Shape Optimization Problems Involving Eigenvalues and Their Applications

Since Lord Rayleigh conjectured that the disk should minimize the first Laplace-Dirichlet eigenvalue among all shapes of equal area more than a century ago, eigenvalue optimization problems have been active research topics with applications in various areas including mechanical vibration, electromagnetic cavities, photonic crystals, and population dynamics. In this talk, we will review some interesting classical problems and discuss some recent developments.