Flexibility of Steklov eigenvalues via boundary homogenisation

TL;DR

This paper extends the understanding of Steklov eigenvalues' flexibility from planar domains to higher-dimensional manifolds, providing new proofs and applications in eigenvalue optimization under perimeter constraints.

Contribution

It generalizes boundary homogenisation results to higher dimensions and manifolds, offering a new proof framework and applying it to optimize Steklov eigenvalues with sharp bounds.

Findings

Steklov eigenvalues exhibit high flexibility in higher dimensions and manifolds.

Planar domains can saturate upper bounds for normalized Steklov eigenvalues.

Sharp upper bounds are established for the first Steklov eigenvalue of doubly connected planar domains.

Abstract

Recently, D. Bucur and M. Nahon used boundary homogenisation to show the remarkable flexibility of Steklov eigenvalues of planar domains. In the present paper we extend their result to higher dimensions and to arbitrary manifolds with boundary, even though in those cases the boundary does not generally exhibit any periodic structure. Our arguments use framework of variational eigenvalues and provides a different proof of the original results. Furthermore, we present an application of this flexibility to the optimisation of Steklov eigenvalues under perimeter constraint. It is proved that the best upper bound for normalised Steklov eigenvalues of surfaces of genus zero and any fixed number of boundary components can always be saturated by planar domains. This is the case even though any actual maximiser (except for simply connected surfaces) is always far from being planar themselves. In particular, it yields sharp upper bound for the first Steklov eigenvalue of doubly connected planar domains.