Spectral shape optimization for the Neumann traces of the Dirichlet-Laplacian eigenfunctions
Privat Y, Trelat E, Zuazua E (2019)
Publication Type: Journal article
Publication year: 2019
Journal
Book Volume: 58
Article Number: 64
Journal Issue: 2
DOI: 10.1007/s00526-019-1522-3
Abstract
We consider a spectral optimal design problem involving the Neumann traces of the Dirichlet-Laplacian eigenfunctions on a smooth bounded open subset Ω of [InlineEquation not available: see fulltext.]. 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 ∞ -norm of Rellich functions may be large, depending on the shape of Ω , 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.
Authors with CRIS profile
Involved external institutions
Université de Strasbourg (UDS)
France (FR)
University of Paris 7 - Denis Diderot / Université Paris VII Denis Diderot
France (FR)
University of Deusto / Universidad de Deusto
Spain (ES)
France (FR)
University of Paris 7 - Denis Diderot / Université Paris VII Denis Diderot
France (FR)
University of Deusto / Universidad de Deusto
Spain (ES)
How to cite
APA:
Privat, Y., Trelat, E., & Zuazua, E. (2019). Spectral shape optimization for the Neumann traces of the Dirichlet-Laplacian eigenfunctions. Calculus of Variations and Partial Differential Equations, 58(2). https://dx.doi.org/10.1007/s00526-019-1522-3
MLA:
Privat, Yannick, Emmanuel Trelat, and Enrique Zuazua. "Spectral shape optimization for the Neumann traces of the Dirichlet-Laplacian eigenfunctions." Calculus of Variations and Partial Differential Equations 58.2 (2019).
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