On the Minimization of Dirichlet Eigenvalues of the Laplace Operator

Abstract

We study variational problems of the form inf{λ k (Ω) : Ω open in R m , T (Ω) ≤ 1}, where λ k (Ω) is the k'th eigenvalue of the Dirichlet Laplacian acting in L 2 (Ω), and where T is a non-negative set function defined on the open sets in R m , which is invariant under isometries, additive on disjoint families of open sets, and is such that the ball with T (B) = 1 is a minimiser for k = 1. Upper bounds are obtained for the number of components of any bounded minimiser if T satisfies a scaling relation. For example we show that if T is Lebesgue measure and if k ≤ m + 1 then any bounded minimiser has at most 7 components. We also consider variational problems over open sets Ω in R m involving the (m − 1) -dimensional Hausdorff measure of ∂Ω. Mathematics Subject Classification (2000): 49Q10; 49R50; 35P15.

On torsional rigidity and ground-state energy of compact quantum graphs

“…(2) shows that a bound on this quantity cannot simply be obtained as a straightforward corollary of the classical isoperimetric inequality for the ground-state energy of quantum graphs in [48, Théorème 3.1], in the case of domains the celebrated Kohler-Jobin inequality does state that "balls minimize the ground-state energy among sets with given torsional rigidity". Conjectured in [54], it was proved in [35] and extended in [7,13] to rougher domains and nonlinear operators.…”

Existence of minimizers for spectral problems

“…Instead, the existence of a minimizer for λ 3 has been proved only in 2000 by Bucur and Henrot (see [5]), but it is still presently not known which set is the minimizer (and this is a major open problem). For other open problems and partial results see [7,2].…”

Maximal spectral surfaces of revolution converge to a catenoid

“…For example, Bucur et al [1] showed that there is a domain that minimizes the second eigenvalue among bounded planar domains with the same perimeter. This was extended to higher eigenvalues by van den Berg & Iversen [2]. Bucur & Freitas [3] showed that these domains converge to a disc.…”