Critical metrics of eigenvalue functionals via Clarke subdifferential

We set up a new framework to study critical points of functionals defined as combinations of eigenvalues of operators with respect to a given set of parameters: Riemannian metrics, potentials, etc. Our setting builds upon Clarke's differentiation theory to provide a novel understanding of critical metrics. In particular, we unify and refine previous research carried out on Laplace and Steklov eigenvalues. We also use our theory to tackle original examples such as the conformal GJMS operators, the conformal Laplacian, and the Laplacian with mixed boundary conditions.