# Optimization of Steklov-Neumann eigenvalues

## TL;DR

This paper develops an algorithm to optimize the placement of boundary coverings in a container modeled by Laplace's equation with mixed boundary conditions, aiming to control resonance phenomena.

## Contribution

It introduces a novel algorithm based on asymptotic formulas for boundary perturbations to optimize Steklov-Neumann eigenvalues.

## Key findings

- Algorithm effectively determines optimal boundary coverings.
- Established proofs for asymptotic formulas used in optimization.
- Provided bounds and examples for the eigenvalue problem.

## Abstract

This paper examines the Laplace equation with mixed boundary conditions, the Neumann and Steklov boundary conditions. This models a container with holes in it, like a pond filled with water but partly covered by immovable pieces on the surface. The main objective is to determine the right extent of the covering pieces, so that any shock inside the container yields a resonance. To this end, an algorithm is developed which uses asymptotic formulas concerning perturbations of the partitioning of the boundary pieces. Proofs for these formulas are established. Furthermore, this paper displays some results concerning bounds and examples with regards to the governing problem.