Positive Eigenfunctions Above the First Eigenvalue of the Affine 2-Laplacian
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Abstract
For every integer n≥2, we construct a bounded connected nonconvex domain with smooth boundary on which the affine 2-Laplacian has two nonproportional strictly positive first eigenfunctions and a strictly positive eigenfunction above the first eigenvalue. These examples give negative answers at exponent two to both the simplicity and positive-eigenvalue uniqueness questions of Haddad, Jiménez and Montenegro. The construction starts with a planar pentagon pinched into two triangular lobes. A closing odd spectral gap creates negative shear curvature at an exactly isotropic position. A global minimum comparison on the symmetry-preserving linear slice then forces every positive first affine eigenfunction to break a domain symmetry. Parameter-controlled smooth exhaustion and a balanced product lift preserve both properties in every dimension. We also obtain sharp inverse-gap curvature limits and nonparallel-hull isotropic saddles. In contrast, a normalization correspondence and Faifman and Polterovich's convex-position theorem show that on bounded convex domains the positive affine 2-eigenfunction is unique up to scale and belongs to the first affine eigenvalue.
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Research details
- Subjects
- Mathematics · Analysis and Spectral Theory · Differential Equations
- Keywords
- affine Laplacian, positive eigenfunction, Dirichlet eigenvalue, spectral shape optimization, nonconvex polygon, second variation
- Manuscript date
- 2026-10-20
- Reader reuse
- none
Scope
Smooth bounded connected nonconvex examples in every dimension n ≥ 2 with nonproportional positive first affine 2-eigenfunctions and a positive eigenfunction above the first eigenvalue; curvature and saddle refinements contrasted with positive-eigenfunction uniqueness on bounded convex domains.
Disclosures
AI tools assisted research, proof development, writing, and verification. This is an author-published preprint; internal AI scrutiny is not external human refereeing.
Reproduction
The 15-page manuscript and complete editable source are supplied in a 14-member public package. README.txt gives build instructions, dependencies and reproduction scope. Build reproduction is qualified by the compiler, support files and timestamp settings specified in the supplied instructions. The mathematical conclusions have analytic proofs; accompanying finite checks, where present, retain the limited scopes stated in the package.
References and related resources
- 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.
- D. Daners, Dirichlet problems on varying domains , Journal of Differential Equations 188 (2003), 591–624. doi:10.1016/S0022-0396(02)00105-5.
- D. Faifman and I. Polterovich, The Faber–Krahn position of convex bodies and Gaussian measure inequalities, preprint, arXiv:2607.21539v2, 5 August 2026.
- 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.
- 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.
- 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.
- F. Marín Sola, Affine variational functionals, isoperimetric problems and stability , preprint, arXiv:2610.05403v1, 4 October 2026.
- 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.