# From Steklov to Neumann via homogenisation

## TL;DR

This paper establishes a novel connection between Steklov and Neumann eigenvalues through homogenisation of perforated domains, leading to new bounds and domain constructions with extremal spectral properties.

## Contribution

It introduces a homogenisation approach linking Steklov and Neumann eigenvalues, providing new bounds and explicit domain examples with maximal eigenvalues.

## Key findings

- Recovered isoperimetric bounds for Neumann eigenvalues from Steklov bounds
- Constructed planar domains with unprecedented first perimeter-normalized Steklov eigenvalues
- Developed a homogenisation technique for a transmission problem involving spectral parameters

## Abstract

We study a new link between the Steklov and Neumann eigenvalues of domains in Euclidean space. This is obtained through an homogenisation limit of the Steklov problem on a periodically perforated domain, converging to a family of eigenvalue problems with dynamical boundary conditions. For this problem, the spectral parameter appears both in the interior of the domain and on its boundary. This intermediary problem interpolates between Steklov and Neumann eigenvalues of the domain. As a corollary, we recover some isoperimetric type bounds for Neumann eigenvalues from known isoperimetric bounds for Steklov eigenvalues. The interpolation also leads to the construction of planar domains with first perimeter-normalized Stekov eigenvalue that is larger than any previously known example. The proofs are based on a modification of the energy method. It requires quantitative estimates for norms of harmonic functions. An intermediate step in the proof provides a homogenisation result for a transmission problem.