Shape derivative of the Laplacian eigenvalue problem

The Laplacian eigenvalue problem with two different densities is investigated. By the squeeze theorem, the shape derivative of the least eigenvalue on the interface of two different density subdomains is derived. As an application, the minimization of the least eigenvalue with area constraint is considered. The shape derivative of the objective functional is applied as the velocity for the level set method to involve the interface. The numerical results validate that the proposed method is effective to capture the final optimized distribution of two different densities.