# P3128: Hot spots conjecture for convex planar domains

- ID: `P3128`
- Reference: `planar-convex-hot-spots`
- Page: https://theoremdb.org/statements/P3128
- Record maturity: Reviewed problem with recorded work

## 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 \(\partial\Omega\)?

### Context

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.

### Problem setup

- **Definition (first Neumann eigenfunction).** An eigenfunction for the smallest positive Neumann Laplacian eigenvalue λ₁.
- **Definition (hot spot).** A point attaining the global maximum or minimum of u.
- **Remark.** 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.

### What 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.

## 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. [1](#reference-1) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (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.

The problem was checked as open on 2026-08-01.

The strongest neighboring result found in the cited sources is: Many planar classes are proved; convex counterexamples now exist in sufficiently high dimension.

The exact unresolved remainder is: No convex planar counterexample or general proof was located.

A complete resolution must meet the following acceptance conditions:
- 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.

### Background and intake notes

- 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.
- The release review checked 2 structured sources on 2026-08-01.
- 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.

### Other known results

- **Claim 2** (supported): Many planar classes are proved; convex counterexamples now exist in sufficiently high dimension. [1](#reference-1) [2](#reference-2)

### Prior approaches

- **Route 1** (supported): The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: Many planar classes are proved; convex counterexamples now exist in sufficiently high dimension. Unresolved remainder: No convex planar counterexample or general proof was located. [1](#reference-1) [2](#reference-2)

### Open directions

- **Route 2** (reported): No convex planar counterexample or general proof was located.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `planar-convex-hot-spots`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Lawford Hatcher, “The hot spots conjecture for some non-convex polygons”. arXiv:2405.19508 (2024). introduction and main theorems https://arxiv.org/abs/2405.19508
   - Also cited at Lawford Hatcher, “The hot spots conjecture for some non-convex polygons,” arXiv:2405.19508v3. introduction and main theorems
   - preprint; primary source; arXiv:2405.19508, checked 2026-08-01; checked 2026-08-01
   - Source use: original_summary
   - Proves the hot-spots conclusion for new classes of planar nonconvex polygons and records the surrounding planar problem.
   - 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.
2. <a id="reference-2"></a>Jaume de Dios Pont, “Convex sets can have interior hot spots”. arXiv:2412.06344 (2024). main theorem https://arxiv.org/abs/2412.06344
   - preprint; primary source; arXiv:2412.06344, checked 2026-08-01; checked 2026-08-01
   - Source 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.