On a Spectral Shape Optimization for Nonlinear Eigenvalue Problems Governed by the <i>p</i> ‐Laplacian With Robin Boundary Conditions

Abstract

ABSTRACT This paper is devoted to a numerical resolution of spectral shape optimization problems governed by the Robin ‐Laplacian operator under a volume constraint. We deal notably with the numerical computation of optimal shapes and corresponding higher eigenvalues of the problem governed by the ‐Laplacian operator with Robin boundary conditions, in two and three dimensions. In this regard, it is worth mentioning that to our knowledge, no existing work has yet addressed this numerical investigation. So, we establish first the existence of the shape derivative for simple eigenvalues, providing both volume and boundary shape derivative formulas. Then we develop a numerical approach using gradient descent methods to approximate minimizers of higher Robin eigenvalues. This is based on a finite element discretization, combined with a Picard iteration scheme, designed to compute higher order Robin eigenvalues for various values of . Finally, several numerical experiments in 2D and 3D are presented to demonstrate the effectiveness and robustness of the proposed approaches, and new conjectures are stated, based on these numerical simulations.