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 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.
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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
- No additional permission granted
Scope
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.
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 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.
References and related resources
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