On the Exponential Decay of Laplacian Eigenfunctions in Planar Domains with Branches
Binh Nguyen, Denis S. Grebenkov, Andrey L. Delytsin · Contemporary mathematics - American Mathematical Society · 2014
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.