A mixed eigenvalue problem on domains tending to infinity in several directions

TL;DR

This paper investigates how the eigenvalues of elliptic operators behave asymptotically on cylindrical domains that become unbounded in multiple directions, revealing dependence on eigenvalue problems in simpler unbounded domains.

Contribution

It provides a detailed analysis of the asymptotic behavior of eigenvalues and eigenfunctions for elliptic operators on complex unbounded domains, extending previous work.

Findings

Eigenvalues' asymptotics depend on eigenvalue problems in one unbounded direction.

Eigenfunctions' behavior is characterized in the asymptotic limit.

Results apply to domains with mixed boundary conditions.

Abstract

The aim of this article is to analyze the asymptotic behaviour of the eigenvalues of elliptic operators in divergence form with mixed boundary type conditions for domains that become unbounded in several directions, while they stay bounded in some directions (cylindrical domains). The limiting behavior of such eigenvalues is shown to depend on an ensemble of eigenvalue problems defined on a domain that is unbounded only in one direction. The asymptotic behavior of the eigenfunctions are also discussed. This work is a continuation of the work done in [6].