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Positive Eigenfunctions Above the First Eigenvalue of the Affine 2-Laplacian

Scott Schubr

Luminary Oddities

  • Mathematics
  • Analysis and Spectral Theory
  • Differential Equations
Primary subject
Mathematics
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
3 · Permanent link
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Overview

Abstract

For every integer n ≥ 2, we construct a bounded connected nonconvex domain in Rⁿ with smooth boundary on which the affine 2-Laplacian admits strictly positive weak Dirichlet eigenfunctions at two distinct eigenvalues: its first affine variational value and a value strictly above it. This gives a smooth all-dimensional negative answer at exponent two to the positive-eigenfunction question of Haddad, Jiménez and Montenegro. The construction starts with an affine-isotropic saddle on a planar pentagon pinched into two triangular lobes. A nearly coincident odd mode creates a negative shear variation, while the even sector controls the diagonal normalization. We preserve this saddle by smooth inner exhaustion with second-parameter-derivative control and a symmetry-balanced product construction. The planar examples also give sharp inverse-gap curvature limits, persist under nonparallel-hull perturbations, and exhibit distinct isotropic positions on one linear orbit. These quantitative and geometric refinements are retained alongside the smooth existence theorem.

Keywords affine Laplacian · positive eigenfunction · Dirichlet eigenvalue · spectral shape optimization · nonconvex polygon · second variation

Scope of the work

For every dimension n ≥ 2, a bounded connected smooth nonconvex domain admits positive weak affine 2-Laplacian Dirichlet eigenfunctions at its first variational eigenvalue and at a strictly larger value; planar saddle, curvature and orbit refinements accompany the construction.

Reproduction notes

The 13-page manuscript and complete editable LaTeX sources are supplied in a 13-member package. README.txt gives build instructions for manuscript/main.tex. With Python 3.9 or later, code/verify_constants.py reproduces exact limiting-triangle integrals, curvature coefficients and rational matrix pullbacks using the standard library. The smooth existence, limiting-space and continuation arguments are proved analytically; the small-height and perturbation thresholds are qualitative.

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Author disclosures

AI tools assisted research, proof development, writing, and verification. This is an author-published preprint; internal AI scrutiny is not external human refereeing.

References and related publications

  1. B. Bogosel and D. Bucur, On the polygonal Faber–Krahn inequality , Journal de l’École polytechnique — Mathématiques 11 (2024), 19–105. doi:10.5802/jep.250.
  2. D. Daners, Dirichlet problems on varying domains , Journal of Differential Equations 188 (2003), 591–624. doi:10.1016/S0022-0396(02)00105-5.
  3. D. Faifman and I. Polterovich, The Faber–Krahn position of convex bodies and Gaussian measure inequalities, preprint, arXiv:2607.21539v2, 5 August 2026.
  4. J. Haddad, C. H. Jiménez and M. Montenegro, From affine Poincaré inequalities to affine spectral inequalities, Advances in Mathematics 386 (2021), article 107808. doi:10.1016/j.aim.2021.107808.
  5. Q. Huang and A.-J. Li, Optimal Sobolev norms in the affine class , Journal of Mathematical Analysis and Applications 436 (2016), 568–585. doi:10.1016/j.jmaa.2015.11.063.
  6. D. Langharst and M. Roysdon, Affine isoperimetric inequalities for the first eigenvalue of the mth order Affine p-Laplace Operator, preprint, arXiv:2512.17237v2, 25 September 2026.
  7. F. Marín Sola, Affine variational functionals, isoperimetric problems and stability , preprint, arXiv:2610.05403v1, 4 October 2026.
  8. I. Schindler and C. Tintarev, Compactness properties and ground states for the affine Laplacian , Calculus of Variations and Partial Differential Equations 57 (2018), article 48. doi:10.1007/s00526-018-1338-6.

A stable scholarly record

Cite this edition

Scott Schubr. (2026). Positive Eigenfunctions Above the First Eigenvalue of the Affine 2-Laplacian (Version 3). Luminary Research Archive. https://research.luminaryoddities.com/positive-affine-eigenfunctions-above-the-first-eigenvalue-on-nonconvex-pentagons/v/3

BibTeX
@techreport{luminary_positive_affine_eigenfunctions_above_the_first_eigenvalue_on_nonconvex_pentagons_v3,
  title = {Positive Eigenfunctions Above the First Eigenvalue of the Affine 2-Laplacian},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 3},
  url = {https://research.luminaryoddities.com/positive-affine-eigenfunctions-above-the-first-eigenvalue-on-nonconvex-pentagons/v/3}
}

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