OP-00704 Solution claimed

Hot spots on convex planar domains

Let $\Omega \subset \mathbb{R}^2$ be a bounded convex domain with $C^\infty$ boundary, let $\nu$ be its outward unit normal, and let $\mu_1 > 0$ be the first nonzero Neumann eigenvalue of $-\Delta$ on $\Omega$. The question below is the eigenfunction form of Rauch's hot spots conjecture, for planar convex domains. Jerison and Nadirashvili proved that the eigenfunctions of the first nonzero Neumann eigenvalue attain their extrema only on the boundary for convex planar domains with two axes of symmetry (2000), and Judge and Mondal that on a triangle they have no interior critical points (2020, with an erratum in 2022). The conjecture fails for some planar domains with holes (Burdzy and Werner, 1999), and de Dios Pont constructed smooth convex counterexamples in every sufficiently large dimension (2024).

Problem. For every nonzero real solution $u$ of $$-\Delta u = \mu_1 u \ \text{ in } \Omega, \qquad \partial_\nu u = 0 \ \text{ on } \partial\Omega,$$ do all global maxima and all global minima of $u$ on $\overline{\Omega}$ lie on $\partial\Omega$? When $\mu_1$ is multiple, this is asked of every function in its eigenspace.

Source: Rauch (1974, published 1975); for convex domains, Kawohl (1985). Stated by AIM — Open Applied Problems for planar convex domains with smooth boundary.

Imported from github.com/MColbrook/AIM.

Classification

Primary: SP — Spectral Theory
Secondary: AP
Keywords: AIM Open Applied Problems, convex domains, hot spots conjecture, Neumann eigenfunctions

References

@incollection{rauch1975,
  author    = {Rauch, Jeffrey},
  title     = {Five problems: an introduction to the qualitative theory of partial differential equations},
  booktitle = {Partial Differential Equations and Related Topics (Tulane Univ., New Orleans, 1974)},
  editor    = {Goldstein, Jerome A.},
  series    = {Lecture Notes in Math.},
  volume    = {446},
  publisher = {Springer},
  address   = {Berlin},
  year      = {1975},
  pages     = {355--369},
  doi       = {10.1007/BFb0070610},
}
@book{kawohl1985,
  author    = {Kawohl, Bernhard},
  title     = {Rearrangements and Convexity of Level Sets in {PDE}},
  series    = {Lecture Notes in Math.},
  volume    = {1150},
  publisher = {Springer},
  address   = {Berlin},
  year      = {1985},
  doi       = {10.1007/BFb0075060},
}
@article{jerison2000,
  author  = {Jerison, David and Nadirashvili, Nikolai},
  title   = {The ``hot spots'' conjecture for domains with two axes of symmetry},
  journal = {J. Amer. Math. Soc.},
  volume  = {13},
  number  = {4},
  year    = {2000},
  pages   = {741--772},
  doi     = {10.1090/S0894-0347-00-00346-5},
}
@article{judge2020,
  author        = {Judge, Chris and Mondal, Sugata},
  title         = {Euclidean triangles have no hot spots},
  journal       = {Ann. of Math. (2)},
  volume        = {191},
  number        = {1},
  year          = {2020},
  pages         = {167--211},
  doi           = {10.4007/annals.2020.191.1.3},
  eprint        = {1802.01800},
  archivePrefix = {arXiv},
}
@article{judge2022,
  author  = {Judge, Chris and Mondal, Sugata},
  title   = {Erratum: {E}uclidean triangles have no hot spots},
  journal = {Ann. of Math. (2)},
  volume  = {195},
  number  = {1},
  year    = {2022},
  pages   = {337--362},
  doi     = {10.4007/annals.2022.195.1.5},
}
@article{burdzy1999,
  author        = {Burdzy, Krzysztof and Werner, Wendelin},
  title         = {A counterexample to the ``hot spots'' conjecture},
  journal       = {Ann. of Math. (2)},
  volume        = {149},
  number        = {1},
  year          = {1999},
  pages         = {309--317},
  doi           = {10.2307/121027},
  eprint        = {math/9803030},
  archivePrefix = {arXiv},
}
@misc{dediospont2024,
  author        = {de Dios Pont, Jaume},
  title         = {Convex sets can have interior hot spots},
  year          = {2024},
  eprint        = {2412.06344},
  archivePrefix = {arXiv},
}
@misc{aim-planar-convex-hot-spots,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{Hot spots on convex planar domains}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/003-planar-convex-hot-spots.md},
  note         = {Entry 003 at commit 1b7c7fd, 2026-10-13},
}

Resolution claims

Strict hot spots and absence of interior critical points on smooth simply connected planar domains v1 GitHub ↗

OpenAI

We prove the strict hot spots conjecture for smooth bounded simply connected planar domains. More precisely, every nonzero eigenfunction for the first positive Neumann eigenvalue has nonvanishing gradient in the interior, so all its global maxima and minima lie on the boundary. This holds even when the eigenvalue is multiple.

Full resolution · not verified · fidelity not established · posted on GitHub 6 Oct 2026

Status history

When	Change	By	Reason
2026-10-16	Needs verification → Solution claimed	—	Recorded here: math @ 1581d4c, build/main.tex.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00704, "Hot spots on convex planar domains", clopen.solutions.