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

TL;DR

This paper investigates the optimization of boundary energy concentration of Dirichlet-Laplacian eigenfunctions through spectral shape design, analyzing Rellich functions, bounded densities, and the effects of pointwise constraints, supported by numerical simulations.

Contribution

It introduces a new spectral shape optimization framework for Neumann traces of Dirichlet eigenfunctions, analyzing Rellich functions and bounded densities under various constraints.

Findings

Rellich functions maximize the energy concentration under L^1 constraints.

Pointwise constraints influence the optimality of bang-bang functions.

Numerical simulations illustrate the theoretical results and specific geometries.

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.