Optimization of quasi-normal eigenvalues for 1-D wave equations in inhomogeneous media; description of optimal structures

The paper is devoted to optimization of resonances associated with 1-D wave equations in inhomogeneous media. The medium's structure is represented by a nonnegative function B. The problem is to design for a given $\alpha \in \R$ a medium that generates a resonance on the line $\alpha + \i \R$ with a minimal possible modulus of the imaginary part. We consider an admissible family of mediums that arises in a problem of optimal design for photonic crystals. This admissible family is defined by the constraints $0\leq b_1 \leq B (x) \leq b_2$ with certain constants $b_{1,2}$. The paper gives an accurate definition of optimal structures that ensures their existence. We prove that optimal structures are piecewise constant functions taking only two extreme possible values $b_1$ and $b_2$. This result explains an effect recently observed in numerical experiments. Then we show that intervals of constancy of an optimal structure are tied to the phase of the corresponding resonant mode and write this connection as a nonlinear eigenvalue problem.