Positive Affine Eigenfunctions Above the First Eigenvalue on Nonconvex Pentagons
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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.
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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
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.
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References and related resources
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