The localization effect for eigenfunctions of the mixed boundary value problem in a thin cylinder with distorted ends

TL;DR

This paper identifies conditions under which the principal eigenfunction of the Laplace operator in a thin cylinder localizes near the ends, decaying exponentially inside, with similar effects in Dirichlet and Neumann cases.

Contribution

It provides a simple sufficient condition on the curved end of a cylinder that guarantees eigenfunction localization for mixed boundary value problems.

Findings

Eigenfunction concentrates near the ends of the cylinder.

Eigenfunction decays exponentially in the interior.

Similar localization effects occur in Dirichlet and Neumann problems.

Abstract

A simple sufficient condition on curved end of a straight cylinder is found that provides a localization of the principal eigenfunction of the mixed boundary value for the Laplace operator with the Dirichlet conditions on the lateral side. Namely, the eigenfunction concentrates in the vicinity of the ends and decays exponentially in the interior. Similar effects are observed in the Dirichlet and Neumann problems, too.