The Fundamental Gap and One-Dimensional Collapse
Zhiqin Lu, Julie Rowlett · Contemporary mathematics - American Mathematical Society · 2014
The fundamental gap of a bounded, connected domain in R n \mathbb {R}^n is the difference between the first two (positive) eigenvalues of the Euclidean Laplacian with Dirichlet boundary condition. The one dimensional collapse considered here is the degeneration of a family of convex, bounded domains in R n \mathbb {R}^n to a domain in R n − 1 \mathbb {R}^{n-1} . The boundary of the domains need not be smooth, merely Lipschitz continuous. Our results show that the fundamental gap detects the geometry of one-dimensional collapse, and that depending upon the geometry of the collapse, the gap can either diverge or remain bounded.