$ for every Euclidean motion $g$ , then $f\equiv 0$ unless $\Omega$ is a ball.

References

Primary source

arXiv

Additional references

Convex counterexamples to the Schiffer and Pompeiu conjectures in dimensions three, four, six, eight, ten and fourteen — arXiv — Jizhou Guo

harmonic-analysis partial-differential-equations spectral-geometry

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed solved

A new unrefereed computer-assisted preprint claims counterexamples even among convex shapes and in several higher dimensions, but no independent confirmation is reported.

The Schiffer and Pompeiu conjectures ask whether the relevant overdetermined Helmholtz conditions force a domain to be a ball. Recent preprints claim counterexamples, first in the plane and now in convex domains in dimensions $3$ , $4$ , $6$ , $8$ , $10$ , and $14$ .

Known results

Williams, 1976: characterized failure of the Pompeiu property through an overdetermined problem involving $\Delta u+\alpha u=1$ .
Ebenfelt, 1993: proved that the disk is the only quadrature domain in which the Pompeiu property fails.
A 2025 note proves the conjecture for a class of centrally symmetric planar domains, including ellipses.

September 2026 convex counterexamples

Jizhou Guo's preprint claims bounded convex non-ball domains with analytic spherical boundaries satisfying the overdetermined Helmholtz condition in dimensions $3$ , $4$ , $6$ , $8$ , $10$ , and $14$ . Earlier 2026 preprints claimed planar counterexamples and infinitely many planar examples. These are computer-assisted or bifurcation-based claims, not independently verified results.

Current status (as of September 2026): planar, convex, and higher-dimensional counterexamples are claimed in preprints, but none is independently verified, so the conjectures remain unresolved as established mathematics.

Sources

No solutions have been posted yet.