The Fundamental Gap Conjecture on Polygonal Domains
Zhiqin Lu, Julie Rowlett · arXiv (Cornell University) · 2008
Abstract. In 1985, S. T. Yau made the following “fundamental gap conjecture,” [25]. For a convex domain Ω ⊂ R n, (0.1) ξ(Ω): = d 2 (λ2(Ω) − λ1(Ω)) ≥ 3π 2 where d is the diameter of the domain, and 0 < λ1(Ω) < λ2(Ω) are the first two eigenvalues of the Euclidean Laplacian on Ω with Dirichlet boundary condition. The scalar invariant ξ is the gap function. We restrict attention to planar domains. Our main result is a compactness theorem for the gap function when the domain is a triangle in R2. This result shows that for any triangles which collapse to the unit interval, the gap function is unbounded. Due to numerical methods, we expect that the fundamental gap conjecture holds for all triangular domains in R2. We show with examples that the behavior of the gap for collapsing polygonal domains is quite delicate. These examples motivate a technical result for collapsing polygonal domains giving conditions under which the gap function either remains bounded or becomes infinite. Our work initiates a general program to prove the fundamental gap conjecture using convex polygonal domains. 1. Motivation and