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Math-2268 · mathematics · Analysis

Hot spots: high-dimensional refutation, product rules, and heat-flow quantifiers

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Let $\Omega$ be a bounded planar simply connected domain, or a bounded convex domain in $\mathbb R^n$ for $n\geq3$. Must every first nonconstant Neumann eigenfunction of its Laplacian attain its global maximum and minimum on the boundary rather than in the interior? This is the stated eigenfunction form of the hot-spots question.

Formulation references: [S1], [S2].

Short English statement

Let $\Omega$ be a bounded planar simply connected domain, or a bounded convex domain in $\mathbb R^n$ for $n\geq3$. Must every first nonconstant Neumann eigenfunction of its Laplacian attain its global maximum and minimum on the boundary rather than in the interior? This is the stated eigenfunction form of the hot-spots question.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2268-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2268

Strict planar hot spots: The theorem gives nonvanishing interior gradient and strict boundary extrema for every first-positive Neumann eigenfunction on smooth bounded simply connected planar domains, including multiplicity. The catalogue also asks convex domains in n>=3 and a heat-flow formulation, so the planar clause alone is partial.

This page records partial support. Complete resolution of the catalogue question is not established by this record.

Hot spots: high-dimensional refutation, product rules, and heat-flow quantifiers

OpenAI's claim

Strict planar hot spots: The manuscript claims that every nonzero eigenfunction at the first positive Neumann eigenvalue on a smooth bounded simply connected planar domain has no interior critical point. This includes multiple first positive eigenvalues and forces extrema to the boundary.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $\Omega\subset\mathbb R^2$ be nonempty, bounded, simply connected with smooth boundary, and let $\mu_1>0$ be the first positive Neumann eigenvalue. For every nonzero real $u$ in its entire eigenspace, the manuscript claims \[\nabla u(x)\ne0\quad(x\in\Omega),\qquad\min_{\partial\Omega}u<u(x)<\max_{\partial\Omega}u.\] The eigenspace may be multiple; the conclusion concerns every nonzero eigenfunction, not merely one chosen eigenfunction.

Outline of the argument

Strict planar hot spots: Conformal parametrization transfers the divergence--curl variational principle for gradient fields to boundary vector data. A positive boundary kernel expresses the energy of scalar-multiplied boundary gradients. Conditional negativity of a reciprocal kernel supplies multipliers that remove the obstructing sign, yielding a contradiction if the gradient vanished at an interior point.

Formal verification

Family 369 describes Lean coverage for gradient nonvanishing in the interior throughout the first positive Neumann eigenspace in the smooth simply connected planar setting.

Strict planar hot spots: The manuscript links HotSpots.lean.

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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