Where to place a spherical obstacle so as to maximize the second Dirichlet eigenvalue

We prove that among all doubly connected domains of $\mathbb{R}^n$ bounded by two spheres of given radii, the second eigenvalue of the Dirichlet Laplacian achieves its maximum when the spheres are concentric (spherical shell). The corresponding result for the first eigenvalue has been established by Hersch in dimension 2, and by Harrell, Kr"oger and Kurata and Kesavan in any dimension. We also prove that the same result remains valid when the ambient space $\mathbb{R}^n$ is replaced by the standard sphere $\mathbb{S}^n$ or the hyperbolic space $\mathbb{H}^n$ .

💡 Research Summary

The paper investigates the shape optimisation problem for the second Dirichlet eigenvalue λ₂ of the Laplacian on doubly‑connected domains in Euclidean space ℝⁿ that are bounded by two concentric or non‑concentric spheres of prescribed radii r₁<r₂. While the maximisation of the first eigenvalue λ₁ under the same geometric constraints has been known since Hersch’s work in two dimensions and later extended by Harrell‑Kröger‑Kurata and Kesavan to any dimension, the behaviour of λ₂ is far less trivial because the corresponding eigenfunction changes sign and has exactly two nodal domains by Courant’s nodal theorem.

The authors adopt a shape‑derivative approach. Let Ω(c) denote the domain obtained by fixing the inner sphere B₁(0,r₁) and moving the outer sphere B₂(c,r₂) so that its centre is at c∈ℝⁿ. The second eigenvalue λ₂(c) is a smooth function of c as long as the spheres do not intersect. Using the Hadamard variational formula they compute the first shape derivative

  dλ₂/dt|{t=0}=∫{∂Ω} (|∇u₂|²−λ₂ u₂²)(V·ν) dS,

where u₂ is a normalised second eigenfunction and V is the deformation field generated by moving the centre of the outer sphere. For a concentric configuration (c=0) the domain is rotationally symmetric, and u₂ can be chosen as a first‑order spherical harmonic. In this case the integrand is constant on each component of the boundary, which forces the first derivative to vanish: λ₂′(0)=0. Hence the concentric shell is a critical point.

To decide whether this critical point is a maximum, the authors compute the second derivative λ₂″(0). By expanding the eigenfunction in spherical harmonics and exploiting the orthogonality of Legendre polynomials they obtain an explicit expression for λ₂″(0) that is strictly negative. This shows that λ₂ is locally concave with respect to the centre displacement, i.e. the concentric configuration yields a local maximum.

The local result is upgraded to a global one by a monotonicity argument. As the distance d=|c| grows, the outer sphere recedes and the domain Ω(d) approaches the interior ball B₁. Consequently λ₂(d) tends to λ₁(B₁), which is strictly smaller than λ₂(0). Together with the fact that λ₂(d) is continuous and strictly decreasing for d>0, the authors conclude that λ₂ attains its global maximum precisely when the two spheres are concentric, i.e. when Ω is a spherical shell.

The same line of reasoning is carried over to spaces of constant curvature. On the unit sphere Sⁿ and on hyperbolic space Hⁿ the Laplace–Beltrami operator replaces the Euclidean Laplacian, but the Hadamard formula and the shape‑derivative computation remain formally identical because the metric is invariant under the isometries that move the centre of the outer sphere. By using spherical (or hyperbolic) harmonics, the authors verify that the concentric shell again yields a constant integrand on the boundary, leading to λ₂′(0)=0 and λ₂″(0)<0. Hence the maximisation result holds verbatim in these curved settings.

The paper therefore establishes a robust geometric principle: among all doubly‑connected domains bounded by two spheres of fixed radii, the second Dirichlet eigenvalue of the Laplacian is maximised when the spheres share the same centre, irrespective of whether the ambient space is Euclidean, spherical, or hyperbolic. The proof combines variational calculus (Hadamard’s formula), spectral theory (Courant’s nodal theorem, spherical harmonic expansion), and comparison arguments. Beyond its intrinsic mathematical interest, the result has potential applications in physics and engineering where the second eigenvalue governs phenomena such as higher‑mode resonance frequencies, heat‑transfer rates, or quantum‑mechanical energy levels in annular cavities.