Positive Affine Eigenfunctions Above the First Eigenvalue on Nonconvex Pentagons

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)∂qqλ₁ → −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.

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