[#P3128] Hot spots conjecture for convex planar domains
Problem. Let \(\Omega\subset\mathbb R^2\) be a bounded convex domain, and let \(u\) be a nonconstant first Neumann eigenfunction satisfying \(-\Delta u=\lambda_1u\) in \(\Omega\) and \(\partial_nu=0\) on \(\partial\Omega\). Must every global maximum and minimum of \(u\) occur on \(\partial\Omega\)?
1Context
Known frontier: Many planar classes are proved; convex counterexamples now exist in sufficiently high dimension. Open boundary: No convex planar counterexample or general proof was located.
2Problem setup
Definition 1 (first Neumann eigenfunction). An eigenfunction for the smallest positive Neumann Laplacian eigenvalue λ₁.
Definition 2 (hot spot). A point attaining the global maximum or minimum of u.
Remark 1. The first nonzero Neumann mode models the slowest-decaying temperature imbalance in an insulated plate. The original broad conjecture fails in some nonconvex domains and, recently, for convex sets in high dimension, so the packet is restricted to the planar convex case.
3What counts as a solution
- Prove boundary attainment for every bounded convex planar domain, with a stated boundary regularity convention.
- Or give a convex planar domain whose first Neumann eigenfunction has an interior global extremum.
1Status
Current status (Current status and exact unresolved remainder). OPEN as checked on 2026-08-01. Strongest checked neighboring result: Many planar classes are proved; convex counterexamples now exist in sufficiently high dimension. Exact unresolved remainder: No convex planar counterexample or general proof was located.[1][2]
1Records
Notes and companion material
Original intake status. OPEN as checked on 2026-08-01. Strongest checked neighboring result: Many planar classes are proved; convex counterexamples now exist in sufficiently high dimension. Exact unresolved remainder: No convex planar counterexample or general proof was located.
- Equivalent-formulation queries: convex planar hot spots conjecture open 2026; first Neumann eigenfunction convex domain interior maximum dimension two
- Strongest checked neighboring result: Many planar classes are proved; convex counterexamples now exist in sufficiently high dimension.
- Exact unresolved remainder: No convex planar counterexample or general proof was located.
How the 4 records connect
ProblemHot spots conjecture for convex planar domains
2See also
How to cite
TheoremDB contributors, “Hot spots conjecture for convex planar domains,” TheoremDB research memory, snapshot of August 1, 2026. https://theoremdb.org/statements/planar-convex-hot-spotsThis page as plain text: planar-convex-hot-spots.md
This problem includes 4 records joined by 3 typed links, sourced from arxiv.org[1], current as of August 1, 2026.
1References
- Packet source. Lawford Hatcher, “The hot spots conjecture for some non-convex polygons”. arXiv:2405.19508 (2024). introduction and main theorems. ↗preprint · primary source · arXiv:2405.19508, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Proves the hot-spots conclusion for new classes of planar nonconvex polygons and records the surrounding planar problem.Also cited at Lawford Hatcher, “The hot spots conjecture for some non-convex polygons,” arXiv:2405.19508v3. introduction and main theorems.Source used to assess the problem's recorded status.For Hot spots conjecture for convex planar domains: This is the dated publication status for the canonical target Hot spots conjecture for convex planar domains.Source named by the research packet.
- Jaume de Dios Pont, “Convex sets can have interior hot spots”. arXiv:2412.06344 (2024). main theorem. ↗preprint · primary source · arXiv:2412.06344, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Disproves the dimension-free convex version and makes the planar restriction essential.Source used to assess the problem's recorded status.For Hot spots conjecture for convex planar domains: Disproves the dimension-free convex version and makes the planar restriction essential.
Original TheoremDB editorial statement and source synthesis; external works are used for citation only.