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Spectral shape optimization for the Neumann traces of the Dirichlet-Laplacian eigenfunctions

Yannick Privat 1, 2 Emmanuel Trélat 3, 4 Enrique Zuazua 5, 6
2 TONUS - TOkamaks and NUmerical Simulations
IRMA - Institut de Recherche Mathématique Avancée, Inria Nancy - Grand Est
3 CaGE - Control And GEometry
Inria de Paris, LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
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 $\R^n$. 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 {\it 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 {\it 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 {\it bang-bang} functions and {\it 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.
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Submitted on : Sunday, July 28, 2019 - 2:53:36 PM
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Yannick Privat, Emmanuel Trélat, Enrique Zuazua. Spectral shape optimization for the Neumann traces of the Dirichlet-Laplacian eigenfunctions. Calculus of Variations and Partial Differential Equations, Springer Verlag, 2019, 58 (2), pp.64. ⟨10.1007/s00526-019-1522-3⟩. ⟨hal-01872896v2⟩

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