On the exponential decay of Laplacian eigenfunctions in planar domains with branches

TL;DR

This paper proves that Laplacian eigenfunctions in planar domains with elongated branches decay exponentially along the branches when eigenvalues are below a certain threshold, supported by numerical illustrations.

Contribution

It establishes exponential decay of eigenfunctions in domains with branches under Robin boundary conditions, extending understanding of spectral behavior in such geometries.

Findings

Eigenfunctions decay exponentially along branches for eigenvalues below a threshold

Numerical simulations confirm theoretical decay behavior

Results apply to Robin boundary conditions in planar domains

Abstract

We consider the eigenvalue problem for the Laplace operator in a planar domain which can be decomposed into a bounded domain of arbitrary shape and elongated \branches" of variable cross-sectional profiles. When the eigenvalue is smaller than a prescribed threshold, the corresponding eigenfunction decays exponentially along each branch. We prove this behavior for Robin boundary condition and illustrate some related results by numerically computed eigenfunctions.