Geometrical structure of Laplacian eigenfunctions

TL;DR

This paper explores the relationship between the shape of a domain and the geometric structure of Laplacian eigenfunctions, providing a comprehensive overview accessible across multiple scientific disciplines.

Contribution

It offers a summarized, interdisciplinary perspective on the properties and relations of Laplacian eigenvalues and eigenfunctions in various boundary conditions.

Findings

Relations between domain shape and eigenfunction structure

Properties of eigenvalues and eigenfunctions under different boundary conditions

Accessible presentation for diverse scientific fields

Abstract

We summarize the properties of eigenvalues and eigenfunctions of the Laplace operator in bounded Euclidean domains with Dirichlet, Neumann or Robin boundary condition. We keep the presentation at a level accessible to scientists from various disciplines ranging from mathematics to physics and computer sciences. The main focus is put onto multiple intricate relations between the shape of a domain and the geometrical structure of eigenfunctions.