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PreprintCC BY 4.0arXiv

A starlike 3-dimensional counterexample to the hot spots conjecture

  • Rami Atar
  • Krzysztof Burdzy
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Abstract

We construct a bounded Lipschitz domain in ℝ3 that is starlike about the origin and whose first positive Neumann eigenvalue is simple. A corresponding eigenfunction, after a choice of sign, has its unique global maximum at the origin. The construction uses domains with increasingly fine angular structure and convergence to a weighted eigenvalue problem whose first positive eigenfunction is radial.

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Cite

BibTeXRIS
BibTeX
@misc{atar2026starlike,
  title = {{A starlike 3-dimensional counterexample to the hot spots conjecture}},
  author = {Rami Atar and Krzysztof Burdzy},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.19908},
  eprint = {2610.19908},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.19908},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A starlike 3-dimensional counterexample to the hot spots conjecture
AU  - Rami Atar
AU  - Krzysztof Burdzy
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.19908
UR  - https://arxiv.org/abs/2610.19908
AB  - We construct a bounded Lipschitz domain in ℝ^3 that is starlike about the origin and whose first positive Neumann eigenvalue is simple. A corresponding eigenfunction, after a choice of sign, has its unique global maximum at the origin. The construction uses domains with increasingly fine angular structure and convergence to a weighted eigenvalue problem whose first positive eigenfunction is radial.
ER  -