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Pareto optimal structures producing resonances of minimal decay under \(L^1\)-type constraints. (English) Zbl 1290.49097

Summary: Resonances optimization is studied under the constraint \(\| B \|_1 \leq m\) on the nonnegative function \(B \in \mathcal{L}^1(0, \ell)\) representing the resonator structure. The problem is to design for a given frequency \(\alpha \in \mathbb{R}\) a structure that generates a resonance \({\omega}\) on the line \(\alpha + \operatorname{i} \mathbb{R}\) with minimal possible decay rate \(| \operatorname{Im} \omega |\). We generalize the problem replacing \(B\) by a nonnegative measure, and show that optimal measures consist of a finite number of point masses. This yields non-existence of optimizers for the problem over absolutely continuous measures. We derive restrictions on optimal masses and their positions. This reduces the original infinitely-dimensional problem to optimization over four real parameters. For low frequencies, we explicitly find optimizers. The technique is based on the two-parameter perturbation method and the notion of local boundary point, which is introduced as a generalization of local extrema to vector optimization problems. Special attention is paid to multiple and non-differentiable resonances.

MSC:

49R05 Variational methods for eigenvalues of operators
49J20 Existence theories for optimal control problems involving partial differential equations
35B34 Resonance in context of PDEs
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
58E17 Multiobjective variational problems, Pareto optimality, applications to economics, etc.
32A60 Zero sets of holomorphic functions of several complex variables
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