Instant TravelLuminary Oddities — Enter the Portal
Research Archive

Research Article · Preprint Mathematics

Positive Affine Eigenfunctions Above the First Eigenvalue on Nonconvex Pentagons

Scott Schubr

Luminary Oddities LLC

  • Mathematics
  • Analysis and Spectral Theory
  • Differential Equations
Primary subject
Mathematics
Subject subfields
Analysis and Spectral Theory · Differential Equations
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
2 · Permanent link
Read / download PDF Publication filesCite this edition

Overview

Abstract

We construct bounded connected nonconvex planar pentagons for which the affine 2-Laplacian has a strictly positive Dirichlet eigenfunction at an eigenvalue strictly above its first variational eigenvalue. The same domain therefore supports positive affine eigenfunctions at two distinct values. The construction gives a negative answer, for n = p = 2, to the general-domain positive-eigenfunction question of Haddad, Jiménez and Montenegro. The mechanism is an affine-isotropic saddle formed by pinching a pentagon into two right isosceles triangles. A symmetry-preserving diagonal normalization has positive curvature, while a shear couples the ground state to a nearly coincident odd mode. We prove the sharp limit (λ₂ − λ₁)∂²λ₁/∂q² → −32768/81 at the isotropic positions. An area-preserving apex slide has the corresponding limit −50π⁴. The saddles persist under small vertex perturbations, including examples with nonparallel convex-hull sides. Consequences include distinct isotropic positions on one linear orbit and failure of positive-definite-deformation log-convexity outside convexity. We further preserve the saddle under smooth inner exhaustion and use a symmetry-balanced product construction to obtain positive affine 2-eigenfunctions at two distinct eigenvalues on smooth connected nonconvex domains in every dimension n ≥ 2.

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

Scope of the work

Existence at exponent p = 2 on nonconvex planar pentagons and smooth connected nonconvex domains in every dimension n ≥ 2. Small-height, perturbation and smoothing thresholds are qualitative. The larger affine eigenvalue is not assigned a nonlinear minmax index; no convex-domain or p ≠ 2 conclusion is asserted.

Reproduction notes

The thirteen-member package contains the complete 13-page manuscript, all eight LaTeX source files, an optional exact checker, reader instructions, citation metadata and checksums. Build manuscript/main.tex with Tectonic or two pdflatex runs, preserving its input files. With Python 3.9 or later, run python -B code/verify_constants.py in normal and optimized modes. The checker verifies exact auxiliary constants and matrix pullbacks; existence follows the analytic proof, not a numerical eigenpair solver.

The manuscript

Read the paper

Open PDF

The complete manuscript, as published in this edition.

If the reader is unavailable on your device, open or download the PDF directly.

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 Affine Eigenfunctions Above the First Eigenvalue on Nonconvex Pentagons (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/positive-affine-eigenfunctions-above-the-first-eigenvalue-on-nonconvex-pentagons/v/2

BibTeX
@techreport{luminary_positive_affine_eigenfunctions_above_the_first_eigenvalue_on_nonconvex_pentagons_v2,
  title = {Positive Affine Eigenfunctions Above the First Eigenvalue on Nonconvex Pentagons},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 2},
  url = {https://research.luminaryoddities.com/positive-affine-eigenfunctions-above-the-first-eigenvalue-on-nonconvex-pentagons/v/2}
}

Loading discussion…