Illia Karabash: Nonselfadjoint spectral optimization and its applications

During the last 100 years, studies of optimization problems for eigenvalues of selfadjoint differential operators led to many notable achievements including the famous Rayleigh–Faber–Krahn inequality. Optimization problems involving eigenvalues of nonselfadjoint operators have emerged relatively recently in connection with Physics and Engineering applications. It is planned to give a short overview of spectral optimization for dissipative wave equations and to explain the principally new difficulties that emerge due to the nonselfadjoint character of the related spectral problems.

The main attention will be paid to the problem how to create a lossy resonator that has an eigenvalue as close as possible to the real line under certain fabrication constraints. Since the problem is connected with the optimization of photonic crystal designs for high-Q optical cavities, we consider it in the settings involving 3-dimensional Maxwell systems following the joint research with Matthias Eller and for the 1-dimensional case of TEM-modes in layered structures following the joint paper with Herbert Koch and Ievgen Verbytskyi. The 3-D case occurs to be connected with homogenization and (non-)unique continuation, whereas the 1-D problem admits effective analytic and numerical approaches via optimal control reformulations.