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Spectral shape optimization for the Neumann traces of the Dirichlet-Laplacian eigenfunctions. (English) Zbl 1410.35090

Authors’ abstract: We consider a spectral optimal design problem involving the Neumann traces of the Dirichlet-Laplacian eigenfunctions on a smooth bounded open subset \(\Omega\) of \(\mathbb{R}^n\). The cost functional measures the amount of energy that Dirichlet eigenfunctions concentrate on the boundary and that can be recovered with a bounded density function. We first prove that, assuming a \(L^1\) constraint on densities, the so-called Rellich functions maximize this functional. Motivated by several issues in shape optimization or observation theory where it is relevant to deal with bounded densities, and noticing that the \(L^\infty \)-norm of Rellich functions may be large, depending on the shape of \(\Omega\), we analyze the effect of adding pointwise constraints when maximizing the same functional. We investigate the optimality of bang-bang functions and Rellich densities for this problem. We also deal with similar issues for a close problem, where the cost functional is replaced by a spectral approximation. Finally, this study is completed by the investigation of particular geometries and is illustrated by several numerical simulations.

MSC:

35P20 Asymptotic distributions of eigenvalues in context of PDEs
93B07 Observability
58J51 Relations between spectral theory and ergodic theory, e.g., quantum unique ergodicity
49K20 Optimality conditions for problems involving partial differential equations

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