Range of the first two eigenvalues of the laplacian

Abstract

For each planar domain D of unit area, the first two Dirichlet eigenvalues of —∆ on D determine a point (λ 1 ( D ), λ 2 ( D ) in the (λ 1 , λ 2 ) plane. As D varies over all such domains, this point varies over a set R which we determine. Its boundary consists of two semi-infinite straight lines and a curve connecting their endpoints. This curve is found numerially. We also show how to minimize the n th eigenvalue when the minimizing domain is diconnected. For n = 3 we show that the minimizing domain is connected and that λ 3 is a local minimum for D a circular disc.

On the range of the first two Dirichlet and Neumann eigenvalues of the Laplacian

“…Moreover, the tangents at the extremal points are vertical (at the ball) and horizontal (at two balls) (cf. Wolf & Keller 1994). The above results show that the only unknown part of the set E D is the lower part, the curve joining the points corresponding to one ball and two balls.…”

“…The above results show that the only unknown part of the set E D is the lower part, the curve joining the points corresponding to one ball and two balls. In figure 1, we have determined numerically this curve with the same procedure as in Wolf & Keller (1994), solving a minimization problem with a convex combination of l 1 and l 2 . Our results were obtained using the gradient method to solve the minimization problems, as in Alves & Antunes (in preparation).…”

“…. .. Wolf & Keller (1994) and Bucur et al (1999) studied the region E D = {(x, y) ∈ R 2 : (x, y) = (l 1 (U), l 2 (U)), U ⊂ R 2 , |U| = 1}, which is the range of the first two Dirichlet eigenvalues of planar sets with unit area. We also refer to Levitin & Yagudin (2003) for a similar study for the first three eigenvalues.…”

Minimizing the Second Eigenvalue of the Laplace Operator with Dirichlet Boundary Conditions

“…For other recent results on minimization problems for functions of eigenvalues, we refer e.g. to [5], [32] and the review papers [2], [4], [17]. Finally, we point out that the results in section 2 are mostly valid in any dimension while the results of the section 3 are more specifically two-dimensional.…”

Rearrangement inequalities and applications to isoperimetric problems for eigenvalues

“…For instance, the minimum of λ 2 (Ω) (the second eigenvalue of the Laplace operator in Ω under Dirichlet boundary condition) among bounded open sets of R n with given Lebesgue measure is achieved by the union of two identical balls (this result is attributed to Szegö; see [39]). Very few things seem to be known about optimization problems for the other eigenvalues; see [17], [25], [39], [40], and [49]. Various optimization results are also known for functions of the eigenvalues.…”