Some spectral optimization problems


Séminaire Modèles et Algorithmes Déterministes: EDP-MOISE

15/11/2012 - 11:30 Mr Braxton Osting (UCLA) Amphithéâtre - Maison Jean Kuntzmann

I'll discuss the following two spectral optimization problems.

(1) In many optical and quantum systems it is desirable to engineer a
device to spatially confine energy in a particular  mode for a long
period of time. I'll discuss the mathematics of energy-conserving,
spatially-extended systems and present analytical and computational
results on optimal energy confining structures.

(2) In this part of the talk, I'll discuss the shape optimization
problem where the objective function is a convex combination of
sequential Laplace-Dirichlet eigenvalues. We show that as a function
of the combination parameters, the optimal value is non-decreasing,
Lipschitz continuous, and concave and that the minimizing set is upper
hemicontinuous. For star-shaped domains with smooth boundary, we study
combination parameter sets for which the ball is a local minimum. We
propose a method for computing optimal domains and computationally
study several properties of minimizers, including uniqueness,
connectivity, symmetry, and eigenvalue multiplicity. This is joint
work with Chiu-Yen Kao.