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The paper is devoted to optimization of resonances associated with 1-D waveequations in inhomogeneous media. The medium's structure is represented by anonnegative function B. The problem is to design for a given $\alpha \in \R$ amedium that generates a resonance on the line $\alpha + \i \R$ with a minimalpossible modulus of the imaginary part. We consider an admissible family ofmediums that arises in a problem of optimal design for photonic crystals. Thisadmissible 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 ofoptimal structures that ensures their existence. We prove that optimalstructures are piecewise constant functions taking only two extreme possiblevalues $b_1$ and $b_2$. This result explains an effect recently observed innumerical experiments. Then we show that intervals of constancy of an optimalstructure are tied to the phase of the corresponding resonant mode and writethis connection as a nonlinear eigenvalue problem.

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