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Math-2857 · mathematics · Pde

Strict hot spots and absence of critical points for smooth planar Neumann ground modes

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every bounded smooth simply connected planar domain and every nonzero first-positive Neumann Laplacian eigenfunction, including every vector in a multiple first eigenspace, is its gradient nonzero at every interior point and is every interior value strictly between its boundary minimum and maximum?

Short English statement

For every bounded smooth simply connected planar domain and every nonzero first-positive Neumann Laplacian eigenfunction, including every vector in a multiple first eigenspace, is its gradient nonzero at every interior point and is every interior value strictly between its boundary minimum and maximum?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2857-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2857

Strict planar hot spots: The smooth planar domain is bounded and simply connected. Every vector in the first positive Neumann eigenspace is covered, so multiplicity does not create an exception; gradient nonvanishing gives the strict interior separation from the boundary extrema.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

Strict hot spots and absence of critical points for smooth planar Neumann ground modes

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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