In this talk we consider the problem of determining a domain in $\R^N$ that minimizes the first eigenvalue of the elasticity (or Lam\'e) system under a volume constraint. We prove existence of an optimal domain. We derive first and second-order optimality conditions. Leveraging these conditions, we demonstrate that in two dimensions, the disk cannot be the optimal shape when the Poisson ratio is below a specific threshold, whereas above this value, it serves as a local minimizer. We also find some explicit domains that have a lower first eigenvalue than the disk for Poisson ratios $\nu \leq 0.4$.
This is a joint work with Antoine Lemenant and Yannick Privat
