Tuesday, November 28, 2023

Emanuel Indrei
Sam Houston State University

Title: On the first eigenvalue of the Laplacian for polygons

Abstract: In 1947, Polya proved that if n=3,4 the regular polygon P_n minimizes the principal frequency of an n-gon with given area and suggested that the same holds for larger values of n. In 1951, Polya and Szego discussed the possibility of counterexamples. Recently, I constructed explicit (2n-4)-dimensional polygonal manifolds and proved for n large that there exists an explicit non-empty set A_n such that P_n has the smallest principal frequency among n-gons in A_n. The techniques involve a partial symmetrization, tensor calculus, the spectral theory of circulant matrices, and W^{2,p} estimates. An application is given in the context of electron bubbles.

https://wayne-edu.zoom.us/j/99210289207?pwd=NFMvazN4WnZxWEpOQXQ5MXFlYlhLUT09