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International audience; This paper is devoted to the computation of certain directional semi-derivatives of eigenvalue functionals of self-adjoint elliptic operators involving a variety of boundary conditions. A uniform treatment of these problems is possible by considering them as a problem of calculating the semi-derivative of a minimum with respect to a parameter. The applicability of this approach, which can be traced back to the works of Danskin and Zolésio, to the treatment of eigenvalue problems (where the full shape derivative may not exist, due to multiplicity issues), has been illustrated by Zolésio. Despite this, some of the recent literature on the shape sensitivity of eigenvalue problems still continue to employ methods such as the material derivative method or Lagrangian methods which seem less adapted to this class of problems. The Delfour-Zolésio approach does not seem to be fully exploited in the existing literature: we aim to recall the importance and the simplicity of these ideas, by applying it to the analysis of the shape sensitivity for eigenvalue functionals for a class of elliptic operators in the scalar setting (Laplacian or diffusion in heterogeneous media), thus recovering known results in the case of Dirichlet or Neumann boundary conditions and obtaining new results in the case of Steklov or Wentzell boundary conditions.

In this work, we compute the shape derivatives of eigenvalues problem for elliptic operators associated to various boundary conditions, that is Dirichlet, Neumann, Robin, and Wentzell boundary conditions. We also consider the case when the conductivity and the density have jumps, which corresponds to composite structures. The proposed method is based on a result for the derivative of a minimum with respect to a parameter. The main advantage is that the procedure exposed in this work is uniform and efficient with respect to the computations. Indeed, in order to underline this efficiency, we present in the appendix the computation in the case of the mixture of two phases using the classical method based on the material derivative, which turns out to be much more tedious.