Constrained Shape Derivative for the Spectral Laplace-Dirichlet Problem Using the Minimax Method

In shape optimization, we often seek to optimize the shape of a domain in order to achieve a goal. For example, the sound of a drum depends on the shape of its membrane, via the eigenvalues of the Laplacian. We use the shape derivative to adjust this shape. The shape derivative for an eigenvalue problem is frequently used in shape optimisation, in particular when we want to minimise or maximise an eigenvalue, often the first of an elliptic operator, typically the Laplacian, under geometric constraints. This paper gives a simple method for calculating the shape derivative of a Dirichlet eigenvalue, but also the shape derivative of an objective function constrained to an eigenvalue problem.