Precise bounds and asymptotics for the first Dirichlet eigenvalue of triangles and rhombi

Abstract

We study the asymptotic expansion of the first Dirichlet eigenvalue of certain families of triangles and of rhombi as a singular limit is approached. In certain cases, which include isosceles and right triangles, we obtain the exact value of all the coefficients of the unbounded terms in the asymptotic expansion as the angle opening approaches zero, plus the constant term and estimates on the remainder. For rhombi and other triangle families such as isosceles triangles where now the angle opening approaches π , we have the first two terms plus bounds on the remainder. These results are based on new upper and lower bounds for these domains whose asymptotic expansions coincide up to the orders mentioned. Apart from being accurate near the singular limits considered, our lower bounds for the rhombus improve upon the bound by Hooker and Protter for angles up to approximately 22 • and in the range (31 • , 54 • ). These results also show that the asymptotic expansion around the degenerate case of the isosceles triangle with vanishing angle opening depends on the path used to approach it.

Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions

“…The local character of (1.3) rather resembles the problem of a straight narrow strip of variable width studied recently by Friedlander and Solomyak [13,14], and also by Borisov and Freitas [1] -see also [9] for related work. In view of their asymptotics, the spectrum of the Dirichlet Laplacian is basically determined by the points where the strip is the widest.…”

Dirichlet eigenvalues of cones in the small aperture limit

“…1 We can apply our result to obtain asymptotics for geometrical domains close enough to cones, as spherical sectors, for instance. It yields results in the spirit of [17] in a higher dimension: it is the three dimensional equivalent of the circular sector, the Bessel functions playing a similar role as trigonometric functions.…”

“…Nevertheless, even for simple two dimensional domains like triangles this question is still complicated. This specific question is detailed in [17] where a finite term asymptotics is provided in the regime θ goes to 0 (where θ is the aperture of the triangle). More recently [11] gives a complete asymptotics for right-angled triangles.…”

Bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals

“…The lower bound for quadrilaterals is a straightforward consequence of Steiner symmetrization and Hooker and Protter's lower bound for rhombi [7], but we could not find it in the literature. In a similar fashion, it is possible to combine other lower bounds for the first eigenvalue of rhombi with Steiner symmetrization, and we do this in Theorem 4.2 below where we use a bound from [5].…”

“…In the next section we present the known bounds for triangles, together with improvements based on the combination of techniques from [5,14]. Section 3 contains a similar analysis, but now for rhombi.…”