Whispering-gallery type eigenfunctions of the Laplacian

January 05, at 03:15 p.m. (Rio de Janeiro local time)

Local: Room C116 - Bloco C - CT – Instituto de Matemática – UFRJ.

Online Transmission: https://youtube.com/live/ed74-MknUog

Speaker: Sergey Sergeev (PUC-RJ)

Abstract: Maybe, someone encountered the situation when one person stands near the wall of the building and whispers something. And another person can hear him only if he stands also near the wall, but not in the center. This is called a Whispering Gallery and the sound waves propagate along the wall. The most famous example of the Whispering Gallery is the St. Paul Cathedral in London, but such an effect appears in different situations when we discuss the wave propagation. From the mathematical point of view the description of this effect is reduced to the studying of the special part of the spectrum of the Laplacian in some bounded area, which corresponds to the high-frequency waves. The eigenfunctions will be localized near the boundary and will represent the Whispering-Gallery type waves. In the present talk we will discuss the asymptotic (for the high frequencies) approach for the construction of such eigenfunctions and eigenvalues of the Laplacian. We will give a description of localized eigenfunctions for any 2D area which is bounded by smooth and convex boundary.