**Y. Privat, E. Trélat & E. Zuazua.** Spectral shape optimization for the Neumann traces of the Dirichlet-Laplacian eigenfunctions. CALC VAR PARTIAL DIF. Springer Verlag, Vol. 58, No. 2 (2019) pp. 64. 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 IRn. 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 L1 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 densitiesfor this problem. We also deal with similar issues for a close problem, where the cost functional is replaced by a spectral approximations. Finally, this study is completed by the investigation of particular geometries and is illustrated by several numerical simulations.