Schiffer's Conjecture and the Pompeiu Problem
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Statement
If a smooth bounded domain in admits a Neumann eigenfunction of the Laplacian that is constant on the boundary, must the domain be a ball? Pompeiu posed an equivalent integral-equation form in 1929; Schiffer's 1957 reformulation via Neumann eigenfunctions is the version on Yau's 1982 list (Problem 80), and Williams proved the two formulations logically equivalent for simply connected domains in 1976. Cao-Labora and de Dios Pont construct infinitely many planar domains with large -fold symmetry that are not balls and admit such an eigenfunction, disproving Schiffer's conjecture; applying Williams' classical reduction to the same domains (their Corollary 1.2) disproves Pompeiu's problem as well.
Context
Also refutes the 1929 Pompeiu problem: Corollary 1.2 applies Williams' classical 1976 equivalence to the same constructed domains, so this is one construction settling both, not two separate results.
A flagship problem of spectral geometry for seven decades: Problem 80 on Yau's list, equivalent to the 1929 Pompeiu problem, with a partial-results literature running since the 1970s.
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Models with coding harnesses were used in multiple parts of the research: numerically verifying the asymptotic estimates, producing first drafts of the proofs of the Bessel function estimates, and helping with exposition.
The Lean4 verification of the proof was written by GPT 5.6 from an early draft of the paper. The novel construction strategy is the authors' own.
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Machine check · not human verification
machine: correctscope Lean formalization of the result
A day-old preprint. The paper states that a Lean4 verification of the proof was written by GPT 5.6, available at https://github.com/jaumededios/Schiffer. It solves the Pompeiu Problem challenge provided by https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/PompeiuProblem.lean
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