Counterexamples to the Schiffer and Pompeiu conjectures in dimensions four and six

Paper and computer-assisted verification (Arb interval arithmetic) of bounded convex domains in R^4 and R^6, not balls, with real-analytic boundaries diffeomorphic to spheres, carrying nonconstant Neumann eigenfunctions of the Laplacian that are constant on the boundary. This gives counterexamples to the Schiffer conjecture and to the Pompeiu conjecture in dimensions four and six. This work was carried out with the assistance of AI models (Claude Opus 5.5, GPT-6 Astra and GPT-6 Sol).

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-28
DOI
https://doi.org/10.5281/zenodo.23006742
Primary Topic
Analytic and geometric function theory
Type
preprint
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preprint

Counterexamples to the Schiffer and Pompeiu conjectures in dimensions four and six

Jizhou Guo
Zenodo (CERN European Organization for Nuclear Research)
Analytic and geometric function theory
preprint

Counterexamples to the Schiffer and Pompeiu conjectures in dimensions four and six

Jizhou Guo
preprint en

Abstract

Paper and computer-assisted verification (Arb interval arithmetic) of bounded convex domains in R^4 and R^6, not balls, with real-analytic boundaries diffeomorphic to spheres, carrying nonconstant Neumann eigenfunctions of the Laplacian that are constant on the boundary. This gives counterexamples to the Schiffer conjecture and to the Pompeiu conjecture in dimensions four and six. This work was carried out with the assistance of AI models (Claude Opus 5.5, GPT-6 Astra and GPT-6 Sol).

Zenodo (CERN European Organization for Nuclear Research)
Analytic and geometric function theory
DOIOpen Access PDFOpenAlex
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Counterexamples to the Schiffer and Pompeiu conjectures in dimensions four and six — Jizhou Guo · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS