Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems

Abstract

We associate a sequence of variational eigenvalues to any Radon measure on a compact Riemannian manifold. For particular choices of measures, we recover the Laplace, Steklov and other classical eigenvalue problems. In the first part of the paper we study the properties of variational eigenvalues and establish a general continuity result, which shows for a sequence of measures converging in the dual of an appropriate Sobolev space, that the associated eigenvalues converge as well. The second part of the paper is devoted to various applications to shape optimization. The main theme is studying sharp isoperimetric inequalities for Steklov eigenvalues without any assumption on the number of connected components of the boundary. In particular, we solve the isoperimetric problem for each Steklov eigenvalue of planar domains: the best upper bound for the k-th perimeter-normalized Steklov eigenvalue is $$8\pi k$$ 8 π k , which is the best upper bound for the $$k^{\text {th}}$$ k th area-normalised eigenvalue of the Laplacian on the sphere. The proof involves realizing a weighted Neumann problem as a limit of Steklov problems on perforated domains. For $$k=1$$ k = 1 , the number of connected boundary components of a maximizing sequence must tend to infinity, and we provide a quantitative lower bound on the number of connected components. A surprising consequence of our analysis is that any maximizing sequence of planar domains with fixed perimeter must collapse to a point.

Flexibility of Steklov eigenvalues via boundary homogenisation

“…The optimisation problem for other topologies of domains in R 2 remains unsolved. At the same time, if one does not impose any assumptions on the topology of the planar domain, then the optimal upper bound for all normalised Steklov eigenvalues is for all k ∈ N σ k ( , g) ≤ 8πk, was found in [10]. The main goal of the present paper is to apply the ideas of [4] to the optimisation problem for planar domains of fixed topology.…”

“…, see [10,14] for a discussion around the naturality of that normalisation. The case β ≡ 1 is of particular interest and is referred to as the Steklov problem.…”

From Steklov to Laplace: free boundary minimal surfaces with many boundary components

“…Inequality (1.4) is a quantitative improvement over (1.1). The only other known result of this type is [GKL,Theorem 1.8], where the correction term decays exponentially with k. We note that a variant of (1.3) also holds for the conformally-constrained maximization problem (see Proposition 4.1), and the corresponding variant of the upper bound (1.4) holds for many non-maximizing conformal classes-e.g., for any conformal class admitting a minimal immersion to S n by first eigenfunctions (see Remark 5.14).…”

“…[GKL]. While many examples of admissible measures lead to interesting eigenvalue problems [GKL,Section 4], the following are the only examples used in the present paper.…”

Tubular excision and Steklov eigenvalues

“…In the Euclidean space M = R m this question is equivalent to the maximization of σ 1 among domains with prescribed boundary measure |∂Ω|. For m = 2 the optimal upper bound is known thanks to [19,17,13], while for m ≥ 3 it is known that σ 1 (Ω)|∂Ω| 1/(m−1) is bounded above [4], but the optimal bound remains unknown. For domains Ω in a compact manifold of dimension m ≥ 3, the situation is completely different: it was proved in [14] that in that case σ 1 (Ω)|∂Ω| 1/(m−1) is not bounded above.…”