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

Abstract

Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditionsAntoine HENROT, Edouard OUDET AbstractIn this paper, we are interested in the minimization of the second eigenvalue of the Laplacian with Dirichlet boundary conditions amongst convex plane domains with given area. The natural candidate to be the optimum was the "stadium", convex hull of two identical tangent disks. We refute this conjecture. Nevertheless, we prove existence of a minimzer. We also study some qualitative properties of the minimizer (regularity, geometric properties).

Numerical minimization of eigenmodes of a membrane with respect to the domain

“…the convex hull of two tangent identical disks, looks very close to the optimal set. In fact, as reported in [11] and [12], the stadium does not minimizes λ 2 under convexity and volume constraints. The aim of this work is to present a new technique enabling to approximate numerically the solutions of such optimizations problems.…”

Dirichlet eigenvalue sums on triangles are minimal for equilaterals

“…For the second eigenvalue, figure 6 shows that λ 2 A and λ 2 L 2 are not minimal among isosceles triangles at the equilateral triangle. This fact is analogous to the situation for convex domains, where the minimizers are certain "stadium-like" sets rather than disks (Henrot et al [8,15]). Table 1: Pairs of integers (m, n) giving the first 90 eigenvalues λ j along with pairs [m, n] giving the first 90 antisymmetric eigenvalues λ a j , for an equilateral triangle.…”

Counterexamples to symmetry for partially overdetermined elliptic problems

“…It is proved that Ω * is not a "stadium" (the convex envelope of two identical tangent balls), see [11]. However, it is expected that it looks like a stadium (see [11]).…”

“…It is proved that Ω * is not a "stadium" (the convex envelope of two identical tangent balls), see [11]. However, it is expected that it looks like a stadium (see [11]). If it is the case, as explained in [8], then the first order optimality condition would lead to an overdetermined problem in which the expected overdetermined part Γ would be the strictly convex part of ∂Ω * .…”