Topology optimization of the two smallest p-Laplacian eigenvalues by a level set method

This paper considers optimization of the two smallest Dirichlet $p$-Laplacian eigenvalues subject to geometric volume or perimeter constraint. A relaxed topology optimization model with shape sensitivity analysis is presented. A level set method and finite element discretization are employed to numerically solving the model problems. The $p$-Laplacian eigenvalue is efficiently solved using quasi-Newton method, which achieves accurate results even for extreme values of $p$. Numerical results are provided to indicate that for the first eigenvalue, the optimal shapes are disks in 2D and balls in 3D under both constraints, independent of $p$. For the second eigenvalue, optimizers have two identical disks or balls independent of $p$ under volume constraint, while optimized shapes evolve from ellipsoidal to flatter configurations under perimeter constraint as $p$ increases.