| Abstract: |
| We study a volume-constrained shape optimization problem for the eigenvalues of the Stokes operator in two and three dimensions, aiming to identify domains that minimize individual eigenvalues. While the ball is known to satisfy optimality conditions in two dimensions (Henrot, Mazari-Fouquer and Privat), this is no longer the case in three dimensions.
We propose a numerical framework based on the Method of Fundamental Solutions (MFS) to approximate the solutions of the Stokes eigenvalue problem. The validity of this approach is supported by density results established in this work.
The optimization is carried out using a gradient-type method. To address the difficulties associated with eigenvalue multiplicities, we adopt a smoothing strategy based on lower-order eigenvalues. Instead of computing the shape gradient of the target eigenvalue directly, we consider the gradients of a set of preceding eigenvalues. This approach allows us to define a smooth objective functional, in contrast with the standard formulation, which is typically non-differentiable in a neighborhood of the optimal shape due to eigenvalue multiplicities.
The effectiveness of the method is demonstrated through numerical experiments, which highlight qualitative differences between optimal shapes in two and three dimensions.
This is a joint work with Nuno Martins (Universidade Nova de Lisboa) |
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